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Quantum Mechanics 3:
the quantum mechanics of many-particle systems
W.J.P. Beenakker
Academic year 2017 – 2018
Contents of the common part of the course:
1) Occupation-number representation
2) Quantum statistics (up to § 2.5.3)
Module 1 (high-energy physics):
2) Quantum statistics (rest)
3) Relativistic 1-particle quantum mechanics
4) Quantization of the electromagnetic field
5) Many-particle interpretation of the relativistic quantum mechanics
The following books have been used:
F. Schwabl, “Advanced Quantum Mechanics”, third edition (Springer, 2005);
David J. Griffiths, “Introduction to Quantum Mechanics”, second edition
(Prentice Hall, Pearson Education Ltd, 2005);
Eugen Merzbacher, “Quantum Mechanics”, third edition (John Wiley & Sons, 2003);
B.H. Bransden and C.J. Joachain, “Quantum Mechanics”, second edition
(Prentice Hall, Pearson Education Ltd, 2000).
Contents
1 Occupation-number representation 1
1.1 Summary on identical particles in quantum mechanics . . . . . . . . . . . . 1
1.2 Occupation-number representation . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Construction of Fock space . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Switching to a continuous 1-particle representation . . . . . . . . . . . . . 12
1.3.1 Position and momentum representation . . . . . . . . . . . . . . . . 14
1.4 Additive many-particle quantities and particle conservation . . . . . . . . . 15
1.4.1 Additive 1-particle quantities . . . . . . . . . . . . . . . . . . . . . 16
1.4.2 Additive 2-particle quantities . . . . . . . . . . . . . . . . . . . . . 19
1.5 Heisenberg picture and second quantization . . . . . . . . . . . . . . . . . 20
1.6 Examples and applications: bosonic systems . . . . . . . . . . . . . . . . . 23
1.6.1 The linear harmonic oscillator as identical-particle system . . . . . 23
1.6.2 Forced oscillators: coherent states and quasi particles . . . . . . . . 26
1.6.3 Superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.6.4 Superfluidity for weakly repulsive spin-0 bosons (part 1) . . . . . . 33
1.6.5 Intermezzo: the Bogolyubov transformation for bosons . . . . . . . 38
1.6.6 Superfluidity for weakly repulsive spin-0 bosons (part 2) . . . . . . 40
1.6.7 The wonderful world of superfluid 4He: the two-fluid model . . . . 42
1.7 Examples and applications: fermionic systems . . . . . . . . . . . . . . . . 44
1.7.1 Fermi sea and hole theory . . . . . . . . . . . . . . . . . . . . . . . 44
1.7.2 The Bogolyubov transformation for fermions . . . . . . . . . . . . . 46
2 Quantum statistics 49
2.1 The density operator (J. von Neumann, 1927) . . . . . . . . . . . . . . . . 52
2.2 Example: polarization of a spin-1/2 ensemble . . . . . . . . . . . . . . . . 55
2.3 The equation of motion for the density operator . . . . . . . . . . . . . . . 58
2.4 Quantum mechanical ensembles in thermal equilibrium . . . . . . . . . . . 59
2.4.1 Thermal equilibrium (thermodynamic postulate) . . . . . . . . . . . 61
2.4.2 Canonical ensembles (J.W. Gibbs, 1902) . . . . . . . . . . . . . . . 61
2.4.3 Microcanonical ensembles . . . . . . . . . . . . . . . . . . . . . . . 65
2.4.4 Grand canonical ensembles (J.W. Gibbs, 1902) . . . . . . . . . . . . 66
2.4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.5 Fermi gases at T=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.5.1 Ground state of a Fermi gas . . . . . . . . . . . . . . . . . . . . . . 74
2.5.2 Fermi gas with periodic boundary conditions . . . . . . . . . . . . . 76
2.5.3 Fermi-gas model for conduction electrons in a metal . . . . . . . . . 77
2.5.4 Fermi-gas model for heavy nuclei . . . . . . . . . . . . . . . . . . . 80
2.5.5 Stars in the final stage of stellar evolution . . . . . . . . . . . . . . 84
2.6 Systems consisting of non-interacting particles . . . . . . . . . . . . . . . . 87
2.7 Ideal gases in three dimensions . . . . . . . . . . . . . . . . . . . . . . . . 90
2.7.1 Classical gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
2.7.2 Bose gases and Bose–Einstein condensation (1925) . . . . . . . . . 93
2.7.3 Fermi gases at T 6= 0 (no exam material) . . . . . . . . . . . . . . . 98
2.7.4 Experimental realization of Bose–Einstein condensates . . . . . . . 101
3 Relativistic 1-particle quantum mechanics 103
3.1 First attempt: the Klein–Gordon equation (1926) . . . . . . . . . . . . . . 104
3.2 Second attempt: the Dirac equation (1928) . . . . . . . . . . . . . . . . . . 109
3.2.1 The probability interpretation of the Dirac equation . . . . . . . . . 111
3.2.2 Covariant formulation of the Dirac equation . . . . . . . . . . . . . 112
3.2.3 Dirac spinors and Poincare transformations . . . . . . . . . . . . . 113
3.2.4 Solutions to the Dirac equation . . . . . . . . . . . . . . . . . . . . 116
3.2.5 The success story of the Dirac theory . . . . . . . . . . . . . . . . . 118
3.2.6 Problems with the 1-particle Dirac theory . . . . . . . . . . . . . . 122
4 Quantization of the electromagnetic field 123
4.1 Classical electromagnetic fields: covariant formulation . . . . . . . . . . . . 123
4.1.1 Solutions to the free electromagnetic wave equation . . . . . . . . . 125
4.1.2 Electromagnetic waves in terms of harmonic oscillations . . . . . . . 127
4.2 Quantization and photons . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.2.1 Particle interpretation belonging to the quantized electromagnetic field130
4.2.2 Photon states and density of states for photons . . . . . . . . . . . 131
4.2.3 Zero-point energy and the Casimir effect . . . . . . . . . . . . . . . 132
4.2.4 Continuum limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.3 A new perspective on particle –wave duality . . . . . . . . . . . . . . . . . 134
4.4 Interactions with non-relativistic quantum systems . . . . . . . . . . . . . 135
4.4.1 Classical limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.4.2 Spontaneous photon emission (no exam material) . . . . . . . . . . 139
4.4.3 The photon gas: quantum statistics for photons . . . . . . . . . . . 141
5 Many-particle interpretation of the relativistic QM 144
5.1 Quantization of the Klein–Gordon field . . . . . . . . . . . . . . . . . . . . 145
5.2 Quantization of the Dirac field . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.2.1 Extra: the 1-(quasi)particle approximation for electrons . . . . . . . 153
A Fourier series and Fourier integrals i
A.1 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
A.2 Fourier integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
A.2.1 Definition of the δ function . . . . . . . . . . . . . . . . . . . . . . . iii
B Properties of the Pauli spin matrices iv
C Lagrange-multiplier method v
D Special relativity: conventions and definitions v
E Electromagnetic orthonormality relations ix
1 Occupation-number representation
In this chapter the quantum mechanics of identical-particle systems will be
worked out in detail. The corresponding space of quantum states will be con-
structed in the occupation-number representation by employing creation and
annihilation operators. This will involve the introduction of the notion of quasi
particles and the concept of second quantization.
Similar material can be found in Schwabl (Ch. 1,2 and 3) and Merzbacher
(Ch. 21,22 and the oscillator part of Ch. 14).
1.1 Summary on identical particles in quantum mechanics
Particles are called identical if they cannot be distinguished by means of specific intrinsic
properties (such as spin, charge, mass, · · · ).
This indistinguishability has important quantum mechanical implications in situations
where the wave functions of the identical particles overlap, causing the particles to be ob-
servable simultaneously in the same spatial region. Examples are the interaction region of
a scattering experiment or a gas container. If the particles are effectively localized, such as
metal ions in a solid piece of metal, then the identity of the particles will not play a role.
In those situations the particles are effectively distinguishable by means of their spatial
coordinates and their wave functions have a negligible overlap.
For systems consisting of identical particles two additional constraints have to be imposed
while setting up quantum mechanics (QM).
• Exchanging the particles of a system of identical particles should have no observ-
able effect, otherwise the particles would actually be distinguishable. This gives
rise to the concept of permutation degeneracy, i.e. for such a system the expecta-
tion value for an arbitrary many-particle observable should not change upon inter-
changing the identical particles in the state function. As a consequence, quantum
mechanical observables for identical-particle systems should be symmetric functions
of the separate 1-particle observables.
• Due to the permutation degeneracy, it seems to be impossible to fix the quantum
state of an identical-particle system by means of a complete measurement. Nature
has bypassed this quantum mechanical obstruction through the
symmetrization postulate: identical-particle systems can be described by means
of either totally symmetric state functions if the particles are bosons or totally
antisymmetric state functions if the particles are fermions. In the non-relativistic
QM it is an empirical fact that no mixed symmetry occurs in nature.
1
In this chapter the symmetrization postulate will be reformulated in an alternative
way, giving rise to the conclusion that only totally symmetric/antisymmetric state
functions will fit the bill.
Totally symmetric state functions can be represented in the so-called q-representation by
ψS(q1, · · · , qN , t), where q1 , · · · , qN are the “coordinates” of the N separate identical par-
ticles. These coordinates are the eigenvalues belonging to a complete set of commuting
1-particle observables q . Obviously there are many ways to choose these coordinates. A
popular choice is for instance qj = (spatial coordinate ~rj , magnetic spin quantum number
msj ≡ σj , · · · ), where the dots represent other possible internal (intrinsic) degrees of free-
dom of particle j . For a symmetric state function we have
∀P
P ψS(q1, · · · , qN , t) = ψS(qP (1), · · · , q
P (N), t) = ψS(q1, · · · , qN , t) , (1)
with P a permutation operator that permutes the sets of coordinates of the identical
particles according to
q1 → qP (1)
, q2 → qP (2)
, · · · , qN → qP (N)
. (2)
Particles whose quantum states are described by totally symmetric state functions are
called bosons. They have the following properties:
- bosons have integer spin (see Ch. 4 and 5);
- bosons obey so-called Bose–Einstein statistics (see Ch. 2);
- bosons prefer to be in the same quantum state.
Totally antisymmetric state functions can be represented in the q-representation by
ψA(q1, · · · , qN , t), with
∀P
P ψA(q1, · · · , qN , t) = ψA(qP (1), · · · , q
P (N), t) =
+ ψA(q1, · · · , qN , t) even P
− ψA(q1, · · · , qN , t) odd P. (3)
A permutation P is called even/odd if it consists of an even/odd number of two-particle
interchanges. Particles whose quantum states are described by totally antisymmetric state
functions are called fermions. They have the following properties:
- fermions have half integer spin (see Ch. 5);
- fermions obey so-called Fermi–Dirac statistics (see Ch. 2);
- fermions are not allowed to be in the same quantum state.
2
If the above (anti)symmetrization procedure results in spatial symmetrization, the particles
have an increased probability to be in each other’s vicinity. Note that this can apply to
identical fermions if they happen to be in an antisymmtric spin state. On the other hand,
spatial antisymmetrization gives rise to a decreased probability for the particles to be in
each other’s vicinity. Note that this can apply to identical spin-1 bosons if they happen to
be in an antisymmtric spin state.
Isolated non-interacting many-particle systems: many-particle systems with negi-
gible interactions among the particles are called non-interacting many-particle systems.
The properties of such systems are determined by the type(s) of particles involved and the
possible influence of external potentials on the system (e.g. caused by a magnetic field).
We speak of an ideal gas if the number of non-interacting particles is very large and if we
place the system in a macroscopic finite enclosure.
Isolated non-interacting many-particle systems (such as free-particle systems,
ideal gasses, . . . ) play a central role in this lecture course, both for determining
the physical properties of identical-boson/fermion systems and for setting up
relativistic QM.
Since we are dealing here with isolated systems, the total energy is conserved and the state
function can be expressed in terms of stationary states
ψE(q1, · · · , qN , t) = exp(−iEt/~)ψE(q1, · · · , qN) ,
with H ψE(q1, · · · , qN) = E ψE(q1, · · · , qN) . (4)
The Hamilton operator describing a non-interacting many-particle system comprises of
pure 1-particle Hamilton operators, i.e. Hamilton operators that depend exclusively on
observables belonging to individual single particles. Denoting the 1-particle Hamilton
operator of particle j by Hj , the N -particle Hamilton operator reads
H =N∑
j=1
Hj , with[Hj , Hk
]= 0 for all j , k = 1 , · · · , N . (5)
This implies that the Hamilton operators of the individual particles are compatible observ-
ables, as expected for non-interacting particles. Suppose now that the 1-particle energy-
eigenvalue equation
Hj ψλj(qj) = Eλj
ψλj(qj) (6)
gives rise to an orthonormal set ψλj(qj) of energy eigenfunctions belonging to the energy
eigenvalues Eλj, which are labeled by a complete set of quantum numbers. Examples of
such complete sets of quantum numbers are for instance λj = nj for a linear harmonic
3
oscillator or λj = (nj , ℓj , mℓj , msj ) for a 1-electron atom. For the orthonormal set of
energy eigenstates of the complete non-interacting N -particle system we can identify three
scenarios.
A) The particles are distinguishable. The orthonormal set of N -particle energy eigen-
states comprises of states of the form
ψE(q1, · · · , qN) = ψλ1(q1)ψλ2(q2) · · · ψλN(qN) , with E =
N∑
j=1
Eλj. (7)
Such so-called product functions describe an uncorrelated system for which the prop-
erties of a specific particle can be measured without being influenced by the other
particles.
The complete set of all product functions spans the space of all possible
N-particle states involving distinguishable particles.
B) The particles are indistinguishable bosons. The orthonormal set of N -particle en-
ergy eigenstates comprises of totally symmetric states of the form
ψS(q1, · · · , qN) =1
NS
∑
diff.perm.
ψλ1(qP (1))ψλ2(qP (2)
) · · · ψλN(q
P (N)) ,
with NS
=√
number of different permutations of λ1, · · · , λN . (8)
The possible energy eigenvalues are the same as in equation (7).
The space of bosonic N-particle states is spanned by a reduced set of linear
combinations of product functions. As expected, this describes a correlated
system for which the measurement of the properties of a specific particle is
influenced by the other particles.
C) The particles are indistinguishable fermions. The orthonormal set of N -particle en-
ergy eigenstates comprises of totally antisymmetric states of the form
ψA(q1, · · · , qN) =1√N !
∑
perm.
(−1)P ψλ1(qP (1))ψλ2(qP (2)
) · · · ψλN(q
P (N))
=1√N !
∣∣∣∣∣∣∣∣∣∣∣
ψλ1(q1) ψλ2(q1) · · · ψλN(q1)
ψλ1(q2) ψλ2(q2) · · · ψλN(q2)
......
. . ....
ψλ1(qN) ψλ2(qN) · · · ψλN(qN)
∣∣∣∣∣∣∣∣∣∣∣
, (9)
where the determinant is known as the Slater determinant. The possible energy
eigenvalues are again the same as in equation (7), bearing in mind that λ1, · · · , λN
4
have to be all different. If two complete sets of quantum numbers λj and λk coin-
cide, then two of the columns in the Slater determinant are identical and the totally
antisymmetric eigenfunction ψA vanishes. This is known as the
Pauli exclusion principle for identical fermions: no two identical fermions can be in
the same fully specified 1-particle quantum state.
Also the space of fermionic N-particle states is spanned by a reduced set
of linear combinations of product functions. This too describes a correlated
system for which the measurement of the properties of a specific particle is
influenced by the other particles.
1.2 Occupation-number representation
Our aim: we want to construct the space of quantum states (Fock space) for a
many-particle system consisting of an arbitrary number of unspecified identical
particles. This Fock space makes no statements about the physical scenario that
the considered particles are in, such as being subject to interactions, external
influences, etc.. It simply is the complex vector space (Hilbert space, to be more
precise) that includes all possible many-particle states, on which the quantum
mechanical many-particle theory should be formulated. The actual construction
of Fock space involves finding a complete set of basis states for the decomposition
of arbitrary many-particle state functions. The properties of these basis states
will fix the properties of the Fock space.
The general rules for the construction of Fock space are:
• states should not change when interchanging particles;
• in order to guarantee the superposition principle, Fock space should not change when
changing the representation of the basis states.
1.2.1 Construction of Fock space
Consider identical particles of an unspecified type and assume q to be a corresponding
complete set of commuting 1-particle observables. Take the fully specified eigenvalues of
these observables to be exclusively discrete: qj labeled by j = 1 , 2 , · · · . The correspond-
ing 1-particle basis of normalized eigenstates of q is indicated by |qj〉, j = 1 , 2 , · · · .Subsequently we span the space of many-particle states by means of (special) linear com-
binations of product functions constructed from these 1-particle basis states, in analogy
to the 1-particle energy eigenfunctions that were used in § 1.1 to span the space of non-
interacting many-particle states. Since we are dealing with identical particles, a legitimate
5
many-particle state can make no statement about the identity of a particle in a specific
1-particle eigenstate. Such a many-particle state can at best make statements about the
number of particles nj that reside in a given fully specified 1-particle eigenstate belong-
ing to the eigenvalue qj . These numbers are called occupation numbers and can take the
values 0 , 1 , · · · (if allowed).
Postulate (replacing the symmetrization postulate): the collective set of hermitian
number operators n1 , n2 , · · · , which count the number of identical particles in each of the
1-particle quantum states |q1〉 , |q2〉 , · · · , form a complete set of commuting many-particle
observables. The employed complete set of 1-particle observables q can be chosen freely.
The hidden postulate aspect is that the state space for interacting particles can be
constructed from non-interacting building blocks that are based on 1-particle ob-
servables. Since the complete set of 1-particle observables can be chosen freely,
the superposition principle is automatically incorporated without any restric-
tions. This will guarantee that no mixed symmetry will occur in Fock space.
Hence, the corresponding set of normalized eigenstates |n1, n2, · · ·〉 will span Fock space
completely:
0-particle state : |Ψ(0)〉 ≡ |0, 0, · · ·〉 ≡ vacuum state ,
1-particle states : |Ψ(1)j 〉 ≡ |0, · · · , 0, nj=1, 0, · · ·〉 ≡ |qj〉 ,
. . .
(10)
with
nj | · · · , nj, · · ·〉 = nj | · · · , nj , · · ·〉 , (11)
where · · · indicates the other occupation numbers. The representation of states corre-
sponding to this type of basis is called the occupation-number representation.
We will build up Fock space from the vacuum by adding particles to the basis states step
by step. To this end we introduce the creation operator a†j , which adds a particle with
quantum number qj to a basis state | · · · , nj , · · ·〉 according to
a†j | · · · , nj , · · ·〉 =√nj+1 exp
(iαj(· · · , nj, · · · )
)| · · · , nj+1, · · ·〉 . (12)
In principle the phase factor exp(iαj(· · · , nj, · · · )
)could depend on the chosen basis state
| · · · , nj , · · ·〉 as well as the label j . From the orthonormality of the many-particle basis it
6
follows automatically that
〈· · · , nj, · · · | a†j | · · · , n′j, · · ·〉
(12)====
√
n′j+1 exp
(iαj(· · · , n′
j, · · · ))δnj ,n′
j+1
definition==== 〈· · · , n′
j , · · · | aj | · · · , nj, · · ·〉∗ ,
such that
aj | · · · , nj, · · ·〉 =√nj exp
(−iαj(· · · , nj−1, · · · )
)| · · · , nj−1, · · ·〉 (13)
holds for the annihilation operator aj . This definition of the creation and annihilation
operators is consistent with the notion of the vacuum as a state with no particles, since
aj |Ψ(0)〉 (10),(13)==== 0 ⇒ 〈Ψ(0)| a†j = 0 . (14)
Note that a†j 6= aj , which implies that the creation and annihilation operators themselves
are not observables. At this point we can fix the first part of the phase convention of
the many-particle basis by specifying the action of a†j and aj on the basis states |Ψ(0)〉respectively |Ψ(1)〉 :
a†j |Ψ(0)〉 ≡ |Ψ(1)j 〉 , aj |Ψ(1)
k 〉 ≡ δjk |Ψ(0)〉 ⇒ exp(iαj(0, 0, · · · )
)= 1 . (15)
For arbitrary basis states we then have
[nj , a
†k 6=j
]| · · · , nj, · · · , nk, · · ·〉
(11),(12)====
(nj − nj
)a†k 6=j | · · · , nj , · · · , nk, · · ·〉 = 0 ,
[nj , a
†j
]| · · · , nj , · · ·〉
(11),(12)====
([nj+1]− nj
)a†j | · · · , nj , · · ·〉 = a†j | · · · , nj , · · ·〉 ,
resulting in the commutator relations
[nj , a
†k
]= δjk a
†k
herm. conj.======⇒
[nj, ak
]= − δjk ak . (16)
The above-defined creation and annihilation operators can be linked directly to the associ-
ated number operators. To this end we consider the following matrix elements for arbitrary
basis states:
〈· · · , nj , · · · | a†j aj | · · · , n′j , · · ·〉
(13)==== nj δnj ,n′
j.
This matrix is duly diagonal and has the correct occupation numbers as eigenvalues, i.e.
nj = a†j aj . (17)
7
Switching to another discrete 1-particle representation.
Next we switch from the 1-particle basis |qj〉, j = 1 , 2 , · · · belonging to the complete
set of observables q to a 1-particle basis |pr〉, r = 1 , 2 , · · · belonging to the alternative
complete set of observables p. From the completeness relations it follows that
|qj〉 =∑
r
|pr〉〈pr|qj〉 and |pr〉 =∑
j
|qj〉〈qj|pr〉 , (18)
where the transformation matrix (crj) ≡ (〈pr|qj〉) satisfies the unitarity conditions
∑
r
〈qk|pr〉〈pr|qj〉 = 〈qk|qj〉 = δjk and∑
j
〈pr|qj〉〈qj |pv〉 = 〈pr|pv〉 = δrv . (19)
Fock space can be set up equally well in terms of these 1-particle eigenstates |pr〉 and
corresponding creation and annihilation operators b†r and br . If we take the two vacuum
states in both representations to be identical, the following relations should hold for the
old 1-particle basis states |Ψ(0)〉 , |Ψ(1)j 〉 and the new ones |Φ(0)〉 , |Φ(1)
r 〉 :
vacuum state : |Φ(0)〉 ≡ |0, 0, · · ·〉 = |Ψ(0)〉 ,
1-particle states : a†j |Ψ(0)〉 = |Ψ(1)j 〉 = |qj〉
(18)====
∑
r
|pr〉〈pr|qj〉 (20)
=∑
r
〈pr|qj〉|Φ(1)r 〉 =
∑
r
〈pr|qj〉 b†r |Φ(0)〉 =∑
r
b†r 〈pr|qj〉|Ψ(0)〉 .
Without loss of generality, the (1-particle) relation among the two sets of creation and
annihilation operators can be extended to the entire Fock space:
a†j =∑
r
b†r 〈pr|qj〉herm. conj.======⇒ aj =
∑
r
br 〈qj|pr〉 ,
b†r =∑
j
a†j 〈qj |pr〉herm. conj.======⇒ br =
∑
j
aj 〈pr|qj〉 . (21)
Just as in ordinary 1-particle quantum mechanics, the creation of a particle with quantum
number qj is equivalent to a linear superposition of independently created particles with
all possible quantum numbers pr , each with their own amplitude 〈pr|qj〉 .
An important cross check of the procedure so far is provided by the total number operator
N =∑
j
nj . (22)
This operator counts the total number of particles of a given identical-particle system,
which should be invariant under an arbitrary basis transformation. Indeed we find
N(17)
====∑
j
a†j aj(21)
====∑
j,r,v
b†r bv 〈pr|qj〉〈qj|pv〉(19)
====∑
r,v
b†r bv δrv =∑
r
b†r br .
8
Construction of Fock space: part one.
For an arbitrary many-particle state |Ψ〉 it should hold that a†j a†k |Ψ〉 and a†k a
†j |Ψ〉 fun-
damentally describe the same state. This implies a relation of the type(
a†j a†k − exp
(iβ(Ψ, j, k)
)a†k a
†j
)
|Ψ〉 = 0 ,
where the phase factor exp(iβ(Ψ, j, k)
)can depend on the many-particle state |Ψ〉 as well
as the labels j and k .
Next we invoke the superposition principle to prove that exp(iβ(Ψ, j, k)
)has to be in-
dependent of the many-particle state |Ψ〉 . Consider to this end two different arbitrary
many-particle basis states |Ψ1〉 and |Ψ2〉 in the q-representation. The superposition
principle then states that also |Ψ〉 = c1 |Ψ1〉 + c2 |Ψ2〉 with c1,2 ∈ C should describe a
legitimate many-particle state. From this it follows that
c1 exp(iβ(Ψ, j, k)
)a†k a
†j |Ψ1〉 + c2 exp
(iβ(Ψ, j, k)
)a†k a
†j |Ψ2〉
= exp(iβ(Ψ, j, k)
)a†k a
†j |Ψ〉 = a†j a
†k |Ψ〉 = a†j a
†k (c1 |Ψ1〉+ c2 |Ψ2〉)
= c1 exp(iβ(Ψ1, j, k)
)a†k a
†j |Ψ1〉 + c2 exp
(iβ(Ψ2, j, k)
)a†k a
†j |Ψ2〉.
Here a†k a†j |Ψ1〉 and a†k a
†j |Ψ2〉 are again two different basis states. Therefore
∀Ψ1,Ψ2
exp(iβ(Ψ1, j, k)
)= exp
(iβ(Ψ2, j, k)
)≡ exp
(iβ(j, k)
),
which implies that
∀Ψ
(
a†j a†k − exp
(iβ(j, k)
)a†k a
†j
)
|Ψ〉 = 0 ⇒ a†j a†k − exp
(iβ(j, k)
)a†k a
†j = 0 .
Subsequently the representation independence of Fock space can be used by rewriting this
operator identity according to equation (21) in terms of an arbitrary alternative 1-particle
representation, for which a similar operator identity holds:
∑
r,v
〈pr|qj〉〈pv|qk〉[b†r b
†v − exp
(iβ(j, k)
)b†v b
†r
]= 0 and b†r b
†v − exp
(iβ ′(r, v)
)b†v b
†r = 0 .
For each unique (r, v)-combination in the p-representation we find
〈pr|qj〉〈pv|qk〉[exp(iβ ′(r, v)
)− exp
(iβ(j, k)
) ]
+ 〈pv|qj〉〈pr|qk〉[1 − exp
(iβ(j, k)
)exp(iβ ′(r, v)
) ]= 0 .
Since this has to hold true for an arbitrary basis transformation, we eventually obtain
exp(iβ(j, k)
)= exp
(iβ ′(r, v)
)= ± 1 ≡ exp(iβ) .
9
In this way we have found the phase factor exp(iβ) to be universal and to give rise to two
branches of solutions:
∀Ψ
[a†j , a
†k
]|Ψ〉 = 0 ⇒ commutation relations :
[a†j , a
†k
]=[aj , ak
]= 0
or (23)
∀Ψ
a†j , a
†k
|Ψ〉 = 0 ⇒ anticommutation relations :
a†j , a
†k
=aj , ak
= 0 ,
introducing the anticommutator
A, B
≡ AB + BA . (24)
Construction of Fock space: part two.
Finally this result can be combined with the general commutation identities (16). To this
end we use the fact that a commutator of the type[AB, C
]can be written both in terms
of commutators and anticommutators:
[AB, C
]= A
[B, C
]+[A, C
]B = A
B, C
−A, C
B . (25)
In this way we find for the commuting set of solutions in equation (23) that
[nj , a
†k
] (17)====
[a†j aj, a
†k
]= a†j
[aj , a
†k
]+[a†j , a
†k
]aj
(23)==== a†j
[aj, a
†k
] (16)==== δjk a
†j ,
and for the anticommuting set of solutions that
[nj , a
†k
] (17)====
[a†j aj , a
†k
]= a†j
aj , a
†k
−a†j, a
†k
aj
(23)==== a†j
aj, a
†k
(16)==== δjk a
†j .
All in all the creation and annihilation operators should thus satisfy either
commutation relations :[a†j , a
†k
]=[aj , ak
]= 0 ,
[aj, a
†k
]= δjk 1 , (26)
the corresponding identical particles are called bosons ,
or
anticommutation relations :a†j, a
†k
=aj, ak
= 0 ,
aj , a
†k
= δjk 1 , (27)
the corresponding identical particles are called fermions .
In the next chapter it will be shown that these two branches of creation and annihilation
operators for identical many-particle systems give rise to utterly different statistical the-
ories. In the exercise class it will be proven that the (anti)commutation relations retain
their form under an arbitrary change of representation, which implies that to a given type
10
of particle belongs a given type of statistics and that mixed statistics is indeed excluded.
In addition it will be shown that the many-particle states
|n1, n2, · · ·〉 ≡ (a†1)n1
√n1!
(a†2)n2
√n2!
· · · |Ψ(0)〉 ≡∏
j
(a†j)nj
√nj !
|Ψ(0)〉 (28)
form the orthonormal basis of Fock space that we set out to construct. This universal basis
applies to bosons and fermions alike. Note, though, that the order of the creation operators
matters in the fermionic case and does not matter in the bosonic case!
Properties of bosonic many-particle systems (see exercise 1):
• The basis given above corresponds to the explicit phase convention
exp(iαj(· · · , nj, · · · )
)≡ 1 (29)
in equation (12), such that
aj | · · · , nj, · · ·〉 =√nj | · · · , nj−1, · · ·〉 ,
a†j | · · · , nj, · · ·〉 =√nj+1 | · · · , nj+1, · · ·〉 .
(30)
• For k-fold annihilation (k = 0 , 1 , · · · ) this becomes
(aj)k | · · · , nj, · · ·〉 =
√
nj(nj−1) · · · (nj−k+1) | · · · , nj−k, · · ·〉 ,
which only leads to a sensible result if the series terminates as soon as nj − k < 0.
From this we can deduce straightaway that nj can exclusively take on the values
nj = 0 , 1 , 2 , · · · , (31)
as expected for the eigenvalues of a number operator.
• The order in which the creation operators occur in the basis states is irrelevant. As
a result, the state functions will be totally symmetric in this version of Fock space,
as expected for bosonic states. This aspect will be worked out in § 1.3.
Properties of fermionic many-particle systems (see exercise 2):
• As a result of equation (27) we know that (a†j)2 = 1
2
a†j, a
†j
= 0. This we recognize
as the anticipated Pauli exclusion principle: no two fermions can occupy the same
fully specified 1-particle state.
11
• The occupation numbers can take on only two values,
nj = 0 , 1 . (32)
This follows from the fact that the number operators are projection operators in this
case:
n2j
(17)==== a†j aj a
†j aj
(27)==== a†j (1− a†j aj ) aj
a2j = 0==== a†j aj = nj ,
concurring with Pauli’s exclusion principle.
• The basis given above corresponds to the explicit phase convention
exp(iαj(· · · , nj , · · · )
)≡ (−1)N<j , N<j ≡
j−1∑
k=1
nk (33)
in equation (12), such that
aj | · · · , nj, · · ·〉 = δnj ,1 (−1)N<j | · · · , nj−1, · · ·〉 ,
a†j | · · · , nj, · · ·〉 = δnj ,0 (−1)N<j | · · · , nj+1, · · ·〉 .(34)
• This time the order in which the creation operators occur in the basis states does
matter and the state functions will be totally antisymmetric in this version of Fock
space, as expected for fermionic states. This latter aspect will be worked out in § 1.3
The state functions of arbitrary identical-particle systems can now be expressed in terms
of the basis states (28). Usually these state functions will describe a system with a fixed
number of particles. However, in particular in relativistic quantum mechanical situations
this may not be true, for instance when particles decay (such as n → p e− νe ) or when a
photon is emitted during de-excitation of an excited atom (see Ch. 4). Maser and laser
systems fall into this category as well, as in that case the radiation is built coherently
(i.e. in phase). An example of such a coherent state will be worked out in § 1.6. Finally,
as artefacts of certain approximation methods situations may arise in which the number
of particles is effectively not conserved anymore (see § 1.6 and § 1.7).
1.3 Switching to a continuous 1-particle representation
Since Fock space does not depend on the chosen 1-particle representation, we could also
have opted for a complete set of commuting 1-particle observables k with a continuous
eigenvalue spectrum k . Examples are the position and momentum operators. The cor-
responding “orthonormal” 1-particle basis is given by |k〉 . In this case it makes no
sense to use an occupation-number representation based on counting, unless the continu-
ous k-space is partitioned in small cells.
12
In analogy with the previous discussion we will introduce creation and annihilation oper-
ators
a†k ≡ a†(k) and ak ≡ a(k) , (35)
with the usual properties
a†(k)|Ψ(0)〉 ≡ |k〉 and a(k)|Ψ(0)〉 = 0 . (36)
The properties of these creation and annihilation operators follow directly from the disrete
case by performing a 1-particle basis transformation
a†j =
∫
dk a†(k)〈k|qj〉herm. conj.======⇒ aj =
∫
dk a(k)〈qj |k〉 ,
a†(k) =∑
j
a†j 〈qj |k〉herm. conj.======⇒ a(k) =
∑
j
aj 〈k|qj〉 , (37)
which can be derived in analogy to equation (21) from
a†j |Ψ(0)〉 = |qj〉compl.====
∫
dk |k〉〈k|qj〉(36)
====
∫
dk 〈k|qj〉 a†(k)|Ψ(0)〉 .
For this basis transformation the following unitarity conditions hold:
∑
j
〈k|qj〉〈qj|k′〉compl.==== 〈k|k′〉 = δ(k − k′) ,
∫
dk 〈qj|k〉〈k|qj′〉compl.==== 〈qj|qj′〉 = δjj′ . (38)
Subsequently equations (26) and (27) for a discrete 1-particle representation can be used
to obtain
[a†(k), a†(k′)
]=[a(k), a(k′)
]= 0 ,
[a(k), a†(k′)
]= δ(k − k′)1 (39)
for a bosonic system in a continuous 1-particle representation and
a†(k), a†(k′)
=a(k), a(k′)
= 0 ,
a(k), a†(k′)
= δ(k − k′)1 (40)
for a fermionic system in a continuous 1-particle representation.
Proof: the derivation of the vanishing (anti)commutators is trivial. In addition
a(k) a†(k′) ∓ a†(k′) a(k)(37)
====∑
j, j′
〈k|qj〉〈qj′|k′〉(aj a
†j′ ∓ a†j′ aj
) (26),(27)====
∑
j
〈k|qj〉〈qj|k′〉 1
(38)==== δ(k − k′)1 ,
13
where the minus (plus) sign refers to bosonic (fermionic) systems.
The representation-independent total number operator N can in this case be expressed in
terms of the number density operator
n(k) ≡ a†(k) a(k) (41)
according to
N(22)
====∑
j
nj(17)
====∑
j
a†j aj(37)
====
∫
dk
∫
dk′ a†(k) a(k′)∑
j
〈k|qj〉〈qj |k′〉
(38)====
∫
dk
∫
dk′ a†(k) a(k′)δ(k − k′)(41)
====
∫
dk n(k) . (42)
The basis of N -particle state functions takes the following form in the k-representation:
|k1, · · · , kN〉 ≡ 1√N !
a†(k1) · · · a†(kN)|Ψ(0)〉 : k1 , · · · , kN ∈ k
. (43)
This is a direct consequence of the fact that∫
dk1 · · ·∫
dkN |k1, · · · , kN〉〈k1, · · · , kN | = 1N (44)
is the unit operator in the N -particle subspace, which will be proven in exercise 3. An
arbitrary N -particle state function |Ψ〉 can now be represented in the k-representation by
the function
ψ(k1, · · · , kN) ≡ 〈k1, · · · , kN |Ψ〉 . (45)
As expected, this function is totally symmetric for bosonic systems and totally antisym-
metric for fermionic systems. This too will be proven in exercise 3.
1.3.1 Position and momentum representation
The previous discussion can be generalized trivially to cases with mixed 1-particle eigen-
value spectra, which are partially discrete and partially continuous. Two examples of this
will be worked out now.
The position representation: as a first example of a mixed representation we consider spin-s
particles in the position representation. This representation will be used in chapter 5 when
setting up relativistic QM. In the previous expressions we now simply have to substitute
k → (~r , Sz/~) , k → ~r ∈ IR3 , σ = ms = −s, −s+1 , · · · , s−1 , s ,∫
dk →∑
σ
∫
d~r , δ(k − k′) → δσσ′ δ(~r − ~r ′) and a(k) → ψσ(~r ) . (46)
14
Quite suggestively the annihilation oparator a(k) is replaced in the position representation
by the so-called field operator ψσ(~r ). This suggestive notation is introduced in order to
facilitate the discussion in § 1.5.
The momentum representation: as a second example of a mixed representation we consider
spin-s particles in the momentum representation. This representation is particularly useful
for systems that (in first approximation) consist of free particles, as will be used in chapter 5
for analysing relativistic quantum mechanical wave equations. In this case we have to
perform the following substitutions:
k → (~p, Sz/~) , k → ~p ∈ IR3 , σ = ms = −s, −s+1 , · · · , s−1 , s ,∫
dk →∑
σ
∫
d~p , δ(k − k′) → δσσ′ δ(~p− ~p ′) and a(k) → aσ(~p ) . (47)
These two representations are connected through a Fourier transformation:
ψσ(~r )(37)
====
∫
d~p 〈~r |~p 〉 aσ(~p ) =
∫
d~pexp(i~p · ~r/~)(2π~)3/2
aσ(~p ) ,
aσ(~p )(37)
====
∫
d~r 〈~p |~r 〉ψσ(~r ) =
∫
d~rexp(−i~p · ~r/~)
(2π~)3/2ψσ(~r ) . (48)
In fact this is identical to the connection between the state functions in position and mo-
mentum space known from 1-particle QM (see the lecture course Quantum Mechanics 2).
1.4 Additive many-particle quantities and particle conservation
Observables: we have succeeded in constructing the state space applicable to
systems consisting of an arbitrary number of identical particles and we have
investigated what the basis states look like if we employ discrete and/or contin-
uous 1-particle representations. As a next step in the construction of the many-
particle toolbox we need to have a closer look at the many-particle observables.
In Fock space various kinds of many-particle observables are possible, as long as they are
duly symmetric under particle interchange. An important class of many-particle observ-
ables consists of observables that belong to so-called additive many-particle quantities.
These quantities can be categorized by the number of particles that participate as a clus-
ter in the physical quantity. Subsequently these clusters will be summed/integrated over.
A characteristic property of these additive many-particle quantities is the existence of a
specific representation that makes the corresponding many-particle observable expressible
solely in terms of number (density) operators. This automatically implies that different
eigenstates of the total number operator N are not mixed by such a many-particle ob-
servable (which entails particle conservation).
15
Additive many-particle quantities play a central role in setting up (free-particle) relativistic
QM and in the description of systems consisting of weakly interacting identical particles
(such as electrons in an atom or conduction electrons in a metal). In lowest-order approx-
imation the particles of a weakly interacting system can be considered non-interacting,
which causes the total energy to be expressible as a sum of individual 1-particle energies
(see § 1.1). As a first-order correction to this, the interactions between pairs of particles can
be added. In this way the particles contribute in sets of two to the energy correction. This
procedure can be extended trivially to allow for (progressively smaller) energy corrections
involving larger clusters of particles.
1.4.1 Additive 1-particle quantities
Consider a 1-particle observable A. By employing the completeness relation for the discrete
q -representation, this observable can de written as
A =∑
j,j′
|qj〉〈qj|A|qj′〉〈qj′| .
Each individual term occurring in this sum brings the particle from a state |qj′〉 to a
state |qj〉 , with the matrix element 〈qj |A|qj′〉 as corresponding weight factor. Using this
1-particle observable a proper many-particle observable can be constructed:
A(1)tot =
∑
α
Aα ,
where Aα is the 1-particle observable belonging to particle α . The action of A(1)tot in Fock
space is to simply annihilate a particle in the state |qj′〉 and subsequently create a particle
in the state |qj〉 with weight factor 〈qj|A|qj′〉 :
A(1)tot =
∑
j,j′
a†j 〈qj |A|qj′〉 aj′ =∑
j,j′
〈qj|A|qj′〉 a†j aj′ . (49)
We speak of an additive 1-particle quantity if the corresponding many-particle observable
can be represented in the form (49). This definition hinges on the fact that the given
expression has exactly the same form for any discrete 1-particle representation:
A(1)tot
(18),(21)====
∑
r,r′, j,j′
b†r 〈pr|qj〉〈qj|A|qj′〉〈qj′|pr′〉 br′compl.====
∑
r,r′
b†r 〈pr|A|pr′〉 br′ .
Many-particle observables belonging to additive 1-particle quantities can also be trans-
lated trivially to a continuous 1-particle representation. By means of the basis transfor-
mation (37) and the unitarity conditions (38), equation (49) changes into
A(1)tot =
∫
dk1
∫
dk2 a†(k1)〈k1|A|k2〉 a(k2) =
∫
dk1
∫
dk2 〈k1|A|k2〉 a†(k1) a(k2) . (50)
16
The general expressions (49) and (50) are valid irrespective of the 1-particle observable A
having discrete or continuous eigenvalues.
Special cases: let’s assume that the 1-particle observable A has discrete eigenvalues and
is part of the complete set of commuting 1-particle observables q . In that case
〈qj|A|qj′〉 ≡ Aj′ 〈qj |qj′〉 = Aj δjj′(49)
===⇒ A(1)tot =
∑
j
Aj nj .
The special feature of this representation is the fact that it merely involves counting par-
ticles, in analogy to the total number operator N . We actually already exploited this
concept in § 1.1 while determining the total energy of a non-interacting many-particle sys-
tem. Apart from additive 1-particle quantities with discrete eigenvalues, also additive
1-particle quantities exist with continuous eigenvalues (see below). In that case a contin-
uous representation can be found such that
〈k1|A|k2〉 ≡ A(k2)〈k1|k2〉 = A(k1)δ(k1 − k2)(50)
===⇒ A(1)tot =
∫
dk1A(k1)n(k1) .
Particle conservation: a characteristic feature of additive many-particle quantities is that
the observables only consist of terms with as many creation as annihilation operators. As
such these observables commute with the total number operator N and therefore do not
mix the different eigenstates of N . In that respect one could speak of “conservation of the
number of particles”.
Additive 1-particle quantities in position and momentum representation.
In view of future applications, we now take a closer look at a few useful additive 1-particle
quantities in the mixed representations belonging to spin-s particles in either position or
momentum space.
The total momentum: in the momentum representation the 1-particle momentum opera-
tor ~p is part of the complete set of observables with respect to which the many-particle
Fock space is formulated. As a result, the total momentum operator takes on a particu-
larly simple form in the momentum representation (i.e. merely involving number density
operators in momentum space):
~P(1)tot =
∑
σ
∫
d~p ~p nσ(~p ) =∑
σ
∫
d~r ψ†σ(~r )
(
−i~~ψσ(~r ))
. (51)
The proof of the last step in this equation involves the Fourier transformation (48) for
switching frommomentum to position representation (see also App.A for additional details):
17
−i~∫
d~r ψ†σ(~r )
~ψσ(~r ) = −i~∫
d~p
∫
d~p ′ a†σ(~p ) aσ(~p′)
∫
d~rexp(−i~p · ~r/~)
(2π~)3~ exp(i~p ′·~r/~)
=
∫
d~p
∫
d~p ′ ~p ′ a†σ(~p ) aσ(~p′)
∫
d~rexp(i[~p ′−~p ] · ~r/~)
(2π~)3
=
∫
d~p
∫
d~p ′ ~p ′ a†σ(~p ) aσ(~p′) δ(~p− ~p ′) =
∫
d~p ~p nσ(~p ) .
The total kinetic energy: the corresponding 1-particle observable T = ~p 2/(2m) is again
diagonal in the momentum representation. In analogy to the previous example we find for
the total kinetic-energy operator:
T(1)tot =
∑
σ
∫
d~p~p 2
2mnσ(~p ) =
∑
σ
∫
d~r ψ†σ(~r )
(
− ~2
2m~2ψσ(~r )
)
. (52)
The total spin in the z-direction: in this case the corresponding 1-particle observable Sz is
part of the complete set of observables of both the position and momentum representation.
The total spin operator in the z-direction trivially reads:
~S(1)tot · ~ez =
∑
σ
∫
d~p ~σ nσ(~p ) =∑
σ
∫
d~r ~σ nσ(~r ) ≡∑
σ
~σNσ , (53)
where Nσ counts the total number of particles with spin component ~σ along the z-axis.
The total potential energy in an external field: assume the particles to be subjected to
an external potential V (~r ). The corresponding 1-particle observable V (~r ) is obviously
diagonal in the position representation, resulting in the following total potential-energy
operator:
V(1)tot ≡
∑
σ
∫
d~r V (~r ) nσ(~r ) =∑
σ
∫
d~k
∫
d~p V(~k ) a†σ(~p ) aσ(~p− ~~k) , (54)
using the definition V (~r ) ≡∫d~k V(~k ) exp(i~k ·~r ). The derivation of the right-hand side
of this equation follows the line of proof worked out in exercise 4 for a related type of
potential.
Second quantization: the additive many-particle observables discussed above resemble
quantized versions of the corresponding expectation values of the 1-particle observables
in ordinary 1-particle QM, i.e. expectation values with ψ → ψ . This formal connection
between 1-particle and many-particle quantum theory is indicated somewhat suggestively
by the term “second quantization”.
18
1.4.2 Additive 2-particle quantities
In the discrete q -representation the ordered 2-particle basis states are given by
|Ψ(2)j,k>j〉 ≡ |0, · · · , 0, nj = 1, 0, · · · , 0, nk = 1, 0, · · ·〉 = a†j a
†k |Ψ(0)〉 ,
|Ψ(2)jj 〉 ≡ |0, · · · , 0, nj = 2, 0, · · ·〉 =
(a†j)2
√2
|Ψ(0)〉 , (55)
where the last pair state only has relevance for bosonic systems. With the help of the
corresponding completeness relation, a 2-particle observable B can be written as
B =∑
j,j′,k,k′
|Ψ(2)jk 〉〈Ψ
(2)jk |B |Ψ(2)
j′k′〉〈Ψ(2)j′k′| .
Each individual term occurring in this sum brings the particle pair from a state |Ψ(2)j′k′〉
to a state |Ψ(2)jk 〉 , with the matrix element 〈Ψ(2)
jk |B |Ψ(2)j′k′〉 as corresponding weight factor.
Using this 2-particle observable a proper many-particle observable can be constructed:
B(2)tot =
1
2
∑
α,β 6=α
Bαβ ,
where Bαβ is the 2-particle observable belonging to particles α and β 6= α . The factor
of 1/2 is introduced here to avoid double counting. The corresponding action of this
observable in Fock space simply reads
B(2)tot =
1
2
∑
j,j′,k,k′
a†j a†k 〈Ψ
(2)jk |B |Ψ(2)
j′k′〉 ak′ aj′ =1
2
∑
j,j′,k,k′
〈Ψ(2)jk |B |Ψ(2)
j′k′〉 a†j a†k ak′ aj′ . (56)
This method of writing all creation operators on the left and all annihilation operators
on the right in a many-particle operator is usually referred to as normal ordering. We
speak of an additive 2-particle quantity if the corresponding many-particle observable can
be represented in the form (56). This type of expression has the same form for any
discrete 1-particle representation, bearing in mind that just like we have seen in § 1.4.1each creation/annihilation operator in the expression is linked to a corresponding annihi-
lation/creation operator that is hidden in one of the basis states in the matrix element.
Switching to a continuous representation yields accordingly
B(2)tot =
1
2
∫
dk1 · · ·∫
dk4 〈k1, k2|B |k3, k4〉 a†(k1) a†(k2) a(k4) a(k3) . (57)
Again the generic expressions (56) and (57) are valid irrespective of the 2-particle observ-
abele B having discrete or continuous eigenvalues.
19
Special cases: let’s assume that the 2-particle observable B has discrete eigenvalues and
is diagonal in the q -representation. Then
〈Ψ(2)jk |B |Ψ(2)
j′k′〉 ≡ Bj′k′ 〈Ψ(2)jk |Ψ
(2)j′k′〉 = Bjk δjj′ δkk′
(56)===⇒ B
(2)tot =
1
2
∑
j,k
Bjk a†j a
†k ak aj =
1
2
∑
j,k
Bjk (nj nk − nj δjk) ,
where we have used that
a†j a†k ak aj
(16),(17)==== a†j aj a
†k ak−a†j δjk aj
(17)==== nj nk−nj δjk = pair number operator . (58)
Of course, additive 2-particle quantites may also have continuous eigenvalues (see below).
In that case a continuous representation exists such that B becomes diagonal, allowing to
express B(2)tot in terms of pair density operators belonging to that representation.
Example: spatial pair interactions in position and momentum representation.
Consider a system comprising of spin-s particles with spatial pair interactions described
by the observable U(~r1− ~r2). This observable is obviously diagonal in the position repre-
sentation, resulting in the following total operator for spatial pair interactions:
U(2)tot ≡ 1
2
∑
σ1,σ2
∫
d~r1
∫
d~r2 U(~r1−~r2) ψ†σ1(~r1)ψ
†σ2(~r2)ψσ2(~r2)ψσ1(~r1) . (59)
In the momentum representation this becomes (see exercise 4)
U(2)tot =
1
2
∑
σ1,σ2
∫
d~k
∫
d~p1
∫
d~p2 U(~k ) a†σ1(~p1) a
†σ2(~p2) aσ2(~p2 + ~~k) aσ1(~p1 − ~~k ) , (60)
using the definition U(~r ) ≡∫d~k U(~k) exp(i~k ·~r ).
1.5 Heisenberg picture and second quantization
In preparation for the relativistic treatment of quantum mechanics in chapter 5,
we now address the time dependence of the creation and annihilation operators
in the Heisenberg picture of quantum mechanics.
Consider to this end an isolated identical-particle system described by a Hamilton operator
of the following form:
Htot = H(1)tot + C 1 (C ∈ IR) , (61)
where the term H(1)tot is an additive observable of the type described in § 1.4.1. Next
we address the time evolution of this many-particle system in the Heisenberg picture.
As explained in the lecture course Quantum Mechanics 2, all state functions are time-
independent in the Heisenberg picture and all time dependence of the system resides in
20
the time evolution of the operators. The differences with the many-particle formulation in
the Schrodinger picture, which we have used so far, can be summarized as follows:
Schrodinger picture : state functions → |Ψ(t)〉 = exp(−iHtot t/~
)|Ψ(0)〉 ,
observables → A(t) ,(62)
Heisenberg picture : state functions → |ΨH(t)〉 = |Ψ(0)〉 ,
observables → AH(t) = exp
(iHtot t/~
)A(t) exp
(−iHtot t/~
),
where we have assumed that both pictures coincide at the time stamp t = 0. For example
we have
HtotH(t) = Htot , (63)
since we are dealing here with an isolated system. In the Heisenberg picture the operators
satisfy the so-called Heisenberg equation of motion
d
dtA
H(t) = − i
~
[A
H(t), Htot
]+( ∂A(t)
∂t
)
H
. (64)
Next we investigate the implications for the creation and annihilation operators.
The discrete case: for a generic discrete 1-particle representation we get
H(1)tot
(49)====
∑
r,r′
〈pr|H |pr′〉 b†r br′(63)
====∑
r,r′
〈pr|H |pr′〉 b†rH(t) br′
H(t) ,
with H being the 1-particle Hamilton-operator belonging to the additive part of the many-
particle operator Htot . From this it follows that
[bv
H(t), Htot
]=
∑
r,r′
〈pr|H |pr′〉[bv
H(t), b†r
H(t) br′
H(t)] (26),(27)
====∑
r,r′
〈pr|H |pr′〉 br′H(t)δvr
=∑
r′
〈pv|H |pr′〉 br′H(t) .
The Heisenberg equation of motion then reads
i~d
dtbv
H(t) =
∑
r
〈pv|H |pr〉 brH(t) , (65)
where we have used that the creation and annihilation operators are time-independent
in the Schrodinger picture. This equation of motion strongly resembles the Schrodinger
equation for a 1-particle state function written in the p-representation:
i~d
dt|ψ(t)〉 = H |ψ(t)〉 〈pv|∗
===⇒ i~d
dt〈pv|ψ(t)〉 compl.
====∑
r
〈pv|H |pr〉〈pr|ψ(t)〉 .
21
The coupled linear differential equations (65) can be decoupled if we choose a representation
for which 〈pv|H |pr〉 = Er 〈pv|pr〉 = Ev δrv , such that
i~d
dtbv
H(t) = Ev bv
H(t) ⇒ bv
H(t) = exp(−iEvt/~) bv
H(0) ≡ exp(−iEvt/~) bv .
Free particles in position space: in the Schrodinger picture the position-space annihi-
lation operator is given by the time-independent field operator ψσ(~r ). The corresponding
time-dependent operator in the Heisenberg picture is concisely written as
(ψσ(~r )
)
H(t) ≡ ψσ(~r, t) , (66)
which can be translated to momentum space by means of the Fourier transformation
ψσ(~r, t) =
∫
d~pexp(i~p · ~r/~)(2π~)3/2
aσH(~p, t) . (67)
For free particles the Hamilton operator is given by the total kinetic-energy operator T(1)tot :
Htot(52)
====∑
σ′
∫
d~p ′ ~p′ 2
2mnσ′(~p ′)
(63)====
∑
σ′
∫
d~p ′ ~p′ 2
2mnσ′
H(~p ′, t) . (68)
In the momentum representation the Heisenberg equation of motion reads
i~∂
∂taσ
H(~p, t)
(64),(68)=======
∑
σ′
∫
d~p ′ ~p′ 2
2m
[aσ
H(~p, t) , nσ′
H(~p ′, t)
] (16)====
~p 2
2maσ
H(~p, t)
⇒ aσH(~p, t) = exp(−iE~p t/~) aσ(~p ) with E~p ≡ ~p 2
2m. (69)
This implies the following for the field operator ψσ(~r, t) in the position representation:
i~∂
∂tψσ(~r, t)
(67),(69)=======
∫
d~pexp(i~p · ~r/~)(2π~)3/2
~p 2
2maσ
H(~p, t)
(67)===⇒ i~
∂
∂tψσ(~r, t) = − ~2
2m~2ψσ(~r, t) . (70)
• In the Heisenberg picture the many-particle field operator ψσ(~r, t) satisfies an equa-
tion that is identical to the free Schrodinger equation for a 1-particle wave function
in position space.1 For this reason ψσ(~r, t) is also known as the Schrodinger field.
Equation (70) can be regarded as a quantized version of the “classical” Schrodinger
equation for quantum mechanical wave functions. Also this direct link between a
1-particle wave equation and a many-particle theory formulated in the Heisenberg
picture is a manifestation of the concept of second quantization.
1This relation extends to situations where potential-energy terms are added to the Hamilton operator.
22
Second quantization will play a crucial role in the transition from non-
relativitic to relativistic QM. We will be forced to abandon the idea of a
1-particle quantum mechanical theory based on a relativistic wave equation
and replace it by a corresponding many-particle formulation that constitutes
the starting point for quantum field theory.
• As we have seen, the above-given equation of motion (wave equation) for ψσ(~r, t)
can be solved (decoupled) straightforwardly by means of a Fourier decomposition
(plane-wave expansion):
ψσ(~r, t)(67),(69)
=======
∫
d~pexp(i~p · ~r/~)(2π~)3/2
exp(−iE~p t/~) aσ(~p ) . (71)
In fact this is a manifestation of particle–wave duality, as each annihilation/creation
of a particle is automatically accompanied by a corresponding plane-wave factor.
At a later stage we will make use of this aspect to come up with an alternative,
many-particle formulation of relativistic QM.
1.6 Examples and applications: bosonic systems
In the preceding discussion we have systematically used the generic term “par-
ticles” to talk about the fundamental quantum-mechanical objects of a specific
quantum system. It seems obvious what these particles should be: intuitively we
expect that the particles belonging to an atomic gas system should be the atoms
themselves. However, in the remainder of this chapter we will see that the
particle interpretation of quantum systems in terms of the corresponding funda-
mental energy quanta can change radically if the interaction parameters of the
system change. This will teach us to be more flexible with the term “particles”
than we are used to. This is actually one of the most profound new aspects of
many-particle quantum mechanics that is covered in this lecture course.
1.6.1 The linear harmonic oscillator as identical-particle system
We kick off with one of the simplest and at the same time one of the most widely used
quantum systems: the linear harmonic oscillator. Why the oscillator has such an important
role to play in quantum physics will become clear at a later stage.
A linear harmonic oscillator is by definition a 1-particle system. However, it also possesses
the hallmark features of an identical-particle system, which will play a crucial role in the
description of particles in the context of relativistic QM. A completely worked out example
along this line will be the photon description of electromagnetic fields in chapter 4. In order
23
to highlight the analogy between a linear harmonic oscillator and an identical-particle
system we first rewrite the Hamilton operator:
H =p2x2m
+1
2mω2 x2 ≡ ~ω
2
(P 2 + Q2
),
with P ≡ px
√
1
m~ωand Q ≡ x
√mω
~. (72)
Subsequently we introduce the raising, lowering en number operator according to
raising operator : a† ≡ 1√2
(Q− iP
),
lowering operator : a ≡ 1√2
(Q+ iP
),
number operator : n ≡ a† a . (73)
The dimensionless position and momentum operators Q and P have the following funda-
mental properties:
Q† = Q =a+ a†√
2, P † = P =
a− a†
i√2
and[Q, P
]=
1
~
[x, px
]= i 1 . (74)
In the last step we have used that the coordinate x and momentum px form a conjugate
pair, which implies that the corresponding operators satisfy the usual (canonical) quantum
mechanical quantization conditions. From this we can deduce the following features:
• a† 6= a, i.e. raising and lowering operators are not hermitian and therefore do not
correspond to an observable quantity.
• n† = (a† a)† = n , i.e. the number operator does correspond to an observable quantity.
• the raising and lowering operators satisfy bosonic commutation relations:
[a, a†
]= 1
2
[Q + iP , Q− iP
] (74)==== 1 and
[a†, a†
]=[a, a]= 0 . (75)
• from the relation
P 2 + Q2 (74)====
( a− a†
i√2
)2
+( a + a†√
2
)2
= a† a + a a†
it follows that the Hamilton operator is linked directly to the number operator:
H =~ω
2
(P 2 + Q2
)=
~ω
2
(a† a+ a a†
) (75)==== ~ω
(n + 1
21). (76)
24
A linear harmonic oscillator effectively behaves like an identical-particle system consisting
of non-interacting bosons. These bosons can take on precisely one value for the 1-particle
energy, i.e. E1 = ~ω . As such they can be regarded as the energy quanta belonging to
the considered vibrational/oscillatory motions. These quanta are created by a†, annihi-
lated by a and counted by n. Except for a constant term (zero-point energy) 12~ω , the
Hamilton-operator takes the form of a many-particle observable belonging to an additive
1-particle quantity.
The fundamental energy quanta of an oscillator system can be interpreted as
non-interacting bosonic particles. Nature displays many different bosonic vibra-
tion/oscillation quanta, in particular those that provide a particle interpretation
to wave phenomena in classical physics. Well-known examples are photons, re-
lated to electromagnetic waves (see Ch. 4), and phonons, related to collective
vibrational modes of lattices and other sound waves (see exercise 5).
In accordance with § 1.2.1 the energy eigenstates of the linear harmonic oscillator can be
written as
|n〉 (28)====
(a†)n√n!
|0〉 , with H |n〉 (11)==== ~ω
(n+ 1
2
)|n〉 (n = 0 , 1 , 2 , · · ·) . (77)
The meaning of the designations “raising” and “lowering” becomes apparent if we consider
the action of the creation and annihilation operators:
a† |n〉 (30)====
√n+1 |n+1〉 ⇒ H a† |n〉 = ~ω
(n+ 1 + 1
2
)a† |n〉 ,
a|n〉 (30)====
√n |n−1〉 ⇒ H a|n〉 = ~ω
(n− 1 + 1
2
)a|n〉 . (78)
The raising (lowering) operator changes a given energy eigenstate into the energy eigenstate
belonging to the next excited (previous de-excited) energy level.
Remark: if we would have considered a fermionic oscillator
H = ~ω(n + C 1
)= ~ω
(a† a+ C 1
)(C ∈ IR) ,
witha, a†
= 1 and
a†, a†
=a, a
= 0 , (79)
Pauli’s exclusion principle would have allowed for only two energy eigenvalues, ~ωC and
~ω (C +1) corresponding to n = 0 and n = 1 respectively. Still, the operators a† and a
would have been raising and lowering operators, since the commutator identity[a, n]= a
holds equally well in the fermionic case.
25
1.6.2 Forced oscillators: coherent states and quasi particles
Let’s now add an interaction to the linear harmonic oscillator of § 1.6.1 according to
H = ~ω(n− λa† − λ∗ a + 1
21)
= ~ω(n + 1
21)− ~ω (Reλ)
(a†+ a
)− ~ω (Imλ)
(ia†− ia
)
(72),(74)====
p2x2m
+1
2mω2 x2 − x Reλ
√2m~ω3 − px Imλ
√
2~ω/m (λ ∈ C) . (80)
Systems of this type are called forced linear harmonic oscillators. They are omnipresent
in nature. For instance, taking λ = qE/√2m~ω3 and E ∈ IR we obtain an interaction
term of the form V = −q xE , which we recognize as the interaction term of a charged
linear harmonic oscillator with charge q subjected to a classical electric field E along the
x-axis. The striking feature of the additional interaction term is that it does not conserve
the number of oscillator quanta, since
[n, H
]= − ~ω
[n, λa† + λ∗ a
] (16)==== − ~ω (λa† − λ∗ a) 6= 0 .
This is a first example of what is referred to as a non-additive interaction, i.e. an interaction
that does not conserve the total number of particles.
A new particle interpretation: nature is riddled with systems that involve non-additive
interactions. For instance, such systems are ubiquitous in particle physics (e.g. through
the production and decay of particles), quantum optics/spectroscopy (e.g. in laser systems
and when photons are being absorbed/emitted by atoms) and quantum electronics (e.g. in
describing charge flows).
In scenarios involving non-additive interactions it usually pays to switch to an
alternative particle interpretation, i.e. an alternative description of the interact-
ing quantum system in terms of non-interacting energy quanta that are called
quasi particles. Expressed in terms of these new energy quanta the Hamilton
operator will take on an additive 1-particle form, which causes the number of
quasi particles to be conserved. Furthermore, this implies the existence of a rep-
resentation that will make the Hamilton operator expressible in terms of number
operators for quasi particles.
To get a feel for what a quasi particle could be, consider a particle that propagates through
a medium while influencing other particles and while being influenced by other particles.
The combination of particle and ambient influences can collectively behave as a separate
free entity. For instance, a conduction electron propagating through a piece of metal can
excite vibrations (phonons) in the metal-ion lattice, while in turn the lattice vibrations can
affect the motions of the conduction electrons. The degree by which the quasi particles
will differ from the original particles depends on the nature and strength of the ambient
interactions.
26
In all examples covered in this lecture course, the original particle interpretation
is retrieved in the limit of vanishing ambient influences. If the original particles
are bosons (fermions) then the same should be true for the quasi particles.
Therefore, while switching from particles to quasi particles, one set of bosonic (fermionic)
creation and annihilation operators will be replaced by another set. As such, it will not
come as a surprise that a particle–quasi particle transition actually boils down to per-
forming a unitary transformation in Fock space, since operator identities such as bosonic
commutation relations (fermionic anticommutation relations) will be invariant in that case.
Quasi particles and coherent states (Roy J. Glauber, 1963): in case of a forced
linear harmonic oscillator we can switch to new bosonic quasi-particle operators c and c†
by means of the transformation
c = a− λ 1 , c† = a† − λ∗ 1 ⇒[c, c†
]=[a, a†
]= 1 . (81)
The number operator for quasi particles ˆn now reads
ˆn = c† c(81)
==== n − λa† − λ∗ a + |λ|2 1 ⇒ H = ~ω(ˆn + 1
21 − |λ|2 1
), (82)
allowing us to express the energy eigenstates of the systen in terms of quasi-particle states:
|n〉λ(28)
====(c†)n√n!
|0〉λ , H |n〉λ = ~ω(n+ 1
2−|λ|2
)|n〉λ (n = 0 , 1 , 2 , · · ·) . (83)
The ground state of the forced linear harmonic oscillator is given by the state without any
quasi particles ( quasi-particle vacuum) |0〉λ ≡ |λ〉 , which satisfies
a |λ〉 (81)====
(c+ λ 1
)|λ〉 vacuum
======= λ|λ〉 . (84)
This quasi-particle vacuum is an eigenstate of the original annihilation operator a be-
longing to the eigenvalue λ . Such a state is called a coherent state, where the term
coherent refers to the fact that all original many-particle states contribute to this state
with related phases, as opposed to incoherent situations where the phases are completely
random (see Ch. 2).
Proof: express the state |λ〉 in terms of the original many-particle basis according to
|λ〉 ≡∞∑
n=0
Cn |n〉(84)
===⇒ 0 =∞∑
n=0
Cn
(a− λ 1
)|n〉 (78)
====∞∑
n=0
Cn
(√n |n−1〉 − λ|n〉
)
basis===⇒ Cn+1 =
λCn√n+ 1
= · · · =λn+1
√
(n + 1)!C0 ,
27
where the summation parameter n is increased by one in the term containing |n − 1〉 .Next we have to normalize the state |λ〉 :
1 ≡ 〈λ|λ〉 = |C0|2∞∑
n,n′=0
λn√n!
(λ∗)n′
√n′!
〈n′|n〉 orth. basis======= |C0|2
∞∑
n=0
|λ|2nn!
= |C0|2 exp(|λ|2).
By choosing C0 = exp(−|λ|2/2) we finally obtain
|λ〉 =∞∑
n=0
exp(−|λ|2/2) λn√n!
|n〉 , (85)
which can be rewritten by employing equation (28):
|λ〉 =∞∑
n=0
exp(−|λ|2/2) λn√n!
(a†)n√n!
|0〉 = exp(−|λ|2/2) exp(λa†)|0〉 . (86)
Indeed we observe that all original many-particle states feature in such a coherent state,
with the expectation value for the occupation number and associated quantum mechanical
uncertainty given by
n ≡ 〈λ|n|λ〉 = 〈λ|a†a|λ〉 (84)==== |λ|2 〈λ|λ〉 = |λ|2 ,
∆n ≡√
〈λ|n2|λ〉 − 〈λ|n|λ〉2 =√
〈λ|(a†a†a a+ a†a)|λ〉 − n2(84)
====√n . (87)
Here we used that n2 = a†a a†a(75)
==== a†a†a a + a†a. The larger the value of |λ| =√n
becomes, the larger the average number will be of original oscillator quanta that are
present in the ground state of the forced oscillator and the smaller the relative error
∆n/n = 1/√n = 1/|λ| on this number will be!
The relation between particle number and phase: writing the complex eigenvalue
λ as λ = |λ| exp(iφλ), the corresponding coherent state has the characteristic feature that
the many-particle basis states |n〉 occur with phases that are related by exp(inφλ):
|λ〉 (86)====
∞∑
n=0
exp(−|λ|2/2)(|λ| a†
)n
n!exp(inφλ)|0〉 = exp(−|λ|2/2) exp
(|λ|eiφλ a†
)|0〉 .
⇒ − i∂
∂φλ
|λ〉 = a† |λ|eiφλ |λ〉 = a†λ|λ〉 (84)==== a†a |λ〉 = n |λ〉 .
Since the coherent states form a complete set (see remark 1), the following relation between
particle number and phase emerges
− i∂
∂φλ= n
28
in the coherent-state representation. This resembles the quantum mechanical relation be-
tween conjugate pairs of coordinates and momenta, such as the coordinate x and the cor-
responding momentum px for which we have the position-space relation px = − i~∂/∂x.
As such we expect to find a kind of particle number–phase uncertainty relation of the
form
∆n∆φλ ≥ 1/2 , (88)
although it is not possible to define a unique phase operator in QM in view of the fact that
φλ is defined on the interval [0, 2π) only. For a fixed-n state we would therefore expect
to have a completely undetermined phase. Actually we have already made use of this in
§ 1.2 by exploiting the freedom to choose the overall phase of the occupation-number basis
states. For an arbitrary coherent state we expect to find a compromise between uncertainty
in particle number and uncertainty in phase (see remark 2)
Pseudo-classical properties of the coherent state |λ〉: we know that
〈λ|(a†)k(a)l|λ〉 = (λ∗)k(λ)l = 〈λ|a†|λ〉k〈λ|a|λ〉l
for normal-ordered operators, seemingly bypassing quantum mechanical commutation re-
lations by allowing the operators to be effectively replaced by complex parameters. Well,
hold your horses · · · this classical behaviour does not persist once we consider operators
that are not normal ordered, such as
〈λ|a a†|λ〉 = 〈λ|(a†a+ 1)|λ〉 = |λ|2 + 1 = 〈λ|a|λ〉〈λ|a†|λ〉+ 1 .
The commutation relations lead to relative corrections of O(1/|λ|2
)= O(1/n) to classi-
cal behaviour. In the high occupation-number limit |λ| ≫ 1 this correction is suppressed
and a classical situation is approached that allows you to effectively replace the rele-
vant many-particle operators by complex numbers in the coherent-state expectation value.
Moreover, it will be shown in exercise 6 that the relative overlap between different coherent
states decreases substantially in that case (see also remark 2), allowing the coherent states
to be viewed as pseudo classical states consisting of an almost fixed, very large number of
oscillator quanta.
Unitary transformation: finally it will be shown in exercise 6 that a coherent state can
be obtained from the ground state of the linear harmonic oscillator. As expected this will
be achieved by means of a unitary transformation in Fock space:
|λ〉 = exp(λa† − λ∗a)|0〉 . (89)
Coherent states are particularly useful in describing electromagnetic radiation
that is produced coherently through stimulated emission. For instance, masers
and lasers fall into this category as well as electromagnetic fields produced by
29
classical electric currents (see Ch. 4). Coherent states also feature prominently
in the description of systems with macroscopically occupied 1-particle states,
such as Bose–Einstein condensates (see Ch. 2) and classical electromagnetic
fields (see Ch. 4).
Remark 1: overcomplete set (no exam material).
The collection of all coherent states
|λ〉 =
∞∑
n=0
exp(−|λ|2/2) λn√n!
|n〉 : λ ∈ C
(90)
forms an overcomplete set. To this end we write the complex eigenvalue λ in terms of
polar coordinates in the complex plane as λ = |λ| exp(iφλ). That allows us to derive the
following generalized completeness relation for coherent states:
∞∫
−∞
d(Reλ)
∞∫
−∞
d(Imλ) |λ〉〈λ| =
∞∫
−∞
d(Reλ)
∞∫
−∞
d(Imλ) exp(−|λ|2)∞∑
n,n′=0
λn (λ∗)n′
√n!n′ !
|n〉〈n′|
polar coord.=======
∞∑
n,n′=0
|n〉〈n′|√n!n′ !
∞∫
0
d|λ| |λ|n+n′+1 exp(−|λ|2)
2π δnn′
︷ ︸︸ ︷2π∫
0
dφλ exp(iφλ[n−n′]
)
z = |λ|2======= π
∞∑
n=0
|n〉〈n|n!
∞∫
0
dz zn exp(−z) = π
∞∑
n=0
|n〉〈n| compl.==== π 1 , (91)
where it is used that∫∞0
dz zn exp(−z) = n! for n = 0 , 1 , · · · . An arbitrary many-
particle state is therefore readily expressible in terms of coherent states!
Remark 2: two more pseudo-classical properties (no exam material).
A coherent state is also an example of a quantum state with minimal uncertainty that is
divided equally over position and momentum operators. By combining the results on the
previous page with equation (74), it can be proven straightforwardly that
(∆P )2 = 〈λ|P 2|λ〉 − 〈λ|P |λ〉2 = − 12〈λ|(a− a†)2|λ〉+ 1
2〈λ|(a− a†)|λ〉2 = 1
2,
(∆Q)2 = 〈λ|Q2|λ〉 − 〈λ|Q|λ〉2 = + 12〈λ|(a+ a†)2|λ〉 − 1
2〈λ|(a+ a†)|λ〉2 = 1
2,
which indeed yields (∆x)(∆px)(72)
==== ~(∆P )(∆Q) = ~/2. The quantum mechanical
expectation values and uncertainties in the (Q,P )-plane are illustrated schematically in
the picture displayed below. From the quantum mechanical uncertainties (i.e. the grey
30
circles with radius 1/√2 ) it can be read off that for increasing |λ| both the relative
uncertainty in |λ| and the absolute uncertainty in the phase φλ get smaller. This is
in marked contrast to the case λ = 0, for which the phase φλ is completely undeter-
mined! As will be shown in exercise 6, for |λ| ≫ 1 a classical situation is approached with
substantially decreased relative overlap between coherent states. In addition, upon close
inspection we indeed observe the anticipated uncertainty relation between particle number
and phase:√2 |λ|∆φλ ≥ 1/
√2
(87)===⇒ ∆n∆φλ ≥ 1/2.
Q
P
λ = 0 : |0〉
Q
P
λ 6= 0 : |λ〉
√2 |λ|
√2Reλ
√2 Imλ
φλ
∆φλ
We finalize by noting that a coherent state is the quantum state that comes closest in
approximating the classical motion of a non-interacting linear harmonic oscillator. Equa-
tion (65) tells us that the annihilation operator a of a linear harmonic oscillator system
is given in the Heisenberg picture by
aH(t) = a exp(−iωt) ⇒ a
H(t)|λ〉 = λ exp(−iωt)|λ〉 = |λ| exp
(i[φλ−ωt]
)|λ〉 ,
implying that
〈λ|QH(t)|λ〉 =
√2 Re
(λ exp(−iωt)
)and 〈λ|P
H(t)|λ〉 =
√2 Im
(λ exp(−iωt)
)
indeed reproduce the time-dependent motion of a linear harmonic oscillator with a mini-
mum of quantum mechanical uncertainty.
1.6.3 Superfluidity
As a second application of the concept of quasi particles, we consider the quantum-
mechanical low-temperature phenomenon of superfluidity. The explicit example that will
be investigated will teach us that a low-energy spectrum of excitations in a bosonic many-
particle system can change radically upon switching from a non-interacting (ideal-gas)
scenario to a scenario in which the particles repel each other weakly. Before we go into the
details, first the concept of superfluidity will be introduced.
31
~u
object at rest
−~u
fluid at rest
We speak of a superfluid flow inside a medium
if it is not subject to energy leakage through
friction. By means of an elegant argument
(formulated by L.D. Landau in 1941) it can
be established when a superfluid flow is possi-
ble. To this end we consider a fluid that flows
at uniform velocity ~u past a macroscopic ob-
ject. Next we switch to a frame of reference in
which the fluid is at rest (at absolute zero of
temperature) and the object moves at veloc-
ity −~u . Friction (viscosity) occurs only then,
when the object can convert part of its kinetic
momentum and energy into excitations in the fluid. Let’s have a closer look at such an ex-
citation with momentum ~p and energy ǫ(~p ). The conserved total momentum and energy
of the combined system, consisting of the fluid and the macroscopic object with mass M ,
is given by
~P = −M~u = ~p +M~u ′ and E =1
2M~u 2 = ǫ(~p ) +
1
2M~u ′ 2 ,
with ~u ′ being the velocity of the object after exciting the fluid. By merging both equations
one obtains
ǫ(~p ) + ~p · ~u = − ~p 2
2M.
From this it follows that the excitation is energetically not possible if the flow velocity of
the fluid satisfies the condition
u ≡ |~u | < ǫ(~p )
|~p | .
If the spectrum of fluid excitations ǫ(~p ) displays a minimum (critical velocity)
uc =( ǫ(~p )
|~p |)
min> 0 , (92)
then superfluidity can set in at velocities u < uc provided that the temperature is suffi-
ciently low. This is the so-called Landau criterion for superfluidity.
We can distinguish a few special shapes (dispersion relations) of the excitation spectrum.
• ǫ(~p ) = ~p 2/(2m): this corresponds to the energy spectrum of free non-relativistic
particles. Example: a gas consisting of non-interacting massive particles. In that
case ǫ(~p )/|~p | = |~p |/(2m) does not possess a positive minimum and according to
Landau’s criterion no superfluidity can occur.
32
• ǫ(~p ) = cs |~p | : this corresponds to the energy spectrum of free massless quanta.
Example: quantized wave phenomena, such as sound waves in a fluid or crystal (see
exercise 5). In that case ǫ(~p )/|~p | equals the speed of propagation cs of the waves
inside the medium. According to Landau’s criterion, superfluidity can occur for flow
velocities u < cs . This example will feature prominently in the explicit calculation
presented below.
• The excitation spectrum has a finite energy gap
∆ = lim|~p |→0
ǫ(~p ) between the ground state and the
first excited state. Example: the energy gap that
features in the energy spectrum of superconductors
as a result of attractive interactions between special
pairs (Cooper pairs) of conduction electrons. In that
case ǫ(~p )/|~p | clearly displays a minimum and super-
fluidity can occur if the corresponding flow velocity
is low enough.
ǫ(~p )ǫ(~p )/|~p |
0 |~p |
1.6.4 Superfluidity for weakly repulsive spin-0 bosons (part 1)
As an example we investigate the low-energy excitation spectrum of weakly repulsive
bosons. To this end we consider a system consisting of a very large, constant number
N of identical spin-0 particles with mass m. The particles are contained inside a large
cube with edges L and periodic boundary conditions, giving rise to a discrete momentum
spectrum:
~p = ~~k : kx,y,z = 0 ,± 2π/L,± 4π/L, · · · . (93)
In addition we assume the temperature to be low enough for the system to be effectively
in the ground state, i.e. effectively we are dealing with a T = 0 system.2
Without interactions among the particles: the total kinetic energy operator of the
non-interacting identical-particle system is diagonal in the momentum representation:
Ttot =∑
~k
~2~k 2
2ma†~k a~k =
1
2
∑
~k 6=~0
~2~k 2
2m
(a†~k a~k + a†−~k
a−~k
), (94)
with kinetic-energy eigenvalues
E(0) =∑
~k
~2~k 2
2mn(0)~k
. (95)
2The relevant details regarding temperature dependence and regarding the quantization aspects of the
container can be found in chapter 2.
33
The second expression for Ttot is given for practical purposes only (see below). It exploits
the symmetry of the momentum summation under inversion of the momenta. By adding
the total momentum operator
~Ptot =∑
~k
~~k a†~k a~k =1
2
∑
~k 6=~0
~~k(a†~k a~k − a†−~k
a−~k
), (96)
we can readily read off the particle interpretation belonging to the creation and annihila-
tion operators used. Particles with energy ~2~k 2/(2m) and momentum ~~k are created by
a†~k , annihilated by a~k and counted by n~k = a†~k a~k . The occupation number n(0)~k
indicates
how many particles can be found in the given momentum eigenstate in the absence of mu-
tual interactions. The ground state of the non-interacting N -particle system automatically
has n(0)~0
= N and n(0)~k 6=~0
= 0.
Including a weak repulsive interaction among the particles: in analogy with ex-
ercise 4, the inclusion of weak pair interactions that depend exclusively on the distance
between the two particles will lead to an additive many-particle interaction term of the
form
V =1
2
∑
~k,~k ′, ~q
U(q) a†~k a†~k ′a~k ′+~q a~k−~q , (97)
where
U(q) ≡ U(|~q |) =1
V
∫
V
d~r U(r) exp(−i~q ·~r ) (98)
is the Fourier transform of the spatial pair interaction per unit of volume. As argued in
exercise 4, the pair momentum of the interacting particles remains conserved under the
interaction (owing to translational symmetry) and U(q) merely depends on the absolute
value of ~q (owing to rotational symmetry).
What do we expect for the energy eigenstates in the ineracting case?
• First of all the state |n~0 = N, n~k 6=~0 = 0〉 will no longer be the ground state of the
interacting many-particle system. By applying V we observe
V |n~0 = N, n~k 6=~0 = 0〉 =1
2
∑
~q
U(q) a†~q a†−~q a~0 a~0 |n~0 = N, n~k 6=~0 = 0〉
=1
2N(N−1)U(0)|n~0 = N, n~k 6=~0 = 0〉
+1
2
√
N(N−1)∑
~q 6=~0
U(q)|n~0 = N−2, n~q = n−~q = 1, other n~k = 0〉 .
Therefore we expect the true ground state to receive explicit (small) contributions
from pairs of particles that occupy excited 1-particle eigenstates with opposite mo-
mentum. In that way the net momentum ~Ptot= ~0 indeed remains unaffected.
34
• In the non-interacting case the low-energy N -particle excitations involve just a few
particles in excited 1-particle states with O(h/L) momenta. Since all particles then
come with O(L) de Broglie wavelengths, we expect the quantum mechanical influ-
ence of the particles to extend across the entire system. Upon including repulsive
effects, we expect a single excitation to involve more than just bringing a single par-
ticle into motion and hence to require more energy than in the non-interacting case.
Approximations for weakly repulsive dilute Bose gases (Bogolyubov, 1947):
if the pair interaction is sufficiently weak and repulsive, we expect for the
low-energy N-particle states that still almost all particles occupy the 1-particle
ground state, i.e. N− 〈n~0〉 ≪ N . This allows us to apply two approximation
steps to simplify the many-particle interaction term, which hold as long as the
gas is sufficiently dilute to avoid too many particles from ending up in excited
1-particle states.
Approximation 1: for obtaining the low-energy N -particle states the effect from inter-
actions among particles in excited 1-particle states can be neglected. This boils down to
only considering interaction terms with two or more creation and annihilation operators
that belong to the 1-particle ground state:
V ≈ 1
2U(0)
n2~0− n~0
︷ ︸︸ ︷
a†~0 a†~0a~0 a~0 + U(0)
n~0︷︸︸︷
a†~0 a~0∑
~k 6=~0
n~k︷︸︸︷
a†~k a~k +1
2
n~0︷︸︸︷
a†~0 a~0∑
~q 6=~0
U(q)(
a†~q a~q + a†−~q a−~q
)
+1
2
∑
~q 6=~0
U(q)(
a~0 a~0 a†~q a
†−~q + a†~0 a
†~0a~q a−~q
)
.
• The first term corresponds to the configuration ~k = ~k ′ = ~q = ~0 , where all creation
and annihilation operators refer to the 1-particle ground state.
• The remaining terms cover situations where only two out of three momenta vanish.
The second term corresponds to the configurations ~k = ~q = ~0 and ~k ′ = ~q = ~0 ,
the third term to ~k ′ = ~k − ~q = ~0 and ~k = ~k ′ + ~q = ~0 , and the last term to~k − ~q = ~k ′ + ~q = ~0 and ~k = ~k ′ = ~0.
Subsequently the number operator n~0 can be replaced everywhere by N − ∑~q 6=~0
n~q and the
total number of particles N can be taken as fixed and very large:
V ≈ N(N−1)
2U(0) +
1
2
∑
~q 6=~0
U(q)(
N a†~q a~q + N a†−~q a−~q + a~0 a~0 a†~q a
†−~q + a†~0 a
†~0a−~q a~q
)
.
35
Approximation 2, traditional approach: in Bogolyubov’s approach it is used that
the operators a~0 a~0 and a†~0 a†~0
can be effectively replaced by N when applied to the
lowest-energy N -particle states. In principle this could involve extra phase factors e2iφ0
and e−2iφ0 , however, these can be absorbed into a redefinition of the remaining cre-
ation and annihilation operators. This approach suggests that we are dealing with an
approximately classical situation, where the fact that the operators a~0 and a†~0 do not
commute only affect the considered N -particle states in a negligible way (as if they were
coherent states with |λ| ≫ 1). This assumption is plausible, bearing in mind that√N −n ≈
√N if N ≫ n. As a result, magnitude-wise the action of a~0 and a†~0 on
states with N−〈n~0〉 ≪ N will be effectively the same. In this way the following effective
approximation is obtained for the total Hamilton operator Htot = Ttot + V applicable to
the lowest lying energy eigenstates:
Htot ≈ N(N−1)
2U(0) +
1
2
∑
~q 6=~0
(~2q2
2m+N U(q)
)(
a†~q a~q + a†−~q a−~q
)
+1
2
∑
~q 6=~0
N U(q)(
a†~q a†−~q + a−~q a~q
)
. (99)
Owing to this second approximation the total number of particles is no longer conserved
under the interactions. The justification for such an approach is purely thermodynamic
by nature (see Ch. 2): “the physical properties of a system with a very large number of
particles do not change when adding/removing a particle”. Since approximations are un-
avoidable for the description of complex many-particle systems, non-additive quantities of
this type are ubiquitous in condensed-matter and low-temperature physics. Note, however,
that the total momentum of the many-particle system remains conserved under the inter-
action, since each term in H adds as much momentum as it subtracts.
Approximation 2, but this time conserving particle number (based on the bachelor
thesis by Leon Groenewegen): in order to avoid that particle number is not conserved
during the second approximation step we can again exploit approximation 1 and write
N ≈ a~0 a†~0. Without loss of accuracy this allows to rewrite the total Hamilton operator
Htot = Ttot + V as
Htot ≈ N(N−1)
2U(0) +
1
2
∑
~q 6=~0
(~2q2
2mN+ U(q)
)(
a~0 a†~0a†~q a~q + a~0 a
†~0a†−~q a−~q
)
+1
2
∑
~q 6=~0
U(q)(
a~0 a~0 a†~q a
†−~q + a†~0 a
†~0a−~q a~q
)
36
≡ N(N−1)
2U(0) +
1
2
∑
~q 6=~0
(~2q2
2m+N U(q)
)(
b†~q b~q + b†−~q b−~q
)
+1
2
∑
~q 6=~0
N U(q)(
b†~q b†−~q + b−~q b~q
)
, (100)
where the operators
b†~q 6=~0
=a~0 a
†~q√N
=a†~q a~0√N
and b~q 6=~0 =a†~0 a~q√N
=a~q a
†~0√N
(101)
have a much clearer physical interpretation. By means of b†~q a particle is excited from
the 1-particle ground state to the 1-particle state with momentum ~~q 6= ~0, whereas b~q
describes the de-excitation of an excited particle to the 1-particle ground state. The total
number of particles is not affected in this way! However, the total Hamilton operator
is identical to equation (99), with the non-additive character of Htot simply following
from the fact that the interaction can both excite pairs of particles from the 1-particle
ground state and make them fall back to the ground state. As a result of the macroscopic
occupation of the 1-particle ground state, the operators b†~q and b~q approximately behave
as ordinary creation and annihilation operators:
[b~q , b~q ′
]= 0 and
[b~q , b
†~q ′
]=
1
N
[a†~0 a~q , a
†~q ′ a~0
]=[a~q , a
†~q ′
] n~0N
−a†~q ′ a~q
N≈[a~q , a
†~q ′
].
More details can be found in the bachelor thesis by Leon Groenewegen.
In analogy with § 1.6.2 it is opportune to switch now to a quasi-particle de-
scription that turns the approximated interacting system into a non-interacting
quasi-particle system, with a corresponding total Hamilton operator that is both
additive and diagonal (i.e. decoupled).
In this quasi-particle description we expect that we have to combine the operator pairs b†~qand b−~q as well as b~q and b†−~q , since
• both operators within such a pair describe the same change in momentum and there-
fore affect the total momentum of the many-particle system in the same way;
• the quasi-particle number operators will then generate the correct terms b†~q b~q , b−~q b†−~q ,
b†~q b†−~q and b−~q b~q .
How to find the correct quasi-particle description will be addressed in the next intermezzo.
37
1.6.5 Intermezzo: the Bogolyubov transformation for bosons
Consider a many-particle system consisting of identical bosons that can occupy two fully
specified 1-particle quantum states |q1〉 and |q2〉 , such as the momentum states |~q 〉 and
|−~q 〉 in § 1.6.4. The corresponding creation and annihilation operators are given by a†1 , a†2
and a1 , a2 . The corresponding Fock space is spanned by the basis states |n1, n2〉 as given
in equation (28), where n1,2 represents the number of identical particles in each of the two
1-particle quantum states. Next we consider a non-additive Hamilton operator of the form
H = E (a†1 a1 + a†2 a2) + ∆(a†1 a†2 + a2 a1) (E > 0 and ∆ ∈ IR) . (102)
Evidently, the total number of particles is not conserved by such an operator, as the
occupation number of each of the two quantum states is raised by one or lowered by one
in the ∆ terms. However, by means of a so-called Bogolyubov transformation it can be
cast into an additive form up to a constant term.3 Such a transformation has the generic
form
c1 ≡ u1 a1 + v1 a†2 , c2 ≡ u2 a2 + v2 a
†1 (u1,2 and v1,2 ∈ IR) . (103)
The real constants u1 , u2 , v1 and v2 will be chosen in such a way that the operators c†1,2 and
c1,2 satisfy the same bosonic commutator algebra as a†1,2 and a1,2 . This will allow us to
formulate a new particle interpretation, where c†1,2 and c1,2 describe the creation and an-
nihilation of quasi particles. In order to guarantee that both a†1,2 , a1,2 and c†1,2 , c1,2
satisfy bosonic commutation relations, the following conditions should hold:
u1v2 − v1u2 = 0 and u21 − v21 = u22 − v22 = 1
⇒ u1 = + u2 , v1 = + v2 and u21 − v21 = 1
or u1 = −u2 , v1 = − v2 and u21 − v21 = 1 .
(104)
Proof: the commutation relations[c1, c1
]=[c2, c2
]=[c1, c
†2
]= 0 follow directly from the
bosonic commutation relations for a†1,2 and a1,2 . The indicated conditions for u1,2 and
v1,2 then simply follow from the fact that the other commutators
[c1, c2
] (103)====
[u1 a1 + v1 a
†2 , u2 a2 + v2 a
†1
] (26)==== (u1v2 − v1u2)1 ,
[c1, c
†1
] (103)====
[u1 a1 + v1 a
†2 , u1 a
†1 + v1 a2
] (26)==== (u21 − v21)1 ,
[c2, c
†2
] (103)====
[u2 a2 + v2 a
†1 , u2 a
†2 + v2 a1
] (26)==== (u22 − v22)1 ,
have to satisfy the usual bosonic commutation relations.
3If the states |q1〉 and |q2〉 carry opposite quantum numbers, such as the momentum in the states |~q 〉and |−~q 〉 in § 1.6.4, then both H and the Bogolyubov transformation conserve these quantum numbers.
38
The sign in this transformation can be chosen freely. Usually one chooses the plus sign,
resulting in the following generic form for a bosonic Bogolyubov transformation:
c1 ≡ u1 a1 + v1 a†2 , c2 ≡ u1 a2 + v1 a
†1 (105)
with inverse
a1 = u1 c1 − v1 c†2 , a2 = u1 c2 − v1 c
†1 . (106)
In the literature one usually opts for a parametrization of u1 and v1 in terms of a real
parameter η according to u1 = cosh η and v1 = sinh η , which automatically satisfies the
condition u21 − v21 = 1.
Bringing the non-additive Hamilton operator of equation (102) in additive form.
To this end we consider two combinations of quasi-particle number operators. First of all
c†1 c1 − c†2 c2(105)==== (u1 a
†1 + v1 a2)(u1 a1 + v1 a
†2) − (u1 a
†2 + v1 a1)(u1 a2 + v1 a
†1)
(26)==== u21 (a
†1 a1 − a†2 a2) − v21 (a1 a
†1 − a2 a
†2)
(26)==== (u21 − v21)(a
†1 a1 − a†2 a2)
(104)==== a†1 a1 − a†2 a2 . (107)
This expression tells us that certain quantum numbers are conserved under the transition
from particles to quasi particles, provided that these quantum numbers take on opposite
values in the states |q1〉 and |q2〉 (see the footnote on p. 38). In § 1.6.6 we will use this
property to conserve the momentum quantum numbers and thereby the total momentum
while performing the transformation, as anticipated in § 1.6.4. Secondly we have
c†1 c1 + c†2 c2 + 1 = (u21 + v21)(a†1 a1 + a†2 a2 + 1) + 2u1v1 (a
†1 a
†2 + a2 a1) . (108)
This implies that for |∆| < E the non-additive Hamilton operator H in equation (102)
can be rewritten as
H = E (a†1 a1 + a†2 a2 + 1) + ∆(a†1 a†2 + a2 a1) − E 1
=√E2−∆2 (c†1 c1 + c†2 c2 + 1) − E 1 . (109)
Proof: based on equation (108) we are looking for a factor C such that C(u21 + v21) = E
and 2Cu1v1 = ∆. This implies that E and C should have the same sign and that
E2 −∆2 = C2(u21 − v21)2 (104)==== C2 E,C > 0
======⇒ |∆| < E and C =√E2−∆2 .
As promised at the start of this intermezzo, the Hamilton operator has been cast into a
form that consists exclusively of a unit operator and number operators for quasi particles.
In exercise 7 it will be shown that the corresponding ground state (i.e. the state without
quasi-particle excitations) is comprised of coherently created pairs of particles. In § 1.7.2the fermionic version of all this will be derived by employing a similar procedure.
39
1.6.6 Superfluidity for weakly repulsive spin-0 bosons (part 2)
The approximation (100) for the total Hamilton operator at the end of § 1.6.4 is exactly of
the form discussed in the previous intermezzo. Therefore, an appropriate set of Bogolyubov
transformations can be performed to recast the Hamilton operator in the additive form
H ≈ N(N−1)
2U(0) − 1
2
∑
~q 6=~0
(~2q2
2m+N U(q)− ǫ~q
)
+1
2
∑
~q 6=~0
ǫ~q (c†~q c~q + c†−~q c−~q)
=N(N−1)
2U(0) − 1
2
∑
~q 6=~0
(~2q2
2m+N U(q)− ǫ~q
)
+∑
~q 6=~0
ǫ~q c†~q c~q , (110)
with corresponding quasi-particle excitation spectrum
ǫ~q = ǫ−~q =~2q2
2m
√
1 +4mN U(q)
~2q2. (111)
For each pair of momenta ~q and − ~q with ~q 6= ~0 a bosonic Bogolyubov transformation
of the type
c~q ≡ u~q b~q + v~q b†−~q , c−~q ≡ u~q b−~q + v~q b
†~q
should be used, based on the following energy parameters in equations (102) and (109):
E → 1
2
(~2q2
2m+N U(q)
)
, ∆ → 1
2NU(q) .
For the total momentum operator we find with the help of approximation 2 on p. 37 that
~Ptot(96)
====1
2
∑
~q 6=~0
~~q(a†~q a~q − a†−~q a−~q
)≈ 1
2
∑
~q 6=~0
~~qa~0 a
†~0
N
(a†~q a~q − a†−~q a−~q
)(112)
(101)====
1
2
∑
~q 6=~0
~~q(b†~q b~q − b†−~q b−~q
) (107)====
1
2
∑
~q 6=~0
~~q(c†~q c~q − c†−~q c−~q
)=∑
~q 6=~0
~~q c†~q c~q .
Now we can read off the new particle interpretation. Quasi particles with energy ǫ~q and
momentum ~~q are created by c†~q , annihilated by c~q and counted by c†~q c~q . The ground
state of the new many-particle system still contains no quanta with momentum ~~q 6= ~0 .
However, both the composition of the ground state in terms of the original particles and
the shape (dispersion relation) of the elementary excitation spectrum have changed as a
result of the interaction.
40
1) Approximated excitation spectrum for weakly repulsive spin-0 bosons:
• For large excitation energies ~2q2 ≫ mN |U(q)| the quasi-particle excitation spec-
trum ǫ~q ≈ ~2q2/(2m)+NU(q) hardly differs from the non-interacting spectrum. As
such, the quasi particles have the same properties as the original bosons.
• For small excitation energies ~2q2 ≪ mN U(q) the quasi-particle excitation spec-
trum ǫ~q ≈ ~q√
N U(q)/m ≈ ~q√
N U(0)/m changes substantially. The quasi
particles describe massless quanta, i.e. quantized sound waves inside the considered
medium with speed of propagation
cs = limq→0
ǫ~q~q
=
√
N U(0)m
for U(0) =1
V
∫
V
d~r U(r) . (113)
The low-energy quasi-particle interpretation of the
interacting system therefore differs fundamentally
from the original particle interpretation of the non-
interacting system. Moreover, we have gone from a
free-particle system that does not allow for the possi-
bility of superfluidity to an interacting system that can
display superfluidity if u < uc ≈ cs .
ǫ~q
|~q |
linear
quadratic
2) Approximated ground state for weakly repulsive spin-0 bosons: another thing that has
changed substantially is the composition of the ground state of the interacting many-
particle system in terms of the original particles (see exercise 7).
• In the non-interacting case all particles occupy the 1-particle ground state with mo-
mentum ~0 and energy 0 (i.e. n(0)~0
= N and n(0)~k 6=~0
= 0). This condensate has spatial
correlations on all distance scales in view of the corresponding macroscopic de Broglie
wavelength.
• The latter still holds in the interacting case. However, the particle excitations for~k 6= ~0 are replaced by quasi-particle excitations with the same momentum. In the
new ground state none of these quasi-particle excitations are occupied (i.e. n~k=~0 ≈ N
and for the quasi particles n~k 6=~0 = 0). This situation without excited quasi particles
differs markedly from the situation without excited particles. Written in terms of
the original particle interpretation the new condensate actually contains particles
that are not in the ground state. To be more precise, it contains coherently excited
particle pairs with opposite momenta. The increase in kinetic energy is in that case
compensated by the decrease in repulsive interaction energy.
41
Remark: if the average spatial pair interaction U(0) would have been attractive, i.e. if
U(0) < 0, then the ground state would have been unstable. This can be read off directly
from the spectrum (111), which would become complex at very low energies. In that case
it would be energetically favourable for the system to have a large number of low-energy
particle pairs outside of the 1-particle ground state, which invalidates the assumptions of
the previous approach. The fermionic version of such a pairing effect and the ensuing
“pair-bonding” instability play a crucial role in the quantum mechanical description of
superconductivity. A first exposure to this phenomenon can be found in exercise 9.
1.6.7 The wonderful world of superfluid 4He: the two-fluid model
|~q | [A−1
]
ǫ~q ∗ k−1B
[K]
phononsrotons
The many-particle system worked out above is
often used to model the low-energy excitations
in liquid 4He, which becomes superfluid be-
low Tλ = 2.18K (P.L. Kapitsa, J.F. Allen and
A.D. Misener, 1937). At this point we have to
add a critical note. The pair interactions cannot
be truly weak, as required by the approximation
method, as we are dealing with a fluid rather than
a gas. As a consequence, a second branch of ex-
citations enters the spectrum at higher energies,
lowering the critical velocity for superfluidity.
fraction helium I
fraction helium IIAt the absolute zero of temperature 4He is com-
pletely superfluid, i.e. no low-energy excitations
are excited thermally. At a temperature around
0.9K a noticable influence of thermally induced
excitations sets in. On the temperature inter-
val 0 < T < Tλ there are effectively two fluids.
On the one hand there is the excitation-free con-
densate. This fluid, referred to as helium II, is
superfluid and carries no thermal energy. On the
other hand there is the collection of thermally induced quasi-particle excitations. This
fluid, referred to as helium I, carries the thermal energy and gives rise to friction. For
T > Tλ the influence of the helium II component is negligible. By means of this so-called
two-fluid model a few surprising phenomena can be understood.
How to recognize Tλ (P.L. Kapitsa, 1937): a remarkable superfluid phenomenon is that
the boiling of liquid helium abruptly subsides the moment that the temperature drops be-
low Tλ . Helium I will flow away from any spot where the fluid is locally warmer, thereby
carrying away thermal energy, whereas non-thermal helium II will flow towards that spot in
42
order to compensate for the drop in mass density. This superfluid helium II is characterized
by an infinitely efficient heat conductance, making it effectively impossible to have a tem-
perature gradient in the fluid. Already for temperatures just below Tλ the heat transport
becomes a millionfold more efficient and gas bubbles will have no time to form.
Friction for T < Tλ : objects moving through 4He at temperatures below Tλ experience
friction exclusively from the helium I component and not from the helium II component.
This allows us to experimentally determine the helium I and helium II fluid fractions (see
the picture on the previous page). For temperatures below about 0.9K the fluid behaves
almost entirely as a superfluid, giving rise to a persistent current once the fluid is set in
motion (e.g. at higher temperature).
Frictionless flow through porous media: as a result of friction the helium I component can
not flow through very narrow capillary channels. However, the superfluid helium II compo-
nent can do this without the need for a pressure difference between the two ends of such a
channel. So, 4He can flow through porous surfaces if T < Tλ . This flow will be completely
frictionless. Such a situation where a selective superfluid flow occurs is called a super leak.
Fisher and Pickett, Nature 444, 2006The fontain effect (J.F. Allen and H. Jones, 1938):
consider an experimental set-up consisting of two
containers with 4He that are connected by means
of a porous plug. Both containers are cooled down
to the same temperature below Tλ . If the tem-
perature in one of the containers is raised slightly,
e.g. by shining a pocket torch on it, then the num-
ber of thermally induced excitations will increase
in that container. As a result, the helium I com-
ponent increases at the expense of the helium II
component. In order to compensate for the differ-
ence in helium II concentration, helium II from the
other container will pass through the porous plug.
However, a compensatory reverse flow of helium I
to the other container, where the helium I concen-
tration is lower, is hampered by friction. Conse-
quently, an increased fluid concentration will ac-
cumulate in the heated container (heat pump). By providing the heated container with a
capillary safety-relief outlet, a spectacular helium fountain (fontain effect) is produced.
43
Creeping helium film: 4He has as an additional
property that the mutual van der Waals bond-
ing is weaker than the van der Waals bonding
to other atoms. Because of this a 30 nm thick
2-dimensional helium film (Rollin film) attaches
itself to the entire wall of a closed helium con-
tainer. If part of this helium film could flow or
drip to a lower level within the container (see
picture), then a superfluid helium II flow will
start that terminates only when the helium level
reaches its energetic optimum everywhere in the
container. During this process the helium liquid is seemingly defying gravity.
In case you want to see some video evidence for the bold statements that were made about
the weird and wacky world of superfluid 4He, then you are advised to have a look at
http://www.youtube.com/watch?v=2Z6UJbwxBZI or to do your own YouTube search.
1.7 Examples and applications: fermionic systems
1.7.1 Fermi sea and hole theory
Consider a fermionic many-particle system consisting of a very large, constant number N
of electrons with mass m. The electrons are contained inside a large cube with edges L
and periodic boundary conditions, giving rise to a discrete momentum spectrum:
~p = ~~k : kx,y,z = 0 ,± 2π/L,± 4π/L, · · · . (114)
Without mutual interactions among the electrons the total kinetic-energy operator
of the non-interacting identical-particle system can be written in diagonal form in the
momentum representation:
Ttot =∑
~k
∑
ms=±1/2
~2~k 2
2ma†~k,ms
a~k,ms, (115)
with kinetic energy eigenvalues
E =∑
~k,ms
~2~k 2
2mn~k,ms
. (116)
By adding the total momentum operator and total spin operator along the z-direction
~Ptot =∑
~k,ms
~~k a†~k,msa~k,ms
and ~Stot · ~ez =∑
~k,ms
ms~ a†~k,ms
a~k,ms(117)
44
we can trivially read off the particle interpretation belonging to the creation and an-
nihilation operators used. Particles with kinetic energy ~2~k 2/(2m), momentum ~~k and
spin component ms~ along the z-axis are created by a†~k,ms, annihilated by a~k,ms
and
counted by n~k,ms= a†~k,ms
a~k,ms. The occupation number n~k,ms
indicates how many parti-
cles can be found in the given momentum and spin eigenstate in the absence of mutual
interactions.
Ground state : for the ground state of the non-interacting N -electron system this auto-
matically implies that n~k,ms= 1 if |~k | ≤ kF and n~k,ms
= 0 if |~k | > kF . Such a system is
called a completely degenerate electron gas. The fully specified 1-particle energy levels are
occupied from the bottom up by one electron up to the Fermi-energie EF = ~2k2F/(2m).
In that case
N =∑
~k,ms
n~k,ms= 2
∑
|~k |≤kF
and Eground = 2∑
|~k |≤kF
~2~k 2
2m. (118)
The corresponding ground state is given by
|Ψground〉 =∏
|~k |≤kF
a†~k, 12
a†~k,− 12
|Ψ(0)〉 ⇒ a~k,ms|Ψground〉 = 0 if |~k | > kF . (119)
This ground state, better known as the Fermi sea , has vanishing total momentum and
spin, but a substantial kinetic energy. Even in the ground state the electron gas exerts
pressure (see Ch. 2)!
kF
empty
occupied
ground state:
full Fermi sea
~k1
~k2
electron–hole
excitation
Excited states : the simplest excited states
can be obtained by exciting a single electron
in the Fermi sea to an energy that places it
above the Fermi sea (see picture). As the
states inside the Fermi sea are all occupied,
excitations are (energetically) easier from en-
ergy levels near the edge of the Fermi sea than
from lower-lying energy levels. This aspect
plays a decisive role in the thermal properties
of fermionic many-particle systems at moderate temperatures (see Ch. 2). An excitation
of this type can be represented in creation and annihilation language in the following way:
|Ψex〉 = a†~k2,ms2
a~k1,ms1|Ψground〉 if |~k1| ≤ kF < |~k2| . (120)
Since the excitation creates a hole in the Fermi sea, it also goes under the name of
electron–hole excitation. In the quantum mechanical approach of hole theory this aspect
45
is highlighted by employing the Bogolyubov transformations
a~k,ms≡ c†−~k,−ms
if |~k | ≤ kF and a†~k,ms≡ c†~k,ms
if |~k | > kF (121)
to switch to a description where a hole in the Fermi sea is upgraded to the status of
quasi particle with opposite quantum numbers. In that case the Fermi sea constitutes
a state without any holes in the sea or excitations above it, i.e. the Fermi sea is the cor-
responding quasi-particle vacuum state! An excited state of the above type simply boils
down to quasi-particle pair creation. The hole-theory approach is particularly useful for
the description of electrical conduction and for Dirac’s attempt to set up relativistic QM.
This approach is only possible for fermions: we have actually made use of
Pauli’s exclusion principle, which states that n~k,ms∈ [0, 1], in order to in-
terchange the role of empty and occupied in the particle interpretation of hole
theory. The advantage of doing this is that the Fermi sea (ground state) is the
vacuum state within this alternative particle interpretation and that an excited
state corresponds to the creation of a quasi-particle pair. In the hole-theory ap-
proach conservation of the number of (quasi) particles is sacrificed deliberately
for achieving ease of use!
Weak interactions among the electrons: for weakly repulsive interactions we can in
general use perturbation theory, utilizing the above-mentioned non-interacting situation
as unperturbed starting point. According to hole theory the Fermi sea can be treated as
vacuum state. For weakly attractive interactions the situation changes substantially. In
exercise 9 it will be shown how the presence of the Fermi sea of occupied states can give
rise to the so-called Cooper instability. This entails that two electrons that occupy states
above a Fermi sea of occupied states will always form a “bound” pair with finite binding
energy as long as their mutual interaction is attractive. In contrast to a 1-particle bound
state in a potential well, this many-particle binding effect occurs irrespective of the strength
of the interaction. Moreover, it cannot be described by means of perturbation theory.
1.7.2 The Bogolyubov transformation for fermions
Also a fermionic version of the Bogolyubov transformation of § 1.6.5 exists. To this end
we consider exactly the same 2-level system as described on page 38. Only after equation
(103) things will start to deviate. So, also here the transformation will be defined by
c1 = u1 a1 + v1 a†2 and c2 = u2 a2 + v2 a
†1 .
In contrast to the bosonic case, both a†1,2 , a1,2 and c†1,2 , c1,2 have to satisfy fermionic
anticommutation relations. That implies the following conditions for the real constants
46
u1,2 and v1,2 :
u1v2 + v1u2 = 0 and u21 + v21 = u22 + v22 = 1
⇒ u1 = + u2 , v1 = − v2 and u21 + v21 = 1
or u1 = −u2 , v1 = + v2 and u21 + v21 = 1 .
(122)
Proof: the anticommutation relationsc1, c1
=c2, c2
=c1, c
†2
= 0 follow directly
from the fermionic anticommutation relations for a†1,2 and a1,2 . The indicated conditions
for u1,2 and v1,2 then simply follow from the fact that the other anticommutators
c1, c2
(103)====
u1 a1 + v1 a
†2 , u2 a2 + v2 a
†1
(27)==== (u1v2 + v1u2)1 ,
c1, c
†1
(103)====
u1 a1 + v1 a
†2 , u1 a
†1 + v1 a2
(27)==== (u21 + v21)1 ,
c2, c
†2
(103)====
u2 a2 + v2 a
†1 , u2 a
†2 + v2 a1
(27)==== (u22 + v22)1 ,
have to satisfy the usual fermionic anticommutation relations. The sign in this transforma-
tion can be chosen freely. Usually one chooses the upper sign (i.e. u1 = u2 and v1 = − v2),
resulting in the following generic form for a fermionic Bogolyubov transformation:
c1 ≡ u1 a1 + v1 a†2 , c2 ≡ u1 a2 − v1 a
†1 (123)
with inverse
a1 = u1 c1 − v1 c†2 , a2 = u1 c2 + v1 c
†1 . (124)
In the literature one usually opts for a parametrization of u1 and v1 in terms of a real
parameter θ according to u1 = cos θ and v1 = sin θ , which automatically satisfies the
condition u21 + v21 = 1.
Special case: if u1 = u2 = 0, then the signs of both v1 and v2 can be chosen indepen-
dently. Usually one will choose v1 = v2 = 1, as we did in § 1.7.1 while describing the quasi
particles in hole theory.
The fermionic quasi-particle vacuum : removing a particle from a particular state is in
the fermionic case equivalent with the creation of a hole in the occupation of that state. The
type of quasi particle we are dealing with here is therefore a linear combination of a particle
and a hole. This type of quasi particle is used abundantly in condensed-matter physics for
describing systems of interacting fermions such as conduction electrons. In exercise 8 the
fermionic quasi-particle vacuum |0〉 will be expressed in terms of the original basis states
47
|n1, n2〉 , resulting in the compact expression
|0〉 up to a phase factor============= u1 |0, 0〉 − v1 |1, 1〉 =
(u1 − v1 a
†1 a
†2
)|0, 0〉 . (125)
Bringing the non-additive Hamilton operator of equation (102) in additive form.
To this end we consider two combinations of quasi-particle number operators. First of all
c†1 c1 − c†2 c2(123)==== (u1 a
†1 + v1 a2)(u1 a1 + v1 a
†2) − (u1 a
†2 − v1 a1)(u1 a2 − v1 a
†1)
(27)==== u21 (a
†1 a1 − a†2 a2) − v21 (a1 a
†1 − a2 a
†2)
(27)==== (u21 + v21)(a
†1 a1 − a†2 a2)
(122)==== a†1 a1 − a†2 a2 . (126)
This expression tells us again that certain quantum numbers are conserved under the
transition from particles to quasi particles, provided that these quantum numbers take on
opposite values in the states |q1〉 and |q2〉 .
Secondly we have
c†1 c1 + c†2 c2 − 1 = (u21 − v21)(a†1 a1 + a†2 a2 − 1) + 2u1v1 (a
†1 a
†2 + a2 a1) . (127)
The non-additive Hamilton operator H in equation (102) can then be rewritten as
H = E (a†1 a1 + a†2 a2 − 1) + ∆(a†1 a†2 + a2 a1) + E 1
= ±√E2+∆2 (c†1 c1 + c†2 c2 − 1) + E 1 . (128)
Proof: based on equation (127) we are looking for a factor C such that C(u21 − v21) = E
and 2Cu1v1 = ∆. From this it follows that E2+∆2 = C2(u21+ v21)2 (122)==== C2 , where the
sign of C is determined by the sign of (u21 − v21)/E .
As desired, the Hamilton operator has been cast into a form that consists exclusively of
a unit operator and number operators for quasi particles. For the description of electron
holes in § 1.7.1 we have actually used a special version of this where c1 = a†2 en c2 = a†1 .
In that case E (a†1 a1 + a†2 a2) = −E (a1 a†1 + a2 a
†2 − 2) = −E (c†2 c2 + c†1 c1 − 2). From the
perspective of the Fermi energy, the creation of a hole in the Fermi sea results in a negative
energy contribution. As noted in the proof given above, the sign of the energy of the quasi
particle is fixed by the sign of (u21− v21)/E, which indeed indicates whether the considered
type of quasi particle is more particle than hole (positive sign) or more hole than particle
(negative sign).
48
2 Quantum statistics
In this chapter the concept of mixed quantum mechanical ensembles will be
introduced. Based on this concept the various types of quantum statistics will
be derived, which govern many-particle systems that are in thermal equilibrium.
Similar material can be found in Griffiths (the quantum statistics part in Ch. 5),
Merzbacher (Ch. 15,16,22) and Bransden & Joachain (Ch. 10,14).
Up to now we have been working with pure quantum mechanical ensembles belonging to a
pure quantum state, i.e. a collection of identical, independent, identically prepared systems
that can be described by a single state function |ψ〉 . This state function can be determined
up to a phase by performing a complete measurement on a complete set of commuting ob-
servables. The systems that make up the ensemble are independent of each other and can
consist of individual particles, interacting particle clusters (such as atoms/molecules) or
even entire many-particle systems (such as gases). These pure quantum states can be
realized experimentally by employing dedicated filters, such as a Stern–Gerlach filter for
selecting a pure spin state.
Question: What happens if not all the degrees of freedom that are relevant for the consid-
ered systems are taken into account, thereby leaving out certain quantum information?
cooled gas reservoir:
O(1023) particles
our favourite system
cooling elements
This situation actually reflects the unavoidable
reality of the quantum mechanical world that we
experience. During a quantum experiment the stud-
ied system is for instance prepared by filters and
subsequently subjected to the environment of the
experimental set-up (gases, electromagnetic fields,
etc.). All these environmental influences caused
by the contact with the macroscopic outside world
in fact determine the properties of the quantum
mechanical ensemble that eventually will be mea-
sured. Examples of this are quantum systems
that have their themodynamical properties fixed by the contact with a macroscopic
gas reservoir present within the measurement apparatus, or a particle beam that
is produced by a particle source and subsequently passes through a set of filters.
It is not feasible to determine the state function of the entire experimental apparatus by
means of a complete measurement. The macroscopic outside world simply has way too
many degrees of freedom. In such situations the only practical approach is to explic-
itly integrate out (read: leave out) the degrees of freedom of the outside world, thereby
reducing the quantum mechanical description to the space spanned by the degrees of free-
49
dom of the studied system itself. Note: sometimes it is not really necessary to integrate
out certain degrees of freedom, but it simply happens to be more convenient to do so. An
example is the polarization of a particle beam, for which it is convenient not to have to
deal with the spatial degrees of freedom (see § 2.2).
Integrating out degrees of freedom: to get an idea of what happens when integrating
out quantum mechanical degrees of freedom, we consider a spin-1/2 system described by
the following normalized pure state function in the position representation:
χ(~r ) =
(
ψ+(~r )
ψ−(~r )
)
= ψ+(~r )χ 12, 12+ ψ−(~r )χ 1
2,− 1
2,
with
∫
d~r ψ∗λ(~r )ψλ′(~r ) = Wλδλλ′ (λ, λ′ = ±) and W+ +W− = 1 .
For convenience we have taken ψ±(~r ) to be orthogonal. Next we integrate out the spatial
degrees of freedom in order to reduce the quantum mechanical description to the two-
dimensional spin space. Consider to this end an arbitrary spin observable A, which acts as
a 2×2 matrix A in spin space and which has no action in position space. The corresponding
expectation value for the state χ(~r ) then reads
〈A〉 =
∫
d~r χ†(~r )Aχ(~r ) = Tr(
A
∫
d~r χ(~r )χ†(~r ))
≡ Tr(Aρ) ,
with ρ =
~r integrated out︷ ︸︸ ︷∫
d~r 〈~r |χ〉〈χ|~r 〉 =
(
W+ 0
0 W−
)
= density matrix in spin space
and Tr = trace in spin space .
Here we used that A commutes with ~r and we exploited the fact that for D-dimensional
vectors v and D×D matrices M the following holds:
D∑
i,j=1
v∗iMij vj =
D∑
i,j=1
Mij (vjv∗i ) ≡
D∑
i,j=1
MijWji = Tr(MW ) .
The expectation value for the complete system splits up according to
〈A〉 = W+ Tr(Aχ 1
2, 12χ†
12, 12
)+ W−Tr
(Aχ 1
2,− 1
2χ†
12,− 1
2
)= W+ 〈A〉+ + W− 〈A〉−
into two independent expectation values, one for the pure spin state χ 12, 12
with weight
W+ ≥ 0 and one for the pure spin state χ 12,− 1
2with weight W− = 1 − W+ ≥ 0. The
50
quantum mechanical description in spin space is now determined by the density matrix ρ,
which is composed of statistical weights and projection operators on pure states:
ρ = W+ χ 12, 12χ†
12, 12
+ W− χ 12,− 1
2χ†
12,− 1
2
= W+
(
1 0
0 0
)
+ W−
(
0 0
0 1
)
.
Mixed ensembles : integrating out quantum mechanical degrees of freedom gives rise
to so-called mixed ensembles. These ensembles are incoherent mixtures (i.e. statistical
mixtures) of pure subensembles, implying double statistics:
• on the one hand the pure quantum mechanical state functions that describe the
pure subensembles have a statistical interpretation;
• on the other hand the complete mixed ensemble comprises of a statistical mixture
of such subensembles.
The quantum physics in such scenarios is determined by the density operator, which
contains the relevant properties of the mixed ensemble. These relevant properties are
the pure quantum mechanical state functions that describe the pure subensembles and
the corresponding statistical weights.
You might be inclined to think that a mixed ensemble somehow corresponds to
a kind of superposition of pure state functions. As will become clear from the
example given below, this is not the case.
Mixed ensembles vs pure ensembles: consider the previous example with equal statis-
tical weights W± = 0.5, resulting in a statistical mixture with density matrix ρ = 12I for
which 50% of the particles are in the spin state χ 12, 12and 50% in the spin state χ 1
2,− 1
2.
Such an incoherent mixture of pure spin states χ 12,± 1
2cannot be represented as a linear
combination of states such as (χ 12, 12+ χ 1
2,− 1
2)/√2, in spite of the fact that such a pure
state indeed corresponds to a 50% chance to measure the particles in each of the two spin
states. However, other measurable quantities exist that will result in completely different
measurement results in both situations. For instance, the state (χ 12, 12+ χ 1
2,− 1
2)/√2 is a
pure eigenstate of the spin operator Sx belonging to the eigenvalue ~/2. Upon measuring
the spin in the x-direction the outcome of the measurement will be fixed: 100% of the
particles will correspond to the measurement value ~/2. For the mixed ensemble, however,
we have
〈Sx〉 =~
2Tr(σxρ)
ρ = 1
21
=======~
4Tr(σx) = 0 6= ~
2,
since none of the two types of particles on which the measurement is performed actu-
ally is in an eigenstate of Sx . An incoherent mixture in fact implies that the phases of
51
the separate components in the mixture are unrelated, which excludes a linear combina-
tion of pure state functions. The ensemble splits up in independent subensembles , each
with a specific spin state and statistical weight.
Density operators find applications in many branches of physics. In this chapter we will
encounter a few characteristic examples from scattering theory and statistical physics
(with applications in solid-state physics, nuclear physics, astrophysics and low-temperature
physics). Density operators are also the tools of choice for the description of the state-
function collapse of a microscopic system as caused by the influence of a macroscopic piece
of measurement equipment.
2.1 The density operator (J. von Neumann, 1927)
Consider an ensemble consisting of a statistical mixture of N independent subensembles,
each described by a normalized pure state function
|ψ(α)〉 ≡ |α〉 , with 〈α|α〉 = 1 (α = 1 , · · · , N ) . (129)
Each subensemble is a collection of identical, independent, identically prepared systems
described by the corresponding state function |α〉 . The independent systems represented
by the various state functions |α〉 are of identical composition, but they differ by the state
they are in. There is no need for these states to be eigenstates of one and the same observ-
able, so that in general 〈α|α′〉 6= 0 if α 6= α′. An example could be an electron beam of
which 40% of the particles are polarized along the z-axis, 30% along the y-axis and 30%
along the x-axis. This beam is composed of three independent subensembles that are each
described by a pure spin function. The three types of systems described by these pure spin
functions all consist of one electron, but the spin is quantized in a different direction in
each of the three cases.
From expectation value to ensemble average: take |n〉 to be an orthonormal set
of eigenstates belonging to a complete set of commuting observables of the considered type
of system. For discrete values of n we then have
〈n|n′〉 = δnn′ and∑
n
|n〉〈n| = 1 . (130)
For n-values belonging to the continuous part of the spectrum, as usual δnn′ should be
replaced by δ(n−n′) and the sum by an integral. Next we can decompose the normalized
pure state functions |α〉 in terms of this basis according to
|α〉 (130)====
∑
n
|n〉〈n|α〉 ≡∑
n
c(α)n |n〉 , with 〈α|α〉 =∑
n
|c(α)n |2 = 1 . (131)
52
Let A be an observable pertaining to the considered type of system, with corresponding
(classical) dynamical variable A . For each of the normalized states |α〉 the expectation
value (= quantum mechanical average) of the dynamical variable A is given by
〈A〉α ≡ 〈α|A|α〉 (130)====
∑
n,n′
〈α|n′〉〈n′|A|n〉〈n|α〉 (131)====
∑
n,n′
c(α)n c(α) ∗n′ 〈n′|A|n〉 . (132)
Subsequently we take into account that we are dealing with a statistical mixture of pure
subensembles. To this end we indicate the statistical weight of the pure state |α〉 in the
ensemble by Wα , corresponding to the fraction of systems in the state |α〉. In short, the
factor Wα is the probability for the system to be in the state |α〉 . The ensemble average
(= statistical average) [A] of the dynamical variable A can be expressed in a representa-
tion independent way as
[A] =
N∑
α=1
Wα 〈A〉α , with Wα ∈ [0, 1] and
N∑
α=1
Wα = 1 . (133)
Definition of the density operator: by means of equation (132) the expression for the
ensemble average can be rewritten as
[A] =
N∑
α=1
∑
n,n′
Wα 〈n|α〉〈α|n′〉〈n′|A|n〉 ≡∑
n,n′
〈n|ρ|n′〉〈n′|A|n〉 , (134)
where the density operator ρ of the ensemble is composed of projection operators on the
pure states:
ρ =
N∑
α=1
Wα |α〉〈α| . (135)
Since Wα ∈ IR, this density operator is hermitian and therefore
∀ψ1,ψ2
〈ψ2|ρ|ψ1〉∗(135)====
N∑
α=1
Wα 〈ψ2|α〉∗〈α|ψ1〉∗ =N∑
α=1
Wα 〈ψ1|α〉〈α|ψ2〉 = 〈ψ1|ρ|ψ2〉 .
In the n-representation the density operator is characterized by the so-called density matrix
ρnn′ = 〈n|ρ|n′〉 (131),(135)=======
N∑
α=1
Wαc(α)n c
(α) ∗n′ . (136)
In matrix language the ensemble average of the dynamical variable A simply reads
[A ](134)====
∑
n,n′
〈n|ρ|n′〉〈n′|A|n〉 (130)====
∑
n
〈n|ρA|n〉 = Tr(ρA) = [A ] . (137)
53
Properties of the density operator:
• Since the density operator ρ is hermitian, an orthonormal basis exists of density-
operator eigenstates |k〉 with real eigenvalues ρk . Such an eigenvalue
ρk = 〈k|ρ|k〉 (136)====
N∑
α=1
Wα |c(α)k |2 ∈ [0, 1] (138)
can be interpreted as the probability for the system to be in the pure state |k〉 .
• Conservation of probability implies
Tr(ρ) = 1 . (139)
Proof: Tr(ρ)(137)==== [1]
(133)====
N∑
α=1
Wα 〈1〉α〈α|α〉=1====
N∑
α=1
Wα(133)==== 1 .
• For arbitrary ensembles we have
Tr(ρ2) ≤ 1 . (140)
Proof: in terms of the eigenvalues ρk ∈ [0, 1] of ρ one readily finds
[ ρ ](137)==== Tr(ρ2) =
∑
k
ρ2k ≤∑
k
ρk = Tr(ρ)(139)==== 1 .
• The following criterion holds for a system in a pure state:
Tr(ρ2) = 1 ⇔ ρ2 = ρ ⇔ the system is in a pure state . (141)
Proof (⇐): assume that the system is in the pure state |λ〉 . In that case Wα = δαλ
and the density operator ρ(135)==== |λ〉〈λ| is a projection operator on |λ〉 . This
automatically implies that ρ2 = ρ and Tr(ρ2) = Tr(ρ) = 1.
Proof (⇒): let Tr(ρ2) = Tr(ρ) = 1. In terms of the orthonormal basis |k〉 of
eigenstates of ρ we then have
∑
k
ρ2k =∑
k
ρk = 1ρ2k ≤ ρk===⇒ ∃
λρk = δkλ
⇒ spectral decomposition : ρ =∑
k
ρk |k〉〈k| = |λ〉〈λ| = ρ2 .
Because the density operator can apparently be expressed as a projection operator
ρ = |λ〉〈λ| , it follows automatically that we are dealing with a pure ensemble with
the system being in the pure state |λ〉 .
54
If the system happens to be in a pure state |λ〉 , then
ρ = |λ〉〈λ| ≡ ρλ (142)
and all eigenvalues of ρ equal 0 except for one eigenvalue 1 belonging to the eigen-
state |λ〉 . In this situation the ensemble average of the dynamical variable A obviously
coincides with the expectation value for the state |λ〉 :
[A] = Tr(ρλA)(133)==== 〈A〉λ = 〈λ|A|λ〉 . (143)
2.2 Example: polarization of a spin-1/2 ensemble
As a first application of the density-matrix formalism we consider a beam of spin-1/2 par-
ticles. A good understanding of the properties of such a beam is of particular interest
for scattering experiments in high-energy physics, since at least one of the beams usu-
ally involves spin-1/2 particles (such as electrons, positrons, protons or antiprotons). In
scattering experiments the beams are sufficiently dilute to treat the beam particles as in-
dependent. In this sense the beam should not be regarded as an identical-particle system
anymore, but rather as a (mixed) ensemble of beam particles that can be used to perform
a repeated quantum mechanical scattering experiment.
The pure states of the beam particles can for instance be decomposed in terms of the
basis |~p,ms〉 : ~p ∈ IR3 and ms = ± 1/2 , where ~p is the momentum eigenvalue of
the beam particle and ms~ the spin component along the z-axis. Next we leave out the
momentum variables from the quantum mechanical description and choose to work with
the reduced density matrix in the 2-dimensional spin space. Spin information is actually
an important tool to analyse scattering data in detail.
Preparatory step: combining two pure beams.
Consider two beams. Beam a consists of Na spin-1/2 particles prepared in the pure spin
state |χ(a)〉 and beam b consists of Nb spin-1/2 particles prepared in the pure spin state
|χ(b)〉 . Subsequently both beams are merged. This results in a statistical mixture of the
two pure subensembles with weights Wa = Na/(Na+Nb) and Wb = Nb/(Na+Nb). The
density operator of this mixed ensemble is given by
ρ = Wa |χ(a)〉〈χ(a)| + Wb |χ(b)〉〈χ(b)| . (144)
With respect to the standard basis |χ1〉 ≡ χ 12, 12, |χ2〉 ≡ χ 1
2,− 1
2 we have
|χ(a)〉 = c(a)1 |χ1〉+ c
(a)2 |χ2〉 and |χ(b)〉 = c
(b)1 |χ1〉+ c
(b)2 |χ2〉 , (145)
55
giving rise to the density matrix
ρ(136)====
Wa |c(a)1 |2 + Wb |c(b)1 |2 Wac
(a)1 c
(a) ∗2 + Wbc
(b)1 c
(b) ∗2
Wa c(a) ∗1 c
(a)2 + Wb c
(b) ∗1 c
(b)2 Wa |c(a)2 |2 + Wb |c(b)2 |2
(146)
in spin space. If the original pure beam states happen to coincide with the chosen basis
states of spin space, i.e. |χ(a)〉 = |χ1〉 and |χ(b)〉 = |χ2〉 , then the density matrix simplifies
to a diagonal matrix:
ρ =
(
Wa 0
0 Wb
)
=
Na
Na+Nb0
0Nb
Na+Nb
. (147)
Polarization of an arbitrary spin-1/2 ensemble: consider the 2×2 density matrix ρ
in spin space. This matrix can be decomposed as
ρ = A0 I2 + ~A · ~σ (A0 ∈ C and Ax,y,z ∈ C) , (148)
in terms of the 2-dimensional identity matrix I2 and Pauli spin matrices σx , σy and σz
that together form a basis for 2×2 matrices (see App.B). From the trace identities (B.5)
for these basis matrices we can deduce that
Tr(ρ) = A0Tr(I2) + ~A · Tr(~σ ) (B.5)==== 2A0
(139)==== 1 ,
[σj ] = Tr(ρσj) = A0Tr(σj) +∑
k
AkTr(σkσj)(B.5)====
∑
k
2Ak δjk = 2Aj (j , k = x, y, z ) .
In terms of the spin operator ~S = ~~σ/2 the density matrix is given by
ρ =1
2
(
I2 + ~P · ~σ)
=1
2
1+Pz Px− iPy
Px+ iPy 1−Pz
, ~P = [~σ ] =2
~[ ~S ] ∈ IR3 . (149)
The density matrix is determined by three (real) parameters Px,y,z that together form the
polarization vector ~P of the ensemble. The fact that ρ has three degrees of freedom is
relatively easy to comprehend. A complex 2×2 matrix that is hermitian and has a unit
trace is fixed by 2× (2× 2)× 12− 1 = 3 independent real parameters.
The physical interpretation of the polarization vector ~P : to this end we make use of the
fact that the eigenvalues λ of the matrix 2ρ should satisfy
(1 + Pz − λ)(1− Pz − λ)− P 2x − P 2
y = (1− λ)2 − ~P 2 = 0 ⇒ λ = 1± |~P | . (150)
56
Hence, in diagonal form the density matrix looks as follows:
ρdiag =1
2
(
1 + |~P | 0
0 1− |~P |
)
, (151)
which follows from equation (149) by choosing the spin quantization axis (z-axis) parallel
to ~P . Next define the spin eigenvectors χ↑ and χ↓ for spin quantization along ~P , so that
(~P · ~σ )χ↑ = |~P |χ↑ and (~P · ~σ )χ↓ = − |~P |χ↓ . Then it follows from equation (147) that
1
2(1 + |~P |) = W↑ =
N↑N↑ +N↓
and1
2(1− |~P |) = W↓ =
N↓N↑ +N↓
⇒ |~P | = W↑ −W↓ =N↑ −N↓N↑ +N↓
∈ [0, 1] , (152)
where N↑ and N↓ count the number of spin-1/2 particles in the spin states χ↑ and χ↓ .
In that way |~P | can be readily interpreted as the degree of polarization of the ensemble
and ~eP= ~P/|~P | as the direction of polarization.
Special cases:
• If the ensemble refers to a pure spin state, then we know because of equation (141)
that ρ = ρ2. From this it follows directly that ρ is now determined by two real
parameters and a sign (as a result of squaring): by means of equation (149) as well
as the relation
(~σ · ~A )(~σ · ~B )(B.6)==== ( ~A · ~B )I2 + i~σ · ( ~A× ~B ) ( ~A, ~B ∈ IR3 ) (153)
the condition ρ = ρ2 amounts to
ρ =1
2
(
I2+ ~P ·~σ)
= ρ2 =1
4
(
I2+ ~P ·~σ)2
=1
2
(1+ ~P 2
2I2 + ~P ·~σ
)
⇒ |~P | = 1 .
In diagonal form the density matrix is given by the projection operator
ρdiagpure =
(
1 0
0 0
)
: 100% polarization with quantization axis parallel to ~P .
(154)
This corresponds to maximal order (read: maximal quantum information), with all
particles polarized in the direction of the polarization vector ~P .
• For 0 < |~P | < 1 the ensemble is partially polarized, represented by the inequality12< Tr(ρ2) = 1
2(1 + ~P 2) < 1.
57
• We speak of an unpolarized ensemble if |~P | = 0. In that case
ρ~P=~0
= 12I2 (155)
and Tr(ρ2) takes on its minimal value Tr(ρ2) = 12. In this situation as many parti-
cles are in the spin “↑” state as in the spin “↓” state. As such we are dealing with
an equal admixture of two totally polarized subensembles, one with spin parallel to
the quantization axis and one with spin antiparallel to the quantization axis. Note,
though, that the direction of this quantization axis can be chosen freely! An unpolar-
ized ensemble is in fact an example of a completely random ensemble with maximal
disorder (see § 2.4).
• An ensemble (beam) with degree of polarization |~P | can be viewed as being com-
posed of a totally polarized part and an unpolarized part. In diagonal form this
reads
ρdiag(151)==== |~P |
(
1 0
0 0
)
+1− |~P |
2
(
1 0
0 1
)
.
Freedom to choose: the density matrix (155) of a completely random ensemble does not
depend on the choice of representation for the considered space, in contrast to the density
matrix (154) for a pure ensemble. Moreover, a given mixed ensemble can be decomposed in
different ways in terms of pure ensembles . For instance, a mixture with 20% of the par-
ticles polarized in the positive x-direction, 20% in the negative x-direction, 30% in the
positive z-direction and 30% in the negative z-direction results in a net unpolarized en-
semble, since 0.2 (ρ~P =~ex
+ ρ~P =−~ex
) + 0.3 (ρ~P =~ez
+ ρ~P =−~ez
) = 12I2 .
2.3 The equation of motion for the density operator
As a next step towards quantum statistics we determine the equation of motion for the
density operator in the Schrodinger picture. Consider to this end a statistical mixture of
pure states that is characterized at t = t0 by the density operator
ρ(t0) =
N∑
α=1
Wα |α(t0)〉〈α(t0)| . (156)
Assume the weights Wα of the statistical mixture not to change over time. Then the
density operator evolves according to
|α(t)〉 = U(t, t0)|α(t0)〉 ⇒ ρ(t) =
N∑
α=1
Wα |α(t)〉〈α(t)| = U(t, t0) ρ(t0)U†(t, t0) . (157)
As shown in the lecture course Quantum Mechanics 2, the evolution operator U(t, t0)
satisfies the differential equation
i~∂
∂tU(t, t0) = H(t)U(t, t0) , (158)
58
where H(t) is the Hamilton operator belonging to the type of system described by the
ensemble. From this we obtain the following equation of motion for the density operator:
i~d
dtρ(t) =
[H(t), ρ(t)
], (159)
which is better known as the Liouville equation. This is the quantum mechanical analogue
of the equation of motion for the phase-space probability density in classical statistical me-
chanics, which can be formulated in terms of Poisson brackets as ∂ρcl/∂t = −ρcl ,Hcl .For this reason the name “density operator” was given to ρ .
Note: ρ(t) does not possess the typical time evolution that we would expect for a quantum
mechanical operator. Since ρ(t) is defined in terms of state functions, it is time indepen-
dent in the Heisenberg picture and time dependent in the Schrodinger picture. This is
precisely the opposite of the behaviour of a normal quantum mechanical operator.
Time evolution of the ensemble average of the dynamical variable A: the en-
semble average [A] as defined in § 2.1 satisfies the evolution equation
d
dt[A ]
(137)====
d
dtTr(ρA) = Tr
(ρ∂A
∂t
)+ Tr
( dρ
dtA)
(159)==== Tr
(ρ∂A
∂t
)− i
~Tr(HρA− ρHA
)= Tr
(ρ∂A
∂t
)− i
~Tr(ρAH − ρHA
)
(137)====
[ ∂A
∂t
]− i
~[AH − HA ] =
d
dt[A ] , (160)
where ∂A/∂t refers to the explicit time dependence of A . In the penultimate step we have
used that the trace is invariant under cyclic permutations, i.e. Tr(ABC) = Tr(CAB) =
Tr(BCA). The evolution equation (160) has the same form as the evolution equation for
the expectation value of a dynamical variable in ordinary QM. However, this time we have
averaged twice in view of the double statistics!
2.4 Quantum mechanical ensembles in thermal equilibrium
As we have seen, there are marked differences between pure ensembles (with
maximal order) and completely random ensembles (with maximale disorder).
We are now going to investigate the generic differences a bit closer.
Let |k〉 be an orthonormal set of eigenstates of ρ corresponding to the eigenvalues ρk .As such, these eigenstates together span the (reduced) D-dimensional space on which the
pure state functions of the subensembles are defined. This dimensionality D indicates the
maximum number of independent quantum states that can be identified in the reduced
59
space on which we have chosen to consider the quantum mechanical systems. For instance,
the density matrix in the spin-1/2 spin space has dimensionality D = 2. With respect to
this basis we have
ρpure =
0. . . Ø
01
0Ø
. . .0
vs ρrandom =1
D
11 Ø
. . .Ø 1
1
.
Examples of these extreme forms of density matrices have been given in equations (154)
and (155) in § 2.2. In case of a pure ensemble the density matrix is simply the projection
matrix on the corresponding pure state vector, which depends crucially on the chosen
representation of the D-dimensional space. In case of a completely random ensemble, the
D orthonormal basis states each receive the same statistical weight 1/D to guarantee
that Tr(ρ) = 1. In that case each state is equally probable and the density matrix is
proportional to the D-dimensional identity matrix, which does not depend at all on the
chosen representation.
In order to quantify the differences we introduce the quantity
σ ≡ −Tr(ρ ln ρ)compl.==== −
D∑
k,k′=1
〈k|ρ|k′〉〈k′| ln ρ |k〉 = −D∑
k=1
ρk ln(ρk) ≡ S
kB
, (161)
where ρk represents the probability to find the system in the pure basis state |k〉 . This
quantity is minimal for pure states and maximal for a completely random ensemble :
σpure = 0 vs σrandom = −D∑
k=1
1
Dln(1/D) = ln(D) . (162)
That σrandom is maximal will be proven in § 2.4.3. In accordance with classical thermo-
dynamics, it can be deduced from σ = S/kB and σrandom = ln(D) that the quantity
S should be interpreted as the quantum mechanical entropy4 and kB as the well-known
Boltzmann constant. You could even say that this definition of the entropy is superior
to the classical one. In classical mechanics there is no such thing as counting states.
At best one could work with phase-space volumes that have to be made dimensionless
4This definition of entropy is also used in information theory in the form of the so-called Shannon
entropy −∑k Pk ln(Pk), which is an inverse measure for the amount of information that is encoded in
the probability distribution P1, · · · , PN . Each type of probability distribution in fact corresponds to a
specific type of ensemble. In the lecture course Statistical Mechanics a relation will be established between
the thermodynamical and quantum mechanical definitions of the entropy of a canonical ensemble.
60
by means of an arbitrary normalization factor. For this reason the classical entropy can
only be defined up to an additive constant. In QM, however, there is a natural unit of
counting/normalization in the form of ~ .
2.4.1 Thermal equilibrium (thermodynamic postulate)
A quantum mechanical ensemble is regarded as being in thermal equilibrium if it effectively
satisfies the equilibrium conditions
∂
∂tH = 0 and
d
dtρ = 0
(159)===⇒
[H, ρ
]= 0 (163)
and if the corresponding constant entropy is maximal. This last condition on the entropy
is the quantum mechanical analogue of the classical second law of thermodynamics, which
states that the entropy of macroscopic systems in non-equilibrium configurations always
increases until equilibrium is reached.
In thermal equilibrium all averaged system quantities [A ] are constant, provided that
∂A/∂t = 0. For example this holds for the average energy of the system, i.e. A = H .
Proof :d
dt[A]
(160), ∂A/∂t=0========= Tr
( dρ
dtA) (163)
==== 0 .
In thermal equilibrium the density operator commutes with the Hamilton operator. There-
fore an orthonormal set of simultaneous eigenfunctions |k〉 of both H and ρ exists:
H |k〉 = Ek |k〉 and ρ |k〉 = ρk |k〉 . (164)
Using this basis we will maximize the entropy in three different scenarios in § 2.4.2 – 2.4.4.
2.4.2 Canonical ensembles (J.W. Gibbs, 1902)
heat bathNR , VR , T
systemN , V, T ∆E
(N,V,T)-ensemble closed bottle in a water container
61
Consider a system with a constant number N of identical particles inside a macroscopic
finite enclosure with fixed volume V . Assume the considered system to be in thermal equi-
librium with a very large heat bath (outside world), with which it is in weak energy contact.
Together the heat bath and the embedded system form a closed system (isolated system).
Weak energy contact: by weak energy contact we mean that the embedded system and
heat bath can exchange energy, since only the total energy of the entire isolated system is
conserved, but that they do so without having a noticeable influence on each other’s energy
spectra. Since the mutual interaction is that weak, both the heat bath and the embedded
system can be in a definite energy eigenstate at every instance of time (separation of
variables). The system and heat bath merely depend on each other through the weak
energy contact, which makes the energy of the system time dependent. This weak energy
contact does have a role to play, since it provides the mechanism by which the system can
reach thermal equilibrium with the heat bath. A weak energy contact can for instance be
achieved by separating the system from the heat bath by means of a macroscopic partition
wall. This allows energy to be transferred by means of collisions with the atoms in the
wall. In that case only a negligible fraction of the enclosed particles will feel the presence
of the partition wall if the average de Broglie wavelength of the particles is much smaller
than the dimensions of the enclosure (see § 2.5). For instance this applies to systems with
sufficiently large (macroscopic) enclosures and/or sufficiently high temperatures.
The heat bath determines the mixture: we will see that in thermal equilibrium
all energy eigenstates of a closed system have equal probability, i.e. a closed
system is described by a completely random ensemble. For a fixed total energy
of the entire closed system the number of energy eigenstates of the heat bath is
largest if the energy of the heat bath is highest and therefore the energy of the
embedded N-particle system is lowest. Subsequently the degrees of freedom of
the large heat bath are integrated out, which causes the embedded N-particle
systems with the lowest energies to automatically have the highest statistical
weights. This gives rise to a so-called canonical ensemble, which is sometimes
referred to as (N , V, T )-ensemble. This is a statistical mixture of N-particle
systems in specific energy eigenstates. As a result of the weak energy contact
with the heat bath, such an N-particle system in thermal equilibrium will not
have a fixed energy value. However, the ensemble average E will remain con-
stant in thermal equilibrium.
The canonical density operator: in order to determine the density operator of a canon-
ical ensemble in thermal equilibrium, we have to maximize the entropy under the additional
constraints Tr(ρ) = 1 and Tr(ρH) = [H ] ≡ E = constant. To this end we employ the
Lagrange-multiplier method (see App.C) by varying with respect to the eigenvalues ρk as
62
well as the Lagrange multipliers, i.e.
0 = δ(
σ−βTr(ρH)−E
−λTr(ρ)−1
) (164)==== δ
(
βE+λ−∑
k
ρkln(ρk)+βEk+λ
)
for all variations δρk , δβ and δλ . Note: there is no need to vary with respect to the
energy eigenvalues Ek , as these energy eigenvalues are independent of the statistical mix-
ture in case of a weak energy contact. From the Lagrange-multiplier method it follows
automatically that the constraint conditions have to be satisfied as well as the condition
∀δρk
∑
k
δρkln(ρk) + 1 + βEk + λ
= 0 ⇒ ρk = exp(−1−λ) exp(−βEk) .
The Lagrange multiplier λ can now be eliminated by means of the constraint condition
Tr(ρ) = exp(−1−λ)∑
k′
exp(−βEk′) = 1 ,
which implies for ρk that
ρk =exp(−βEk)
∑
k′exp(−βEk′)
. (165)
In these sums the summation runs over all fully specified N -particle energy eigenvalues,
including the complete degree of degeneracy of these energy levels. Subsequently we use
the Lagrange multiplier β to define the temperature of the considered ensemble:
T ≡ 1
kBβ. (166)
Out of the complete disorder of the closed system, the contact with the heat bath
has created order in the energy distribution of the embedded N-particle system.
Note: if the N-particle ground state is not degenerate, then the canonical en-
semble will change in the low-temperature limit T → 0 (β →∞) into a pure
ensemble where all N-particle systems will be in the ground state.
Finally we introduce the canonical partition function (normalization factor)
ZN(T ) ≡∑
k′
exp(−βEk′) =∑
k′
〈k′| exp(−βH)|k′〉 = Tr(exp(−βH)
). (167)
In spectral decomposition the density operator of a canonical ensemble then reads
ρ(164)====
∑
k
ρk |k〉〈k|(165)====
∑
k
exp(−βEk)
ZN(T )|k〉〈k| =
1
ZN(T )exp(−βH)
∑
k
|k〉〈k|
compl.===⇒ ρ(H) =
1
ZN(T )exp(−βH) =
exp(−βH)
Tr(exp(−βH)
) . (168)
63
The label N in ZN(T ) refers to the number of particles of the considered type of embedded
system. An arbitrary averaged physical quantity of the embedded system is then given by
the ensemble average
[A] =1
ZN(T )Tr(A exp(−βH)
), (169)
allowing us to derive the average system energy from the partition function:
E = [H ] =Tr(H exp(−βH)
)
ZN(T )=
− ∂ZN(T )/∂β
ZN(T )= − ∂
∂βln(ZN(T )
). (170)
Number of particles and the canonical-ensemble concept: the embedded systems of
a canonical ensemble can just as well consist of a single particle as of a macroscopic number
of particles (such as a gas). A popular way to link up with the classical thermodynamics of
ideal gases is to single out one particle in a larger gas system and consider the corresponding
(single-particle) canonical ensemble. This approach of treating the rest of the gas as part of
the heat bath has its limitations, though. The canonical-ensemble concept implies that the
influence exerted by the particles of the heat bath on the particle of the embedded system
is exclusively limited to a weak energy contact. This obviously breaks down if the particle
density is too high, causing the quantum mechanical overlap between the identical particles
in embedded system and heat bath to become relevant. In that case also the effective
quantum mechanical interaction induced by the (anti)symmetrization procedure should
be taken into account. These quantum mechanical effects will become more pronounced
with increasing particle densities (see § 2.5 – 2.7). In the two examples given below we will
nevertheless consider two canonical ensembles of systems that consist of a single particle.
The reason to do so is to first link up with classical statistical physics before zooming in
on the quantum mechanical aspects. In order to study these many-particle aspects, the
grand-canonical ensemble approach will turn out to be more handy.
Equipartition of energy: as a standard example we consider a canonical ensemble of
systems that each consist of a single free spin-0 particle with mass m that is contained in
a macroscopic enclosure (box). Since quantum mechanical many-particle aspects evidently
do not play a role in a 1-particle system, we expect the average energy per particle in that
case to obey the classical principle of equipartition of energy for an ideal gas. This states
that each kinetic and elastic degree of freedom of the considered type of system (i.e. a po-
sition/momentum degree of freedom that contributes quadratically to the Hamiltonian)
will contribute kBT/2 to the average energy. This is indeed confirmed by the quantum
mechanical calculation (see exercise 13), which tells us that the quantum mechanical defi-
nitions of entropy, temperature and Boltzmann constant are consistent with their classical
thermodynamic counterparts.
64
Example: magnetization. Consider a canonical ensemble of systems that each consist
of a single free electron that is experiencing a constant homogeneous magnetic field in the
z-direction, ~B = B~ez . In spin space this corresponds to an interaction
H spinB
= − ~MS · ~B =2µ
BB
~Sz , (171)
in terms of the Bohr magneton µB. The full Hamilton operator H = H spatial + H spin
Bof
the electron commutes with Sz and therefore the same should hold for ρ(H). Hence, the
density matrix is diagonal in spin space if we use a basis of eigenvectors of Sz to describe
it. For a given spatial energy level the density operator has the following form in spin
space:
ρ spin =1
exp(−βµBB) + exp(βµ
BB)
(
exp(−βµBB) 0
0 exp(βµBB)
)
. (172)
This ensemble corresponds to the following polarization vector:
ρ spin =cosh(βµ
BB) I2 − sinh(βµ
BB) σz
2 cosh(βµBB) =
1
2
(
I2 − tanh(βµBB) σz
)
(149)===⇒ ~P = − tanh(βµ
BB)~ez =
2
~[Sz]~ez . (173)
From this we can derive the magnetization, i.e. the average magnetic moment per electron
in the direction of the magnetic field:
MS
≡ 1
B [ ~MS · ~B ] = − 2µB
~[Sz] = µ
Btanh(βµ
BB) . (174)
If βµBB ≪ 1, for instance when the magnetic field is weak or the temperature high, then
we obtain Curie’s law:
MS
≈ µB(βµ
BB) (166)
====µ2
B
kBTB ⇒ χ
S(T ) ≡
( dMS
dB)
B=0=
µ2B
kBT∝ T−1 , (175)
where χS(T ) is called the paramagnetic susceptibility of the spin ensemble.
2.4.3 Microcanonical ensembles
If we do not impose the constraint Tr(ρH) = E in the previous derivation, then we will
obtain the ensemble with maximum disorder once we maximize the entropy. After all,
there is no heat bath to integrate out and create any order in the ensemble. We speak
in that case of a microcanonical ensemble or (N , V, E)-ensemble, which determines the
thermodynamical properties of a closed system in thermal equilibrium. Because of the
absence of energy contact, the system energy is constant and not just the average value of
65
the system energy. If we take the density matrix ρ to have dimensionality D , then we
find by employing the Lagrange-multiplier method that
δ(
σ − λTr(ρ)− 1
) (164)==== δ
(
λ −D∑
k=1
ρkln(ρk) + λ
)
= 0
for all variations δρk and δλ . From this it follows that Tr(ρ) = 1 and ln(ρk)+1+λ = 0.
By combining the two conditions ρk = exp(−1 − λ) and Tr(ρ) = D exp(−1 − λ) = 1,
we arrive at the typical diagonal form ρk = 1/D as given in § 2.4 for a completely
random ensemble. This also proves the conjecture that the density matrix of a com-
pletely random ensemble gives rise to the absolute maximum of entropy. Note also that
from equation (168) it can be read off that a microcanonical ensemble coincides with the
high-temperature limit T→∞ (β→ 0) of a canonical ensemble .
2.4.4 Grand canonical ensembles (J.W. Gibbs, 1902)
reservoirα,VR , T
systemα ,V, T ∆E , ∆N
(α ,V,T) or (µ,V,T)-ensemble open bottle in a water container
Finally, we consider a quantum mechanical ensemble that satisfies the same criteria as for
a canonical ensemble with the exception that the embedded system is allowed to exchange
both energy and particles with a reservoir . The (open) many-particle systems described
by the ensemble have a variable number of particles. However, the total many-particle
Hamilton operator Htot of such a many-particle system satisfies certain particle-number
conservation laws.5 We know that the density operator has to be a constant of motion
and that any order in the ensemble is imparted by the contact with the reservoir. As
such, we expect the density operator to depend exclusively on observables that belong
to conserved quantities that can be exchanged between reservoir and open system. In the
considered case that would be the total many-particle Hamilton operator and the conserved
5For particles that are not constrained by conservation laws, such as thermal photons, only a canonical-
ensemble approach is formally applicable. This will be explained in chapter 4 during the discussion of
photon-gas systems.
66
combinations of total number operators. Let’s for convenience assume that the open system
consists of one type of particle, i.e. we exclude the possibility of particle mixtures, and that
no reactions can occur that affect the number of particles (such as particle decays). In
that case Htot has to be additive, i.e.[Htot, N
]= 0, to guarantee that N is a constant
of motion. In this way the number of particles plays the same role as energy did in the
canonical case, simply because the open system has no fixed number of particles but the
complete closed system including the reservoir does. In thermal and diffusive equilibrium,
the resulting type of ensemble is called a grand canonical ensemble , (α ,V,T)-ensemble, or
(µ ,V,T)-ensemble.
Apart from the average energy we now also have to impose a constraint on the average
number of particles of the open system while maximizing the entropy. For this purpose
we use the total number operator N that counts the total number of particles in the
open system, which has eigenvalues 0 , 1 , 2 , · · · . Apart from the Hamilton operator also
the density operator is additive, in order to guarantee that the entropy is additive. Hence,
the total number operator commutes with both the Hamilton operator and the density
operator. Bear in mind, though, that these two operators will depend on the precise
number of particles present in the open system . This gives rise to the following set of
mutually commuting observables N , Htot(N) and ρ(N), with corresponding orthonormal
basis |k,N〉 :
N |k,N〉 = N |k,N〉 , Htot(N)|k,N〉 = Ek(N)|k,N〉
and ρ(N)|k,N〉 = ρk(N)|k,N〉 . (176)
The grand canonical density operator: in order to determine the density operator
of a grand canonical ensemble in thermal and diffusive equilibrium we have to maximize
the entropy under the additional constraints Tr(ρ) = 1, [Htot(N)] ≡ Etot = constant
and [N ] ≡ N = constant. By means of the Lagrange-multiplier method we find
0 ==== δ(
σ − α[N ]− N
− β
[Htot(N)]− Etot
− λTr(ρ)− 1
)
(176)==== δ
(
αN + βEtot + λ −∑
N
∑
k
ρk(N)ln(ρk(N)
)+ αN + βEk(N) + λ
)
(177)
for all variations δρk(N) , δα, δβ and δλ . From this it follows automatically that the
constraint conditions have to be satisfied as well as the condition
∀δρk(N)
∑
N
∑
k
δρk(N)ln(ρk(N)
)+ 1 + αN + βEk(N) + λ
= 0 .
67
Combined with the constraint Tr(ρ) = 1 this results in
ρk(N) =exp(−βEk(N)− αN
)
∑
N ′
∑
k′exp(−βEk′(N ′)− αN ′
) . (178)
As in equation (166), the temperature T of the ensemble is defined through T ≡ 1/kBβ .
Finally, we introduce the grand canonical partition function
Z(α, T ) ≡∑
N ′
∑
k′
exp(−βEk′(N
′)− αN ′) (167)====
∑
N ′
exp(−αN ′)ZN′(T ) (179)
and rewrite the grand-canonical density operator as
ρ =1
Z(α, T )exp(−βHtot(N)− αN
)= ρ
(Htot(N), N
). (180)
The quantity exp(−α) can be regarded as the fugacity of the ensemble, which describes
how easy it is to move a particle from the reservoir to the open system:
• if exp(−α) is small, than contributions from small N values dominate and quantum
mechanical many-particle aspects are not important;
• if exp(−α) is not small, than particle exchange is easier and quantum mechanical
many-particle aspects are relevant.
Often the fugacity is written as exp(βµ), with µ = −α/β known as the chemical potential.
The average total energy and the average number of particles of the open system can be
extracted directly from the grand canonical partition function:
Etot = − ∂
∂βln(Z(α, T )
)and N = − ∂
∂αln(Z(α, T )
). (181)
Remark: in scenarios that do either involve chemical reactions, particle decays or rela-
tivistic energies, we have to abandon conservation laws for individual types of particles.
However, in those cases certain combinations of particle numbers often will be conserved.
While maximizing the entropy, these generalized conservation laws will have to be imposed
as constraints on the corresponding ensemble averages. As before this can be implemented
by means of the Lagrange-multiplier method, assigning to each conservation law a spe-
cific Lagrange multiplier and a related chemical potential. For instance, for relativistic
energies particle–antiparticle pair creation can occur, so that only the difference between
the number of particles and antiparticles remains conserved (which is equivalent to charge
conservation). Another example is provided by a statistical mixture of A-, B- and C-
particles for which the reversible reaction A B + C is in chemical equilibrium. In that
case the particle-number combination 2NA+N
B+N
Cis conserved.
68
Possible realization: if we are dealing with a vary large number of particles inside a
macroscopic volume, then the grand canonical ensemble can be realized straightforwardly
by partitioning the macroscopic volume in very many identical cells. The inherent condi-
tion is that the cells themselves are again sufficiently macroscopic to guarantee that the
energy contact with the reservoir (read: the other cells) can be regarded as weak. Repeated
measurements can be performed on such an ensemble by simply measuring locally. The
internal structure of a star can, for instance, be determined along these lines by partition-
ing the star in macroscopic volume elements that are of negligible size compared to the
dimensions of the star itself (see exercise 18). The relevant “measurable” quantities are in
that case the local particle densities and the local total energy density.
2.4.5 Summary
The differences between the three types of ensembles reside in the increasing degree of
order that is imparted on the embedded systems by the contact with the reservoir. In the
microcanonical case this contact is extremely weak. It is that weak that we could effectively
speak of the absence of any contact, causing E and N to be effectively fixed. Thermal
equilibrium is in that case equivalent with the absence of any order. In the canonical case
energy transfer occurs between reservoir and embedded system, with the total energy being
conserved. Since the reservoir has so many more degrees of freedom than the embedded
system, in thermal equilibrium order is generated in the energy mix of the embedded system
with a preference for lower energies (i.e. higher reservoir energies). In the grand canonical
case we add to this the possibility of particle exchange, with the corresponding particle con-
servation law translating into extra order in the particle mix of the embedded open system.
The three different types of ensembles give rise to identical physical results if the relative
fluctuations around the equilibrium values Etot and N are very small, causing the energy
and number of particles of the embedded system to be effectively fixed. It is then a matter
of choice which type of ensemble is more suitable for performing calculations (which in
most cases will be the grand canonical ensemble). For instance this applies to systems
consisting of a very large number of particles, in view of the extra factors 1/√N that in
general occur in the average relative fluctuations on the measurements.
2.5 Fermi gases at T=0
In nature many physical systems can be modelled pretty well by means of an N -parti-
cle system consisting of a large number of non-interacting identical fermions inside a finite
enclosure (box) with fixed volume. If the size of the box and the number of particles N
are sufficiently large, then the precise shape of the box will not matter in the analysis. This
type of many-particle system is called a Fermi gas. Examples are conduction electrons in
a metal (see § 2.5.3), heavy nuclei (see § 2.5.4), compact imploded stars (see § 2.5.5), etc.
69
x
z
y
O
L
L
L
For determining the properties of such a Fermi gas
we consider a large cube with edges L and with the
origin in one of the vertices (see picture). The vol-
ume of the cube is then given by V = L3. As a first
step we assume the cube to contain N free identical
fermions with spin s and mass m. Scenarios where
the particles experience a constant attractive poten-
tial or a constant homogeneous magnetic field are
considered in a next step.
The 1-particle Hamilton operator for a free particle in a box is given by
H =~p 2
2m+ V (~r ) , V (~r ) =
0 inside the box
∞ otherwise. (182)
This (kinetic) 1-particle Hamilton operator does not depend on the spin of the particle.
Therefore, we start by solving the spatial eigenvalue equations and subsequently take into
account the spin degeneracy. The spatial 1-particle energy eigenfunctions have to satisfy
the position-space equation
~p 2
2mψ~ν(~r ) = − ~2
2m~2ψ~ν(~r ) = Eνψ~ν(~r ) ,
with ψ~ν(~r ) = 0 outside and on the edge of the box . (183)
For the cubic box the problem reduces to solving the well-known eigenvalue problem for a
particle in a 1-dimensional infinite potential well (a.k.a. infinite square well). The normal-
ized spatial energy eigenfunctions form standing waves inside the cube (see App.A.1):
ψ~ν(~r ) =( 2
L
)3/2
sin( νxπ
Lx)
sin( νyπ
Ly)
sin( νzπ
Lz)
, (184)
Eν =~2π2
2mL2(ν2x + ν2y + ν2z ) =
~2π2
2mL2~ν 2 ≡ ~2π2
2mL2ν2 (νx,y,z = 1 , 2 , · · ·) .
Since Eν only depends on ν = |~ν | and not on νx,y,z separately, almost all kinetic energy
eigenvalues are degenerate. Adding spin to the game finally leads to
ψ~ν,ms(~r, σ) = ψ~ν(~r )χs,ms(σ) (ms = −s, −s+1 , · · · , s−1 , s) , (185)
where the 1-particle spinvector χs,ms(σ) as usual refers to a spin-s particle with spin
component ms~ along the z-axis. As a result, the degree of degeneracy of the energy
eigenvalues Eν is increased by a factor 2s+1.
70
νz
νy
νx
1
2
3
4
1 2 3 4 51
23
45
0
Next we make use of the fact that the box has macroscopic dimensions. As a result, the
1-particle energy levels are very closely spaced and it makes good sense to switch to a
continuous density of states D(Ekin) that counts the number of 1-particle quantum states
per unit of kinetic energy. In the continuum limit the number of 1-particle quantum states
with kinetic energy on the interval (Ekin , Ekin + dEkin) is simply given by D(Ekin)dEkin .
In order to determine D(Ekin) it will prove handy to introduce the quantized wave vector
~k = π~ν/L (νx,y,z = 1 , 2 , · · ·) , (186)
to determine the corresponding density of states D(k), and finally to insert the expression
Ekin(k) = ~2k2/(2m) for the kinetic energy in terms of the wave vector. As indicated
in the picture given above, a unit cube in ~ν-space contains exactly one spatial quantum
state and therefore 2s+1 fully specified quantum states (including spin). In view of the
positivity conditions νx,y,z > 0, all possible ~ν-values are in one octant of ~ν-space.
νx
νyν = 4
νx
νyν = 11
71
For ν-values that are not too small, the number of states N(k) with wave vector smaller
than k = πν/L is in good approximation related to the corresponding volume in ~ν-space
(as illustrated in the preceding picture for a 2-dimensional lattice):
N(k) =
k∫
0
dk′D(k′) ≈ (2s+ 1)1
8
( 4
3πν3)
(186)====
2s+ 1
6π2V k3 . (187)
By taking a derivative we obtain
D(k) ≈ d
dkN(k) =
2s+ 1
2π2V k2 . (188)
This expression is actually valid for an arbitrary spin-s particle in a box. Only
when we switch to D(Ekin) differences will occur between for instance relativis-
tic and non-relativistic scenarios, since in those cases the dispersion relation
between Ekin and k will be different (see exercise 17).
In the non-relativistic case considered here the dispersion relation reads k =√
2mEkin/~2
and therefore
D(Ekin) =( dk
dEkin
)
D(k) =2s+ 1
4π2
( 2m
~2
)3/2
V E1/2kin . (189)
The approximation that we just used is obviously not correct for very low kinetic 1-particle
energies. For instance, for k < π/L there are no kinetic energy levels possible at all. In
fact, in the low-energy regime the discrete character of the energy levels manifests itself.
This is caused by the fact that the box has finite dimensions that cannot be considered
macroscopic for de Broglie wavelengths λ(k) = 2π/k = O(L). As we will see later, such low
kinetic energies play a relevant role in bosonic gases at very low temperatures (see § 2.7).They do not play an important role in a Fermi gas in view of the small number of different
low-energy states N(k)(187)==== O[V/λ3(k)] (see later).
Geometrical aspects: the number of quantum states per unit of kinetic energy and per
unit of volume of the box is given in the continuum limit by D(Ekin)/V , which does not
depend on the volume! In the continuum limit the precise shape of the box in general
does not matter. If we indicate the surface area of the macroscopic box by S , then the
influence of the edge of the box on D(Ekin) is suppressed by a relative factor S/(kV )
(see exercise 15). This simply underlines the quantum mechanical expectation that the
presence of the edge of the box is felt up to order λ(k) distances away. The corresponding
suppression factor Sλ(k)/V is small if the kinetic energy is sufficiently high, as is true (on
average) for a Fermi gas that has a sufficiently large number of particles.
72
Fermi gas in a constant attractive potential: as we will see in § 2.5.3 and § 2.5.4,in certain physical scenarios the Fermi gas is bound by a constant attractive potential
V = −V0 < 0 inside the box. This has no bearing on the kinetic energy levels nor on the
corresponding energy eigenfunctions that were derived previously. However, the 1-particle
energy levels will shift according to
Eν
V0 6=0−−−−→ Eν − V0 , (190)
where Eν is the kinetic contribution to the total 1-particle energy. By means of the
definition
E ≡ Ekin − V0 ≥ −V0 (191)
the 1-particle density of states of the bound system is expressible in terms of the 1-particle
density of states (189) for free particles inside the box:
DV0(E) = D(Ekin) = D(E + V0) . (192)
Fermi gas in a constant homogeneous magnetic field: assume the Fermi gas to be
subjected to a constant homogeneous magnetic field in the z-direction, ~B = B ~ez . In spin
space this gives rise to an additional interaction
H spinB
= C Sz (C ∈ IR , C ∝ B) . (193)
Since this operator acts exclusively on spin space and the Hamilton operator (182) ex-
clusively on position space, the 1-particle energy eigenfunctions are still given by equa-
tion (185). However, the (2s+1)-fold spin degeneracy of the kinetic 1-particle energy
levels will be lifted according to
Eν
B 6=0−−−−→ Eν +ms~C ≡ Eν,ms , (194)
where Eν is the kinetic contribution to the total 1-particle energy Eν,ms for a particle
with spin component ms~ along the z-axis. For each value of the (magnetic) spin quantum
number ms this is equivalent to a constant potential shift V = ms~C. By defining
Ems ≡ Ekin +ms~C ≥ ms~C , (195)
the density of states for each of the 2s+1 individual spin states is expressible in terms of
the 1-particle density of states (189) for free particles inside the box:
Dms(Ems) =1
2s+ 1D(Ekin) =
1
2s+ 1D(Ems−ms~C ) . (196)
73
2.5.1 Ground state of a Fermi gas
In § 2.4.2 it was shown that a gas at temperature T = 0 will be in the ground state
(completely degenerate gas). For gases consisting of bosons or distinguishable particles
this automatically implies that all particles will populate the lowest 1-particle energy
eigenstate. For a Fermi gas this is not a possibility, bearing in mind that Pauli’s ex-
clusion principle tells us that at most one fermion can occupy a fully specified quantum
state. In that case the N particles will occupy the kinetic 1-particle energy eigenstates
one by one from the bottom up, with the highest occupied level given by the Fermi energy
EF ≡ ~2k2F/(2m). States with Ekin > EF will not be occupied (vacant) for a Fermi gas at
T = 0 and can only by excited by increasing the temperature or by including pair inter-
actions into the analysis. The collective set of states with Ekin ≤ EF form the so-called
Fermi sea. The value of the Fermi energy can be extracted readily from equation (187),
since
N = N(kF ) =
kF∫
0
dk D(k)(187)====
2s+ 1
6π2V k3F (197)
implies that
kF =( 6π2
2s+ 1
)1/3
ρ1/3N and EF =
~2k2F2m
=~2
2m
( 6π2
2s+ 1
)2/3
ρ2/3N (198)
in terms of the particle density ρN = N/V . Since N is large, the precise degree of
occupation of the (degenerate) highest energy level is immaterial. Moreover, the fact
that only a limited amount of particles will occupy the lowest kinetic energy eigenstates
also implies that it is a good approximation to replace the density of states by the con-
tinuous function in equation (189). Note that EF ∝ ρ2/3N /m, which tells us that the
Fermi energy increases with increasing particle density and that it will intrinsically be a
factor 103 smaller for protons and neutrons than for electrons. The total kinetic energy
of the ground state of the Fermi gas amounts to
ET=0kin,tot =
kF∫
0
dk~2k2
2mD(k)
(188)====
2s+ 1
10π2
~2
2mV k5F
(197),(198)=======
3
5NEF . (199)
As a consequence, the average kinetic energy per particle is pretty substantial in spite of
the extremely low temperature:
Ekin =ET=0
kin,tot
N=
3
5EF . (200)
74
Quite often a kind of effective (classical) temperature is assigned to a T = 0 Fermi gas.
This is done by invoking the classical principle of equipartition of energy:
Tcl ≡ ET=0kin,tot
3NkB/2
(199)====
2
5
EF
kB
≡ 2
5TF ∝ ρ
2/3N
m, (201)
where TF is called the Fermi temperature. For a given constant number of particles such
a completely degenerate Fermi gas actually exerts a finite pressure (degeneracy pressure)
P = −(
∂ET=0kin,tot
∂V
)
N
(198),(199)=======
2
3
ET=0kin,tot
V
(201)==== kBρNTcl ∝ ρ
5/3N
m, (202)
which obeys a kind of effective (classical) ideal gas law. The higher the particle density of
the Fermi gas becomes, the more the particles lose their individuality and the higher the
effective temperature will be, making it more difficult to excite the fermions. For finite
temperatures T ≪ TF the Fermi gas will actually behave as if we were dealing with a
completely degenerate Fermi gas (see § 2.7).
Calculational tool: to get a feel for the particle-specific aspects of the relation between
EF , TF and ρN , a few characteristic values are listed in the table below. In this table ρrelN
and T relF mark the particle density and Fermi temperature for which EF = mc2, so that
ρN ≪ ρrelN
and TF ≪ T relF are required for the non-relativistic approach to be valid:
EF = mc2(ρN/ρ
relN
)2/3and TF = T rel
F
(ρN/ρ
relN
)2/3. (203)
particle mc2 in MeV T relF in K ρrel
Nin m−3
electron 0.51100 0.5930× 1010 1.6589× 1036
proton 938.27 1.0888× 1013 1.0269× 1046
neutron 939.57 1.0903× 1013 1.0312× 1046
Ground state of a Fermi gas in a constant attractive potential: assume the Fermi
gas to be bound by a constant attractive potential V = −V0 < 0 inside the box. All quan-
tities related to the kinetic part of the energy remain unaffected, such as EF , ET=0kin,tot , Ekin
and the thermal gas pressure P . However, the total energy of the completely degenerate
Fermi gas will undergo a shift:
ET=0tot =
3
5NEF −NV0 ⇒ Eb ≡ V0 − Ekin = V0 − 3
5EF , (204)
with Eb the average binding energy per particle.
75
2.5.2 Fermi gas with periodic boundary conditions
In view of the applications in chapter 1 and the discussion of relativistic quantum theories,
we make a small excursion to non-interacting periodic systems with periodicity length L.
The previously-used rigid-wall boundary condition will now be replaced by a set of spatial
periodicity conditions for the wave function of the system:
ψ(~r ) = ψ(~r + L~ex) = ψ(~r + L~ey) = ψ(~r + L~ez) . (205)
The spatial 1-particle energy eigenfunctions are now periodic plane waves (see App.A.1):
ψ~k(~r ) =1√L3
exp(i[xkx + yky + zkz ]
)=
1√L3
exp(i~k · ~r ) ,
Ek =~2
2m(k2x + k2y + k2z) =
~2~k 2
2m≡ ~2k2
2m, (206)
in terms of the quantized wave vectors
~k =2π
L~ν , with νx,y,z = 0 ,±1 ,±2 , · · · . (207)
The remaining steps closely follow the previous discussion. Each unit cube in ~k-space
now contains precisely (2s+ 1) (L/2π)3 = (2s+ 1) V/(8π3) fully specified quantum states
(including spin). In the continuum limit the number of states N(k) with wave vector
smaller than k is related to the corresponding volume in ~k-space:
N(k) =
k∫
0
dk′D(k′) ≈ (2s+ 1)V
8π3
( 4
3πk3)
=2s+ 1
6π2V k3 . (208)
This expression is identical to the result (187) for a particle inside a cube with edges L
and rigid walls. As such, also the 1-particle density of states is identical for both types of
systems. The most prominent difference between both types of systems is the absence of
finite-size effects in the periodic case. This difference vanishes in the limit L → ∞ . In
that case we are effectively dealing with a system without explicit periodicity or enclosure.
In such scenarios we usually resort to using a periodic system with periodicity length L
and subsequently take the continuum limit L→ ∞ (see App.A.2).
Ground state: in the ground state a Fermi gas with periodic boundary conditions occu-
pies all 1-particle ~k-states inside a Fermi sphere with radius
kF =( 6π2
2s+ 1
)1/3
ρ1/3N (209)
for which N(kF ) = N . On the Fermi surface of the Fermi sphere the energy and momen-
tum are given by the Fermi energy EF and the Fermi momentum pF :
EF =~2k2F2m
and pF = ~kF . (210)
76
2.5.3 Fermi-gas model for conduction electrons in a metal
As a first example of a Fermi gas we consider conduction electrons in a piece of metal. In a
metal the electrons of the metal atoms can be divided into two categories. The bulk of all
the electrons occupy very narrow (nearly atomic) energy bands, being effectively bound to
the atomic nuclei to form metal ions. On the other hand there are the conduction electrons,
which occupy the partially filled highest energy band (conduction band). In an individual
(separate) metal atom these electrons would be the valence electrons situated in the outer-
most atomic shell. In first approximation these conduction electrons can roam freely inside
the metal lattice (nearly-free electron model). This phenomenon occurs when the average
distance between the metal ions is smaller than the average extension of the atomic orbitals
for these conduction electrons. In this oversimplified, semi-classical picture the conduction
electrons are kind of squeezed out of their atomic confinement by the metal lattice, as
sketched in the picture displayed below where the hatched area represents the conduction
band. For characteristic metal densities of ρNat= 1028–1029metal atoms/m3 the effective
volume per metal atom corresponds to the volume of a sphere of radius rs =(3/4πρNat
)1/3,
which is no more than 2– 6 times the Bohr radius a0 = 0.529× 10−10m.
separate atom
energy levels
V (x)
. . .
. . .
macroscopic chain
energy bands (closely spaced energy levels)
V (x)
“edge” “edge”
In first approximation the conduction electrons form a Fermi gas (electron gas), with the
edge of the metal delimiting the macroscopic box (Sommerfeld, 1928).6 In this very crude
approximation the explicit electron–electron as well as electron–ion interactions are rep-
resented by an averaged constant (attractive) potential V = −V0 < 0.
6As a whole, the combination of metal ions and electron gas is neutral. Such a combination of an
electron gas and a complementary collection of positively charged particles is also referred to as a plasma.
77
V
x0 L
EF conduction band
−V0
metal
As we have seen in § 2.5.1 this adds a constant additive contribution −NV0 to the total
energy of the electron gas. This constant term is not essential for the thermal response of
the system. However, it is instrumental in binding the conduction electrons to the metal.
Inside the metal lattice the conduction electrons can be considered approximately free,
with scattering being suppressed due to the fact that scattering into occupied 1-particle
states is forbidden (Pauli-blocked) by the exclusion principle. This eliminates the large
majority of all electron–electron and electron–ion collisions. Pauli-allowed collisions, with
a sufficiently large energy transfer, actually provides the mechanism of weak energy contact
that allows the gas of conduction electrons to achieve thermodynamical equilibrium inside
the metal. This will play a crucial role in understanding the thermal properties of metals,
as we will see shortly.
The heat capacity of metals at room temperature.
Consider the heat capacity(∂Etot/∂T
)
V,Nof an ideal gas consisting of a constant num-
ber N of conduction electrons inside a fixed volume V . In classical statistical mechanics,
equipartition of energy tells us that each free particle of an ideal gas contributes 3kB/2
to the heat capacity. For N conduction electrons this would amount to a total classical
heat capacity of at least 3NkB/2. This is nowhere near the experimentally measured heat
capacities of metals, which turn out to have much smaller values.
Question: can we understand this based on what we have learned so far?
As mentioned above, in first approximation the conduction electrons form a Fermi gas
with the edge of the metal delimiting the macroscopic box. The particle density of the
conduction-electron gas equals the density of metal atoms multiplied by the number of con-
duction electrons Zv per metal atom. For typical conductors such as copper (Cu) or silver
(Ag) we have Zv = 1 and both densities coincide. A conduction-electron gas at T = 0 has
an average kinetic energy per conduction electron of Ekin = 3EF/5, which translates into a
78
classical gas at temperature Tcl = 2EF/(5kB) = 2TF/5. For instance, silver has a particle
density of ρN = 5.8× 1028 particles/m3 for both metal ions and conduction electrons. By
means of the table below equation (203) we find the Fermi energy to be EF ≈ 5.5 eV,
which corresponds to an effective classical temperature of Tcl ≈ 2.5×104K. By increasing
the thermal energy of the system, conduction electrons can start to occupy excited states
with kinetic 1-particle energy Ekin > EF . At room temperature, i.e. Tk ≈ 300K, the
available thermal energy per conduction electron amounts to 3kBTk/2 ≈ 0.039 eV ≪ EF .
As a consequence, only a small fraction of the conduction electrons is liable to be excited,
i.e. conduction electrons with kinetic energy Ekin = EF −O(kBTk). In the figure displayed
below this is illustrated by plotting the degree of occupation n(Ekin) of the fully specified
quantum states with given kinetic energy Ekin .
Ekin
n(Ekin)
0
0.5
1 T = 0
0<T≪ TF
EF
The distributions for T = 0 and 0 < T ≪ TF only differ marginally. Later on in this
chapter it will be proven that at room temperature only a fraction of order Tk/TF of
the conduction electrons will participate in the heat conductivity of a metal. This heat
conductivity is actually largely determined by the thermal motion of the ions in the metal
lattice (phonons) and not so much by the freely moving collective of conduction electrons!
This is the reason why you will not be burning your fingers on a piece of metal at room
temperature, in spite of the fact that the metal possesses certain characteristic properties
of a classical gas at temperature Tcl = O(104K).
Note: the above-given Fermi-gas model does not apply to all properties of a metal. For
instance, in order to adequately describe the electrical conductivity of metals, the inter-
actions between the conduction electrons and the lattice of metal ions are indispensable.
Among other things, these interactions give rise to the band structure in metals and the
Cooper-pair binding in low-temperature superconductors.
79
2.5.4 Fermi-gas model for heavy nuclei
Hahn –Ravenhall –Hofstadter (1956)
nucleon distribution in arbitrary units
r [fm]
Ca
V
Co
InSb
Au
Bi
Up to now it has only been possible to de-
scribe the collective nuclear forces inside
atomic nuclei by means of simplified mod-
els. These models parametrize our present
experimental and theoretical knowledge.
On the one hand, the radius of the charge
distribution inside the nucleus, i.e. the pro-
ton distribution, can be inferred from scat-
tering experiments and spectroscopy. For
example: the scattering of electrons off nu-
clei and the Rontgen spectra of muonic
atoms. In the latter case an atomic elec-
tron is replaced by a muon, which is 200
times heavier and therefore has 200 times
smaller Bohr orbits. On the other hand, information about the radius of the complete
nucleon distribution, involving both protons and neutrons, can be inferred from the scat-
tering of α-particles (helium nuclei) off nuclei (see figure). In that case a relatively sharp
energy threshold is observed, marking the minimum energy that the α-particles need in or-
der to overcome the nuclear Coulomb-repulsion and become sensitive to the nuclear forces.
By means of these experiments we have learned that the nucleon density is almost constant
inside a heavy nucleus:
ρN ≈ 0.17× 1045 nucleons/m3 ⇒ mass density ≈ 2.8× 1017 kg/m3 . (211)
ρ/ρN
nuclear radius
surface thickness
r [fm]
This density only has a mild dependence
on the type of nucleus and falls off rel-
atively quickly in the vicinity of the nu-
clear surface. Therefore it makes sense to
assign to each nucleus a spherical volume
with effective radius R , which is defined
in such a way that it contains all nucleons
when assuming a constant nucleon den-
sity ρN . This nuclear radius roughly sat-
isfies the relation
R ≈ r0A1/3 , with r0 ≡
( 3
4πρN
)1/3
≈ 1.12× 10−15m = 1.12 fm . (212)
The quantity A is the mass number of the nucleus, which is the sum of the number of
protons Z (charge number) and the number of neutrons A−Z .
80
Thomas – Fermi model: these observations can be combined into a simple Fermi-gas
model (Thomas–Fermi model) for heavy atomic nuclei. In this model the nucleus is com-
prised of two independent Fermi gases of protons and neutrons that are bound to a sphere
of radius R by a constant attractive potential V = −V0 < 0 inside the sphere. This po-
tential well replaces the average short-distance interactions with neighbouring nucleons.
The distribution of the two types of nucleons inside the nucleus are in first approximation
taken to be uniform, i.e. finite-size effects are neglected. At first sight it may seem non-
sensical to “leave out” the actual strong interactions among the nucleons. However, a few
properties of nuclei can be explained by Pauli’s exclusion principle alone. By studying the
Fermi-gas model we want to figure out which properties fall into this category and which
properties require detailed knowledge about the dynamics of the nuclear forces. If we wish,
the model can be refined by additionally taking into account the Coulomb repulsion be-
tween protons as well as an approximation for the nuclear forces between two nucleons.
This obviously makes the model less simple, since the interactions are very sensitive to
the actual distance between the two nucleons that participate in the interaction and there-
fore to their 2-particle spin state. After all, spatial symmetrization/antisymmetrization
decreases/increases the average distance between the particles.
V
r0 R
E(n)F
−V0
neutrons
V
r0 R′
E(p)F
∆VCoul/Z
per protonprotons
including Coulomb repulsion
In the ground state both Fermi gases fill all energy levels up to the Fermi energy, causing the
ground-state energy of a heavy nucleus to simply amount to 35
[ZE
(p)F +(A−Z)E(n)
F
]−AV0
in this very simple model. So, if we do not account for Coulomb repulsion between the
protons the total ground-state energy becomes minimal for Z = A/2 (see exercise 16). The
pure Fermi-gas model therefore predicts ground-state nuclei that consist of equal amounts
of protons and neutrons:
ρ(p)N = ρ
(n)N =
ρN
2
(203),(211)======= 8.25× 10−3 ρrelN . (213)
81
The empirical fact that the ratio Z/(A − Z) decreases for increasing A-values can only
be understood if we do account for the Coulomb repulsion between the protons (see exer-
cise 16), making it energetically favourable to have slightly more neutrons than protons in
the nucleus. Since the proton mass (Mp) and neutron mass (Mn) are almost identical, both
Fermi gases will have roughly the same Fermi energy in the absence of Coulomb repulsion:
EF(198)====
~2
2Mp
(
3π2 ρN
2
)2/3 (203),(213)
−−−−−−→ EF ≈ 38MeV , (214)
with corresponding Fermi temperature TF = EF/kB
(203)==== O(1011K). The average kinetic
energy per nucleon then reads
Ekin =3
5EF ≈ 23MeV (215)
and the average binding energy per nucleon
Eb = V0 − Ekin . (216)
By scattering low-energy neutrons off complex nuclei, the depth of the potential well V0 is
measured to be about 40MeV. The Fermi-gas model therefore predicts an average binding
energy per nucleon of 40MeV− 23MeV = 17MeV.
Eb
[MeV]
A
fusion fission
very stable nuclei
asymmetric interactions
surface effects
The experimental data on the binding energy per nucleon (see figure) follows directly from
the difference between the observed mass of the heavy nucleus and the sum of the rest
masses of the individual nucleons:
AEb =[ZMp + (A− Z)Mn − M
]c2 ,
Mpc2 = 938.27MeV and Mnc
2 = 939.57MeV . (217)
82
The corresponding experimental results for the binding energy per nucleon can be cast into
a rather insightful form, known as the semi-empirical mass formula (Bethe –Weizsacker
formula, 1935):
Eb = aV − aSA−1/3 − aC
Z2
A4/3− aa
( 2Z−AA
)2
+ pair interactions ,
aV = 15.8MeV , aS = 18.3MeV , aC = 0.71MeV and aa = 23.2MeV . (218)
The so-called volume coefficient aV is almost completely explained by the exclusion prin-
ciple. Furthermore, it will be shown in the exercise class that the Fermi-gas model will be
less successful in predicting the asymmetry coefficient aa , which quantifies the influence
of Z 6= A/2 on the binding energy. That should not come as a big surprise as we expect
nucleon-sensitive interactions such as Coulomb and isopsin interactions to play a role in
that quantity. By contrast, the binding-energy effects caused by a uniform charge distri-
bution of protons inside the nucleus are found to agree extremely well with the observed
Coulomb coefficient aC.
Additional remarks: the fact that the total binding energy of a heavy nucleus is approx-
imately proportional to the mass number A is called saturation. In fact the first signs
of saturation already set in for relatively light nuclei with A ≥ 12. The physical picture
behind this is that the nucleons of heavy nuclei only interact with a small, fixed number
of nearest neighbours in view of the short-range nature of the nuclear forces, which fall
off exponentially on characteristic distance scales of roughly 1.4 fm. For a closest packing
of nucleons the number of nearest neighbours indeed equals 12. This is in sharp contrast
with long-range interactions, such as Coulomb interactions, which involve the interactions
between all pairs of particles. In that case we expect the binding energy to be propor-
tional to the number of pairs 12A(A− 1), as can be read off from the Coulomb-repulsion
term ∝ Z2 in the semi-empirical mass formula. This nuclear saturation effect legitimizes
the point of view that each nucleon in the inner parts of a large nucleus experiences an
effective (averaged) nuclear binding potential that does not depend on the total number of
nucleons. The saturation effect also predicts the nuclear surface to have a dual influence
on the binding energy (see the surface term ∝ A−1/3 = r0/R in the semi-empirical mass
formula). First of all, there are finite-size effects that affect the density of states of the
Fermi gas, as worked out in exercise 15(ii). Secondly, the particle density has a sharp
drop-off in the vicinity of the nuclear surface, causing the nucleons near the edge of the
nucleus to have fewer nearest neighbours and consequently less binding energy. The latter
aspect gives rise to surface tension, resembling the kind of surface tension that is observed
for liquid droplets. For this reason the Fermi-gas model including finite-size effects is also
referred to as the liquid-drop model.
83
2.5.5 Stars in the final stage of stellar evolution
A star can be regarded as a many-particle system for which the gravitational pressure
is in equilibrium with the thermal and radiation pressure. During the stellar evolution
various nuclear-fusion stages can be identified, with the fusion fuel being consumed step-
by-step (and shell-by-shell) from the inside out. In the final stage of stellar evolution the
fusion fuel gets exhausted in the stellar core, where lowly reactive nuclei such as 12C- and16O-nuclei have amassed. If the fusion reactions effectively cease in the stellar core, then
so does the radiation pressure. With one component of the compensatory pressure gone,
the star will start to implode in an attempt to find a new equilibrium. While the star
implodes the outer layers of stellar matter are ejected, leaving behind a stellar core with
a thin atmosphere. Depending on its mass, this stellar core can go through different phases.
Phase 1: Fermi-gas model for white dwarfs.
During the implosion the thermal (kinetic) pressure increases in the stellar core, where the
matter is being compressed. Assume, for convenience, that the stellar core is comprised
of one characteristic type of atom with charge number Z and mass number A = 2Z . If
the stellar matter is compressed to the point that the distance between the atomic nuclei
becomes smaller than the Bohr radius aZ = a0/Z of a 1-electron atom consisting of a
nucleus with charge Ze and one electron, then the atoms will be compressed into full
ionization (pressure ionization). In first approximation the electrons that are “squeezed
out of the atoms” will form an electron gas inside a spherical volume.7 For atoms with
A = 12 –16 this phenomenon occurs when the mass density inside the stellar core exceeds
the 107 kg/m3 barrier. A star that reaches a new stable equilibrium during this electron-
gas phase is called a white dwarf.
Let’s assume the electron density of the electron gas to be uniform, which can be replaced
by a more realistic density profile in a refined version of the model (see exercise 18). This
constant density can be translated by means of the plasma relation
ρ(e)N = Zρ
(nuclei)N =
ρM
AMpZ (219)
in terms of the uniform mass density ρM of the stellar core. In this expression AMp is the
mass per characteristic atomic nucleus and Z indicates how many electrons are released
per atom. For typical white-dwarf mass densities ρM = 107–1011 kg/m3, the electron den-
sity is given by ρ(e)N = 1033–1037 electrons/m3 and the corresponding Fermi temperature by
TF = 108–1010K (remembering that A = 2Z ). The typical thermodynamic temperature
T <107K of a white dwarf is (much) smaller than the Fermi temperature of the electron
gas, which therefore effectively will be in the ground state.
7Just as in § 2.5.3 the electrons and ions (nuclei) together form a plasma.
84
r
r+drSince the kinetic energy of the electrons substantially
exceeds the thermal energy of the much heavier nu-
clei, we have to look for a stable equilibrium con-
dition based exclusively on the kinetic energy of the
electron gas at T = 0 and the gravitational potential
of the star. The gravitational potential is obtained
straightforwardly by building up the star in spherical layers
Egrav = −R∫
0
dr(4πr2ρM
)( 4
3πr3ρM
)GN
r= − 16
3π2ρ2MGN
R∫
0
dr r4
= −( 4
3πR3ρM
)2 3GN
5R= − 3GNM
2
5R. (220)
Here M and R are the mass and radius of the imploded star, whereas GN is Newton’s
gravitational constant. The total energy of the star is now approximately given by
ET(199)
=======3
5NEF − 3GNM
2
5R
(198)==== N
3~2
10me
(3π2ρ(e)N )2/3 − 3GNM
2
5R
(219)=======
( M
AMpZ) 3~2
10me
( 3π2ZρM
AMp
)2/3
− 3GNM2
5R
ρM =M/V=======
[3
10
( 9π
4
)2/3 ~2
me
( Z
AMp
)5/3]M5/3
R2− 3GNM
2
5R≡ a
M5/3
R2− 3GNM
2
5R.
The corresponding equilibrium condition reads dET/dR = 0, which yields
− 2aM5/3
R3+
3GNM2
5R2= 0 ⇒ MR3 =
( 10a
3GN
)3 A=2Z==== 7.30× 1050 kgm3 . (221)
In this way we have established a relation between the mass and radius (∼ 104 km) of
a white dwarf. This relation runs counterintuitive to our gut feeling that a larger star
should necessarily be a heavier star. However, this counterintuitive relation is confirmed
by astronomical observations. In fact we are dealing here with a pure quantum mechani-
cal phenomenon caused by degenerate quantum matter that manifests itself in a directly
measurable way in an object of astronomical dimensions!
Question: Is the non-relativistic approach actually viable here?
For a mass density of ρM = 107 kg/m3 we have ρ(e)N = 3 × 1033 electrons/m3 ≪ ρrel
N, as
can be extracted from the table below equation (203). At “low” mass densities the non-
relativistic approach is indeed perfectly acceptable. However, at “high” mass densities
relativistic corrections are unavoidable. For a mass density of ρM = 1011 kg/m3 the other
85
extreme is reached, since ρ(e)N = 3 × 1037 electrons/m3 ≫ ρrelN . In that case the Fermi gas
should be treated (ultra) relativistically. The ultra-relativistic treatment, which implies
neglecting the electron mass with respect to the kinetic energy, gives rise to a completely
different type of equilibrium condition (see exercise 18): stable white dwarfs have an up-
per limit MC
on the mass. This upper limit of roughly 1.4 solar masses is called the
Chandrasekhar limit.
radius versus mass for celestial bodies
Phase 2: Fermi-gas model for neutron stars.
Heavy stars can pass phase 1 without reaching a stable equilibrium. Such a star will
continue to implode. For increasing mass density electrons can be “captured” by the
protons through the inverse β-decay process
e− + nucleus(A,Z) → νe + nucleus(A,Z−1) , (222)
with the approximately massless neutrinos escaping (evaporating) and thereby cooling the
star. The physical mechanism behind this inverse β-decay process looks as follows. On
the right-hand side of the equation the nuclei have less total binding energy in view of
the disturbed balance between the number of protons and neutrons. For sufficiently high
mass densities this energy increase will be compensated by an energy reduction caused
by lowering the number of electrons in the electron gas. At a mass density of roughly
86
ρM = 1015 kg/m3 the kinetic pressure of the neutrons will become more important than
the pressure of the electron gas. At even higher densities in the range of 1017–1018 kg/m3
this will eventually result in a Fermi gas that is predominantly comprised of neutrons
(neutron gas). A star that reaches stability during this neutron-gas phase is therefore
called a neutron star. The main differences with the Fermi gas of phase 1 are summarized
by two simple substitutions: Z → A and me → Mn ≈ Mp . Apart from that every-
thing else stays the same. As a result, the a-parameter in equation (221) will receive
an extra factor (me/Mp)(A/Z)5/3 = O(10−3). This intrinsic factor of 1000 reduction for
the characteristic radii of neutron stars is indeed confirmed by astronomical observations
(R ∼ 10 km). Again the ultra-relativistic treatment gives rise to an upper limit on the
mass of a neutron star. This upper limit has to be taken with a pinch of salt, though. For
mass densities of the order of the nuclear mass density 2.8 × 1017 kg/m3 we also expect
the dynamics of the strong nuclear force to play an important role.
2.6 Systems consisting of non-interacting particles
As a second application of ensembles in thermal equilibrium, we try to derive the statistical
properties of systems consisting of a large number of non-interacting particles. We assume
the corresponding 1-particle energy spectrum to be discrete, as is for instance the case
for systems that are contained inside an enclosure or trapped in a potential well. Three
fundamentally different scenarios can be identified.
(A) The particles are distinguishable, for example by means of their positions. This will
give rise to the so-called Maxwell–Boltzmann statistics.
(B) The particles are indistinguishable identical bosons. This will give rise to the so-called
Bose–Einstein statistics.
(C) The particles are indistinguishable identical fermions. This will give rise to the so-
called Fermi–Dirac statistics.
For deriving the statistical distributions in scenarios (B) and (C) we will use the grand-
canonical ensemble approach of § 2.4.4. The Maxwell–Boltzmann distribution can then be
obtained by taking the classical limit. Since the particles are assumed to be non-interacting,
we choose to work with an occupation-number representation based on a complete set of
1-particle observables that includes the 1-particle Hamilton operator. The total Hamilton
operator and total number operator of the many-particle system have a simple additive
form in this representation:
N =∑
j
nj and Htot(N) =∑
j
Ej nj , (223)
with E1 ≤ E2 ≤ E3 ≤ · · · the ordered fully specified 1-particle energy spectrum. The
87
statistical properties of the ensemble can be specified by the average occupation numbers
nk = [nk] , since
N =∑
j
nj and Etot =∑
j
Ej nj . (224)
For the grand-canonical density operator ρ = exp(−βHtot(N)−αN
)/Z(α, T ) this implies
ρ a†k = exp(−βEk − α) a†k ρ . (225)
Proof: consider the operator O(λ) ≡ exp(λA) a†k exp(−λA), with the observable in the
exponent given by A ≡ −βHtot(N) − αN(223)==== −∑j (βEj + α)nj . With the help of
[A, a†k
] (16)==== − (βEk + α) a†k we can derive for arbitrary real λ that
d
dλO(λ) = exp(λA)
[A, a†k
]exp(−λA) = − (βEk + α)O(λ)
⇒ O(λ) = exp(−λ [βEk + α]
)O(0) : exp(λA) a†k exp(−λA) = exp
(−λ [βEk + α]
)a†k .
The operator identity in equation (225) is then obtained by simply inserting λ = 1 and
subsequently right-multiplying the result by ρ = exp(A)/Z(α, T ).
The desired statistical distribution for identical particles then becomes
nk(17),(137)==== Tr(ρ a†k ak)
(225)==== exp(−βEk − α) Tr(a†k ρ ak) = exp(−βEk − α) Tr(ρ ak a
†k)
(26),(27)==== exp(−βEk − α) Tr
(ρ[1± nk]
) (139)==== (1± nk) exp(−βEk − α)
⇒ nk =1
exp(βEk + α)∓ 1, (226)
where the minus (plus) sign refers to bosons (fermions). In the limit α→ ∞ the identity of
the particles does not matter anymore and the quantum mechanical distributions smoothly
approach the well-known classical Maxwell–Boltzmann distribution nk = exp(−βEk−α).In this asymptotic limit 1-particle quantum effects dominate the statistics as a result of
the suppression factor exp(−αN) in the grand canonical density matrix.
All this can be summarized by the following distribution for the average occupation number
belonging to a fully specified 1-particle energy level with energy E :
n(E) =1
exp(βE + α) + γ, with γMB = 0 , γBE = −1 and γFD = 1 . (227)
Note: the temperature and fugacity parameters β and α are not independent of each
other in view of the conditions (181) for the ensemble averages Etot and N .
88
0
1
2
3
4
5
6
7
8
-2 -1 0 1 2 3
n(E)
βE+α
Bose–Einstein
Maxwell–Boltzmann
Fermi–Dirac
quantum mechanical domain
classical
domain
The quantum mechanical distributions
The characteristic properties of these distributions can be read off directly from the plots
displayed above.
• The three distributions approach each other for large values of βE + α . For instance
this applies to the high-energy limit or to situations for which α is large. In such situ-
ations the identity of the particles does not matter anymore and quantum mechanical
statistics effectively merges with classical Maxwell–Boltzmann statistics.
• As anticipated n(E) ∈ [0, 1] for the Fermi–Dirac distribution.
• For the Bose–Einstein distribution we have to demand that βE + α > 0 for each
energy E, in order to guarantee that n(E) ≥ 0. Hence, α is bounded from below
by α > −βEground , with Eground the 1-particle ground-state energy.
The Fermi energy: in the low-temperature limit T → 0 ( β → ∞ ) the Fermi–Dirac
distribution approaches the distribution
n(E)β→∞−−→
1 if E < EF
0 if E > EF
,
for which all energy states are occupied up to the maximum energy EF = − limβ→∞
α/β .
For a Fermi gas this maximum energy of course coincides with the definition for the Fermi
energy that was introduced in § 2.5.
89
Freeze out of bosonic degrees of freedom: the fact that βE + α ≥ βEground + α > 0
for the Bose–Einstein distribution has as direct consequence that
n(E)
n(Eground)=
exp(βEground + α)− 1
exp(βE + α)− 1≤ exp
(−β [E−Eground]
).
If an energy gap ∆E is present between the 1-particle ground state and the first excited
state, then thermal excitations are heavily suppressed for temperatures kBT ≪ ∆E . Such
a scenario for instance applies to a harmonic oscillator with characteristic frequency ω for
temperatures kBT ≪ ~ω (see exercise 14). In that case we speak of an effective freeze
out of the corresponding bosonic degree of freedom since the excited states containing one
or more bosonic energy quanta ~ω will be scarcely populated. The effective freeze out of
such a degree of freedom goes hand in hand with a reduced thermal response (e.g. heat
conduction) of the system. Applied to complex systems such as molecular gases or crystal
lattices, this phenomenon can occur step-by-step: for decreasing temperature more and
more vibrational modes of the complex system will freeze out.
Bose –Einstein condensation: if βE + α ↓ 0 in the Bose–Einstein distribution, then
the corresponding average occupation number diverges. This sharp rise in the degree of
occupation is unique for the bosonic distribution and is responsible for the physical phe-
nomenon of Bose–Einstein condensation. The fact that bosons have an increased proba-
bility to occupy the same quantum state can result in a noticeable (macroscopic) fraction
of the particles occupying the ground state already at finite temperatures. Whether this
phenomenon actually occurs or not depends on the properties of the system considered.
2.7 Ideal gases in three dimensions
An important example of non-interacting many-particle systems in thermal equilibrium is
provided by the ideal gases. These systems consist of non-interacting particles of mass m
and spin s that are contained inside a macroscopically large enclosure with fixed volume
V . We assume the enclosure to contain a very large, approximately constant number N
of free identical particles. The 1-particle energy spectrum pertaining to such an enclosed
system is derived in § 2.5. In view of the macroscopic volume and the very large number
of particles, we can in most situations8 resort to taking the continuum limit (189) for the
1-particle density of states D(Ekin):
D(Ekin) =
2s+ 1
4π2
( 2m
~2
)3/2
V E1/2kin if Ekin ≥ 0
0 if Ekin < 0
. (228)
8This actually does not hold for ideal gases consisting of identical bosons at very low temperatures (see
the discussion of Bose–Einstein condensation in § 2.7.2).
90
The average number of gas particles with kinetic energy on the interval [Ekin , Ekin+dEkin]
is then given by n(Ekin, T ) dEkin , with
n(Ekin, T ) = n(Ekin)D(Ekin)(227)====
D(Ekin)
exp(βEkin + α) + γ. (229)
The quantity α is determined by the constant particle density ρN = N/V :
ρN =1
V
∞∫
−∞
dEkin n(Ekin, T )(228),(229)=======
2s+ 1
4π2
( 2m
~2
)3/2∞∫
0
dEkinE
1/2kin
exp(βEkin + α) + γ
x=βEkin , (166)=========
2s+ 1
4π2
( 2mkBT
~2
)3/2∞∫
0
dxx1/2
exp(x+ α) + γ, (230)
where the integral is a strictly decreasing function of α . This last observation will play
an important role in the phenomenon of Bose–Einstein condensation discussed in § 2.7.2.By means of the thermal de Broglie wavelength
λT ≡ h√2πmkBT
= characteristic thermal quantum mechanical length scale (231)
equation (230) can be rewritten as
ρNλ3T/(2s+1) =
2√π
∞∫
0
dxx1/2
exp(x+ α) + γ, (232)
which links α directly to the temperature. The size of the right-hand side of this equation
will determine whether the system is in the quantum mechanical or classical regime.
Quantum mechanically speaking we are dealing with a classical gas if the average
distance between the particles satisfies ρ−1/3N ≫ λT , which implies that the
number of particles in the characteristic thermal volume λ3T is very small and
that the individual wave packets of the particles therefore effectively do not
overlap. For ρ−1/3N = O(λT ) the gas is in the quantum mechanical regime.
In the same way we can obtain the average total energy density U = Etot/V of the
ideal gas:
U =1
V
∞∫
−∞
dEkin Ekin n(Ekin, T )(228),(229)=======
2s+ 1
4π2
( 2m
~2
)3/2∞∫
0
dEkinE
3/2kin
exp(βEkin + α) + γ
x=βEkin , (166)=========
2s+ 1
4π2
( 2mkBT
~2
)3/2
kBT
∞∫
0
dxx3/2
exp(x+ α) + γ,
91
which can be rewritten by means of equation (230) as
U = ρNkBT
( ∞∫
0
dxx3/2
exp(x+ α) + γ
)/( ∞∫
0
dxx1/2
exp(x+ α) + γ
)
. (233)
From this we can derive the heat capacity per unit of volume
CV =( ∂U
∂T
)
V,N. (234)
2.7.1 Classical gases
If α≫ 1, then exp(x+ α) ≫ γ and all three types of distributions coincide. In that case
we speak of classical gases. The corresponding condition (232) for α simplifies to
ρNλ3T/(2s+1) ≈ 2√
π
∞∫
0
dx x1/2 exp(−x− α) = exp(−α) (235)
if we employ the integral identity∞∫
0
dxx1/2 exp(−x) = Γ(3/2) = 12
√π . For α≫ 1 the
right-hand side of this equation indeed becomes small, which duly puts the system in the
classical regime. This scenario occurs when the particle density is sufficiently low and/or
the temperature sufficiently high. For instance, this applies to gases at room temperature
and room pressure, for which quantum mechanical effects due to the identity of the particles
indeed do not matter.9 As anticipated from classical kinetic gas theory, classical gases can
be described perfectly well by Maxwell–Boltzmann statistics. The corresponding average
total energy density of equation (233) then simply reads
U ≈ 3
2ρNkBT , (236)
if we employ the integral identity∞∫
0
dxx3/2 exp(−x) = Γ(5/2) = 34
√π . From this follows
the typical heat capacity per unit of volume of a classical gas:
CV =( ∂U
∂T
)
V,N=
3
2
NkB
V, (237)
in full agreement with the principle of equipartition of energy.
9The Fermi gases as described in § 2.5 do not fall into this category in view of the high particle density.
In the classical era one therefore had no reason to interpret such systems as gases.
92
2.7.2 Bose gases and Bose–Einstein condensation (1925)
For ideal gases that consist of identical bosons we know that there is the additional con-
dition α > −βEground = −βEν=√3 ≈ 0 to be satisfied, in order to make sure that
βEkin + α > 0 and therefore guarantee that n(Ekin, T ) ≥ 0. Equation (232) now implies
that
ρNλ3T/(2s+1) = f1/2(α) , f1/2(α) ≡ 2√
π
∞∫
0
dxx1/2
exp(x+ α)− 1(238)
0
1
2
3
4
0 1 2 3 4 5
f1/2(α)
α
2.612
T <T0
T =T0
T >T0 ρNλ3T
(2s+1)
with the Bose gas being in the quan-
tum mechanical regime if f1/2(α) = O(1).
The integral f1/2(α) is strictly decreasing
(see plotted curve) and assumes its max-
imum value when α becomes minimal.
Due to the condition α ≥ 0 this maxi-
mum occurs for α= 0 with corresponding
finite value f1/2(0) = 2.612. Something
special happens as soon as the tempera-
ture is lowered to the critical temperature
T0 for which equation (238) has a solution
(indicated by the intersection with one of
the horizontal dashed curves in the plot)
that coincides with the minimum value of
α and therefore with the maximum value
of f1/2(α):
ρNλ3T0/(2s+1) = f1/2(0)
(231) , f1/2(0)=2.612=========⇒ T0 =
2π~2
mkB
( ρN
2.612 (2s+ 1)
)2/3
. (239)
For T < T0 suddenly no solution to equation (238) can be obtained anymore,
indicating that apparently something has gone wrong!
In order to properly describe the low-temperature behaviour of Bose gases, a very careful
treatment of the low-energy 1-particle states is paramount. After all, compared with
distinguishable particles the indistinguishable identical bosons have an increased tendency
to occupy the 1-particle ground state. From the continuum limit (189) we can read off
that D(0) = 0, whereas in fact 2s+1 different 1-particle states exist that have the lowest
1-particle energy Eground = Eν=√3 ≈ 0. The minimum requirement for correcting the
93
1-particle density of states D(Ekin) thus reads
D(Ekin) → D(Ekin) = D(Ekin) + (2s+1)δ(Ekin) , (240)
so that
ρN =N0
V+ λ−3
T (2s+1)f1/2(α) , N0 = (2s+1)n(0)(227)====
2s+ 1
exp(α)− 1. (241)
In this expression N0 represents the average number of bosons in the 1-particle ground
state. This ground-state occupation number can be neglected with respect to N, unless
α = O(1/N) ≈ 0. As such, the density of particles in the 1-particle ground state starts
to play a prominent role if the temperature is lowered to the critical temperature T0 for
which α ≈ 0. By taking the ratio with respect to ρN , equation (241) can be rewritten in
terms of the critical temperature:
N0
N= 1 − (2s+1)f1/2(α)
ρNλ3T= 1 − f1/2(α)
f1/2(0)
λ3T0
λ3T= 1 − f1/2(α)
f1/2(0)
( T
T0
)3/2
. (242)
The last term on the right-hand side represents the fraction of bosons that occupy excited
1-particle states.
T < T0 : below the critical temperature a noticeable fraction of all bosons should be in
the ground state, since α cannot become smaller than 0 and therefore at most a fraction
(T/T0)3/2 < 1 of all bosons can occupy excited 1-particle states. In that case we say
that the ground state is occupied macroscopically. According to equation (241), getting a
fraction N0/N = O(1) of all bosons to occupy the ground state requires α = O(1/N) ≈ 0.
To very good approximation the fraction of bosons in the ground state is therefore given by
N0
N≈ 1−
( T
T0
)3/2
. (243)
This observable tendency of the identical bosons to occupy the same quantum state is called
Bose–Einstein condensation (BEC). This condensation takes place in momentum space
and should not be confused with the more traditional meaning of the word condensation as
a transition from a gaseous phase to a liquid phase. To increase the amount of condensation
requires lower temperatures or higher densities .10 The latter statement follows from the
fact that according to equation (239) T0 ∝ ρ2/3N . The condensate of bosons in the ground
state forms a degenerate Bose gas, which will not exert pressure nor contribute to the heat
capacity. At low temperatures only the fraction of bosons outside of the condensate will
10Note: it is far from easy to realize Bose–Einstein condensation in the lab. At high densities any
interaction among the bosons tends to completely swamp the symmetrization effects. The density should
therefore be kept low, causing T0 to approach absolute zero (microkelvins or less).
94
0
0.5
1
1.5
2
0 0.5 1 1.5 2 2.5 3
CV
ρNkB
T/T0
∝ (T/T0)3/2
classical
limit
contribute to the pressure and heat capacity of the Bose gas, giving rise to an additional
factor (T/T0)3/2 in the temperature dependence of these quantities (as the plot for the
heat capacity reveals). More details about this are given in the lecture course on Statistical
Mechanics. In § 2.7.4 we will briefly discuss why it took another 70 years to actually realize
the first gaseous Bose–Einstein condensates in the lab.
T > T0 : in that case α has to differ noticeably from 0 to avoid that the ratio N0/N in
equation (242) becomes negative. Automatically only a negligible fraction of the bosons
occupy the ground state.
Occupation of the first excited state for T <T0 : in order to say something sensible
about this we consider the 1-particle energy spectrum (184) for a particle in a cubic box.
For the energy difference between the ground state with ν =√3 and the first excited state
with ν =√6 we can derive that
β (Eex−Eground) =3~2π2
2mL2kBT
V =L3
==== N−2/3 T0T
3h2ρ2/3N
8mkBT0
(231)==== N−2/3 T0
T
3π
4(ρNλ
3T0)2/3
(239)==== N−2/3 T0
T
3π
4
(2.612 (2s+1)
)2/3= O(N−2/3T0/T ) .
The maximum average occupancy of the first excited energy level is obtained for the
minimum value for α , i.e. for α ↓ − βEground :
n(Eex) ≤ 1
exp(β [Eex−Eground]
)− 1
=1
exp(O[N−2/3T0/T ]
)− 1
≤ O(N2/3) ≪ N .
So, solely the 1-particle ground state can achieve macroscopic occupancy in a 3-dimensional
Bose gas! The adjacent first excited energy level can contain many particles, but never-
theless will not be able to reach macroscopic occupancy.
95
The phase transition at T = T0 : as explained on page 92, the quantum mechanical
domain of an ideal gas is reached if ρNλ3T = O(1). In that case the wave packets of the
individual particles will overlap and therefore the quantum mechanical indistinguishability
of the identical particles becomes relevant. In the vicinity of the critical temperature T0
this is indeed the case. The remarkable aspect of T0 is that the gas ondergoes a rapid
transition (phase transition) at that temperature.
For an arbitrary temperature T < T0 an ideal Bose gas is comprised of a fixed,
saturated number of particles in excited states. The distribution of these par-
ticles has a canonical form: it is described by a Bose–Einstein distribution
with α = 0. The remaining particles occupy the ground state, which thereby
achieves macroscopic occupancy. The presence of the condensate is a direct
consequence of the underlying particle conservation law.
Proof: for T < T0 we know that α vanishes, which implies that the excited states are
maximally occupied according to a canonical distribution. From equation (243) we know
that the number of particles that do not occupy the ground state is given by N (T/T0)3/2.
When increasing the number of particles by a factor f while keeping the temperature
constant, we get
N → fN(239)
===⇒ T0 → f 2/3T0 and N (T/T0)3/2 → N (T/T0)
3/2.
The number of particles outside the condensate indeed remains constant. We could actually
have read this off directly from equation (239), since
N (T/T0)3/2 (231),(239)
======= 2.612 (2s+1)V/λ3T
is indeed independent of the particle density and merely counts the number of times the
characteristic thermal volume λ3T fits into the system volume V .
The fact that we have added additional particles merely shows up in the occupancy of
the ground state! One could say that this condensate “does not occupy volume”, as
it does not exert pressure, and therefore can be adjusted freely to account for the cor-
rect total particle density. On this ground we anticipate that the Bose–Einstein distri-
bution will have a O(N) statistical uncertainty in the occupancy of the ground state,
as will be shown in exercise 19. Moreover, the presence of the condensate manifests
itself in long-range correlations (long-range order) in view of the corresponding macro-
scopic de Broglie wavelength. The high-occupancy condensate behaves approximately as
a (pseudo classical) matter wave that can be described quantum mechanically by means
of a complex function (order parameter) ψ0 =√N0 exp(iφ0).
96
Quantum mechanical indistinguishability and Bose–Einstein condensation:
the reason behind the absence of condensation for distinguishable particles resides in the
number of possible ways to excite particles from the ground state. Assume N0 spin-0
particles to occupy the ground state. Then there are 3N0 different first excited states if
the particles are distinguishable and merely 3 if the particles are indistinguishable.
N0 Eground
0 Eex (3×)
ground state
N0−1 Eground
1 Eex (3×)
excited state
For large enough N0 it becomes entropically more favourable for distinguishable particles
to be excited because of the kB ln(3N0) gain in entropy. For indistinguishable particles the
gain in entropy is substantially smaller and independent of N0 , i.e. kB ln(3), which lifts
the entropic obstruction for having a large number of particles in the ground state.
Bose–Einstein condensation versus D(Ekin) for low energies: the previous indis-
tinguishability argument is actually not able to guarantee the occurrence of a condensate.
If the continuous density of states is not small for low-energy 1-particle states, then there
are nevertheless ample of ways to excite a particle with a minimum gain of energy. This
can keep the system from actually achieving a macroscopic occupation of the ground state.
In the above-given derivation for a 3-dimensional Bose gas it was crucial for the occurrence
of Bose–Einstein condensation that the strictly decreasing integral f1/2(α) remained finite
when α reached its minimal value. In exercise 20 it will be investigated how the formation
of a condensate depends on the dimensionality of the system and on the energy–momentum
relation of the particles. The results of this investigation can be summarized as follows:
Bose–Einstein condensation is indeed not possible if D(Ekin) does not vanish
for Ekin = Eground : in that case∫∞−∞ dEkinD(Ekin)/
[exp(βEkin+α)−1
]is not
finite for α ↓ − βEground and the continuous density of states D(Ekin) does not
require being corrected.
The role of the particle conservation law: in the absence of particle conservation
there is no reason for a condensate to form. A photon gas in thermal equilibrium with an
atomic heat bath is an example of this (see § 4.3.2). As a result of being absorbed by
the atoms, the photons actually do not show up as particles in the atomic heat bath and
therefore they exclusively contribute to the thermodynamic balance of energy. In that case
the entropy of the heat bath actually does not depend on the number of photons.
On the other hand, a canonical ensemble with a fixed number of particles is able to give rise
to Bose–Einstein condensation, as is for instance the case for low-energy photons in the
97
curved-mirror microresonator of exercise 21. In that case the fixed number of particles acts
as a boundary condition on the occupation numbers of the 1-particle energy levels, which
resembles the action of keeping the ensemble average N constant in a grand-canonical
ensemble.
2.7.3 Fermi gases at T 6= 0 (no exam material)
For ideal gases that consist of identical fermions there can be no condensation phenomenon,
since maximally 2s + 1 fermions can occupy the ground state as a consequence of the
exclusion principle. Therefore there is no need to correct the continuous density of states
D(Ekin). We simply obtain
ρNλ3T/(2s+1) = g1/2(α) , g1/2(α) ≡ 2√
π
∞∫
0
dxx1/2
exp(x+ α) + 1. (244)
The low-temperature domain (degenerate Fermi gases): for Fermi gases we speak
of low temperatures if ρNλ3T is large, which implies that α has to be strongly negative in
order to satisfy equation (244). In that case the following integral approximation can be
used for a function G(x) that grows slower than exponentially for large x:
∞∫
0
dxG(x)
exp(x+ α) + 1≈(
1 +O[α−2 ])
−α∫
0
dx G(x) , (245)
provided that the integral on the right-hand side exists. This approximation follows from
the fact that the function n(x) ≡ [exp(x+α)+1]−1 effectively behaves like a step function
for large −α , with a deviation that is symmetric around x = −α (zie plotted curve). This
deviation is only significant for a limited range of x-values: x+ α = (E−µ)/kBT = O(1),
which corresponds to kinetic energies that only differ from the Fermi energy by O(kBT ).
x
n(x)
0
0.5
1step function approximation
actual distribution
−α
x+ α = (E−µ)/kBT = O(1)
98
For a degenerate Fermi gas equation (244) leads to the following approximate relation
between α, ρN and T :
ρNλ3T/(2s+1) ≈ 4
3√π(−α)3/2
(
1 +O[α−2 ])
(231)===⇒ −α ≈ ~2
2mkBT
( 6π2ρN
2s+ 1
)2/3 (198)====
EF
kBT
(201)====
TFT
. (246)
This relation is valid up to O(α−2) = O(T 2/T 2F ). In EF and TF we immediately recognize
the Fermi energy and Fermi temperature as derived for T = 0 Fermi gases in equations
(198) and (201). In accordance with the examples discussed in § 2.5 the domain of degen-
erate Fermi gases corresponds to −α ≈ TF/T ≫ 1, which can be reached by lowering the
temperature or increasing the density and (related to that) TF . Only a O(T/TF ) fraction
of the particles will participate in the thermal response of a degenerate Fermi gas. This
can be read off from equation (233) by employing the integral approximation (245):
U ≈ ρNkBT25(−α)5/2
23(−α)3/2
(
1 +O[α−2 ])
(246)====
3
5ρNkBTF
(
1 +O[T 2/T 2F ])
⇒ CV =( ∂U
∂T
)
V,N= O(ρNkBT/TF ) . (247)
More details will be given in the lecture course on Statistical Mechanics. Contrary to Bose
gases, in a degenerate Fermi gas almost all particles will contribute to the kinetic energy
and pressure of the gas, regardless of the temperature.
Chemical potential: in statistical mechanics α is usually rewritten in terms of the so-
called chemical potential
µ ≡ − α
β
(166)==== −αkBT . (248)
For a degenerate Fermi gas we have µ ≈ EF , so that
n(E)(227)====
1
exp(β [E − µ]
)+ 1
≈ 1
exp(β [E −EF ]
)+ 1
. (249)
For a completely degenerate Fermi gas at T = 0 all states will be occupied up to the
Fermi energy:
n(E) =
1 if E < EF
0 if E > EF
,
in accordance with the definition given in § 2.5.
99
Example: Pauli paramagnetism of a degenerate electron gas.
Consider an ideal electron gas with constant volume V and constant number of electrons N .
Assume that T ≪ TF , which implies that the chemical potential µ of the gas satisfies
βµ = µ/kBT ≫ 1. Subject this system to a very weak constant homogeneous magnetic
field in the z-direction, ~B = B~ez . In spin space this gives rise to an extra interaction
H spinB
= − ~MS · ~B = −B ~MS · ~ez =2µ
BB
~Sz (0 < µ
BB ≪ kBT ≪ µ) ,
in terms of the Bohr-magneton µB. As explained on page 73, the magnetically induced
spin interaction leads to a splitting of the kinetic 1-particle energy levels, thereby lifting the
spin degeneracy. The average number of electrons with spin parallel (N+) and antiparallel
(N−) to the magnetic field are given by
N± =
∞∫
−∞
dE±D±(E±)
exp(β [E± − µ ]
)+ 1
(196)====
∞∫
±µBB
dE±
12D(E± ∓ µ
BB)
exp(β [E± − µ ]
)+ 1
,
with E± the 1-electron energy for states with spin component ± ~/2 along the z-axis and
with D(Ekin) the kinetic 1-particle density of states (228) for m = me and s = 1/2. By
means of the change of variable E± = Ekin ± µBB this can be simplified to
N± =V
4π2
( 2me
~2
)3/2∞∫
0
dEkinE
1/2kin
exp(β [Ekin − µ ± µ
BB ])+ 1
, (250)
with Ekin the kinetic 1-electron energy. This allows us to derive the following additive ex-
pressions for the (constant) electron density and the magnetization density, i.e. the average
magnetic moment per unit of volume in the direction of the magnetic field:
ρN =N
V=
N+ + N−V
,
MS
≡ 1
V B [ ~M(1)Stot
· ~B ] = − 2µB
~V[ ~S
(1)tot · ~ez ]
(53)==== − µ
B
V(N+ − N−) . (251)
In the limit µBB ≪ µ this can be simplified by means of the integral approximation (245):
ρN ≈ 1
6π2
( 2me
~2
)3/2(
[µ− µBB ]3/2 + [µ+ µ
BB ]3/2
)
≈ 1
3π2
( 2meµ
~2
)3/2
,
MS
≈ µB
6π2
( 2me
~2
)3/2(
[µ+ µBB ]3/2 − [µ− µ
BB ]3/2
)
≈ 1
6π2
( 2meµ
~2
)3/2( 3µ2BB
µ
)
.
In lowest-order approximation we have µ ≈ EF and
MS
≈ µ2B
3ρN
2EF
B EF = kBTF======⇒ χS(T ) ≡
( dMS
dB)
B=0= µ2
B
3ρN
2kBTF. (252)
100
The paramagnetic susceptibility χS(T ) of the electron gas is constant for T ≪ TF . This
low-temperature phenomenon is called Pauli paramagnetism. It stresses that a degenerate
Fermi gas does not obey Curie’s classical law χS(T ) ∝ T−1 for weak magnetic fields. This
is caused by the effective quantum mechanical interaction induced by the fermionic anti-
symmetrization procedure. In the classical high-temperature limit the magnetic allignment
of the magnetic moments is counteracted by thermal disorder. This leads to a magneti-
zation density that depends on the ratio of the magnetic interaction energy µBB and the
characteristic thermal energy kBT , resulting in a net 1/T temperature dependence of the
paramagnetic susceptibility. At low temperatures only a O(T/TF ) fraction of the electrons
take part in thermal excitations, effectively replacing the 1/T temperature dependence by
a temperature-independent factor 1/TF . The exclusion principle is more effective at coun-
teracting magnetic allignment than thermal disorder, since the characteristic temperature
TF of a degenerate Fermi gas can largely exceed the (ambient) temperature T .
2.7.4 Experimental realization of Bose–Einstein condensates
Superfluid helium: 4He-atoms consist of two protons, two neutrons and two electrons,
which implies that the corresponding total spin is integer. The 4He-atoms can thus be
regarded as bosons. They satisfy Bose–Einstein statistics and can therefore give rise to the
formation of a Bose–Einstein condensate at low temperatures. As argued in § 1.6.4 –1.6.6,this condensate plays an essential role in the phenomenon of superfluidity in liquid 4He
at temperatures below 2.2K. The rare 3He-isotope consists of two protons, one neutron
and two electrons, which implies that the corresponding total spin is half integer. The3He-atoms can thus be regarded as fermions. They satisfy Fermi–Dirac statistics and can
therefore not give rise to a similar type of condensate as 4He-atoms. Pairs of 3He-atoms,
however, can behave bosonically. As a result of pairing effects, also 3He can display su-
perfluidity, but only at temperatures below 2mK.
Ultracold dilute gases: for long the existence of superfluidity was seen as an indica-
tion that it should be possible to realize a gaseous Bose–Einstein condensate in the lab.
However, that was all that could be said, since the observed superfluidity involved flu-
ids rather than an ideal or weakly-interacting gas. It took another 70 years to realize a
Bose–Einstein condensate in a nearly ideal gas. The main problem was to refrigerate the
gas without creating a liquid or solid and without the atoms binding into molecules. To
achieve this dilute, neutral gases had to be used without contact with any walls in order
to avoid freezing. This presented two new challenges:
• a Bose–Einstein condensate in a low-density gas requires very low temperatures,
since T0 ∝ ρ2/3N ;
• the gas has to be refrigerated without physical contact with the outside world.
101
By means of two advanced cooling techniques, i.e. laser cooling and evaporative cooling
(see the lecture course Quantum Mechanics 2 for more details), Eric Cornell and Carl
Wieman were finally able to produce the first gaseous Bose–Einstein condensate in 1995,
consisting of about a thousand rubidium-87 atoms at a temperature of 170 nK. In the same
year a similar feat was realized for ultracold dilute gases of lithium-7 atoms (Randy Hulet)
and sodium-23 atoms (Wolfgang Ketterle). In the latter case, the number of condensate
particles had already increased a thousandfold. In 2001 Cornell, Wieman and Ketterle
were awarded the Nobel prize for this achievement.
Cornell & Wieman, JILA, 1995: transition from a “normal” gas to a condensate.
Velocity distribution of the gas particles for T > T0 , T < T0 and T ≪ T0 .
R.G. Hulet et al., Rice University, 2005
Application: isotope effects for lithium.
In this experiment ultracold gases were used
that consisted of fermionic 6Li-atoms and
bosonic 7Li-atoms. At very low tempera-
tures we expect that a magnetically trapped6Li-gas has a larger spatial extent than a7Li-gas for the same number of gas atoms,
since in the fermionic case the average ki-
netic energy is larger. As can be observed
in the figure on the right, this is indeed con-
firmed experimentally.
102
3 Relativistic 1-particle quantum mechanics
In this chapter we will attempt to cast non-relativistic quantum mechanics into
a relativistically viable form. To phrase it differently, we will try to find a
relativistic analogue of the Schrodinger equation.
Similar material can be found in Schwabl (Ch. 5–7 and Ch. 10,11), Merzbacher
(Ch. 24) and Bransden & Joachain (Ch. 15).
The desired equation should have exactly the same form for any inertial observer (relativity
principle) and should have solutions that respect the quantum mechanical postulates. As a
starting point for the construction of the relativistic wave equation we will use the standard
connection between plane waves of the type exp(i~k · ~r − iωt) and free particles with
energy E = ~ω and momentum ~p = ~~k (de Broglie hypothesis). The non-relativistic
QM followed from (slightly) modifying the classical Hamilton–Jacobi equation to obtain a
wave equation, which allowed to model particle–wave duality by describing the particles
as a superposition of plane waves. The correct energy and momentum of the particles
could be extracted from the plane waves by means of the position-space operators
E = ~ω → E = i~∂
∂t, ~p = ~~k → ~p = − i~ ~ , ~r → ~r = ~r .
Applied to the classical expression E = Hcl(~r, ~p ) = ~p 2/(2m) + V (~r ) for the energy of a
particle with mass m in a potential field V , the Schrodinger equation
i~∂
∂tψ(~r, t) = Hcl(~r, ~p )ψ(~r, t) =
(
− ~2
2m~2
+ V (~r ))
ψ(~r, t)
is obtained. However, in this equation ~r and t are not treated on equal footing, invalidating
the relativity principle. The plan is now to recast this idea into a relativistic form in flat
spacetime. To this end we will adopt the following plan of attack to construct a relativistic
QM for free particles.
(1) Particle–wave duality: construct a wave equation that incorporates an operator
version of the correct relativistic relation between energy and momentum. This
will imply that any solution to the wave equation should also satisfy the Klein–
Gordon equation, providing a clear link to plane-wave decompositions (see § 3.1)!
(2) Relativity principle and spin: make sure that the wave equation satisfies the rela-
tivity principle. By specifying this for spatial rotations, we will be able to determine
the spin of the particles belonging to the wave equation!
(3) QM: extract a continuity equation from the wave equation and verify whether this
allows a probability interpretation that conforms with the superposition principle.
(4) Check the non-relativistic limit and compare with experimental observations.
103
3.1 First attempt: the Klein–Gordon equation (1926)
First read appendix D, where the conventions and definitions of special relativity are sum-
marized. The classical energy of a free relativistic particle with rest mass m is given by
E =√
m2c4 + ~p 2c2 . (253)
As a first ansatz for a relativistic wave equation in position space, we could start from
this classical expression and simply substitute E → E = i~ ∂∂t
and ~p → ~p = − i~ ~ in
analogy with the non-relativistic case:
i~∂
∂tψ(x) =
√
m2c4 − ~2c2 ~2ψ(x) (x = position 4-vector) . (254)
The “square-root operator” has to be defined by means of a power series (see the lecture
course Quantum Mechanics 2), which effectively results in a differential operator of infi-
nite order. This hugely complicates the discussion. More importantly, in this equation
~r and t are treated asymmetrically, making it not directly suitable as frame-independent
expression . . . however, these drawbacks would be lifted if we would decide to apply the
differential operators twice on the left- and right-hand side of the equation.
The wave equation: as a first serious attempt it therefore makes sense to start from the
squared expression E2 = m2c4+~p 2c2 . Upon performing the substitution we obtain a wave
equation that does treat ~r and t on equal footing: the so-called Klein–Gordon equation
− ~2 ∂
2
∂t2ψ(x) =
(
m2c4 − ~2c2 ~
2 )
ψ(x) , (255)
which is a second order differential equation in time. In contrast to the non-relativistic
case, this implies that two boundary conditions are needed to fully determine the time
evolution of the wave function. It should be noted, though, that by squaring the energy–
momentum relation we have introduced a sign ambiguity for the energy. We will see that
this aspect will come back to haunt us while trying to interpret the wave function.
Implementation of the relativity principle: to guarantee that the form of the Klein–
Gordon equation is the same for each inertial observer, we first introduce a manifest co-
variant notation in terms of covariant and contravariant vectors:
(
~2c2∂µ∂
µ +m2c4)
ψ(x) = 0 ⇒(
+m2c2
~2
)
ψ(x) = 0 , (256)
in terms of the d’Alembertian
≡ ∂µ∂µ (D.11)
====1
c2∂2
∂t2− ~2
. (257)
104
1) Generic approach: as this will be a recurring theme in the subsequent discussions,
we first try to formulate the generic procedure to implement the relativity principle. To
this end we build in the possibility that the wave function ψ(x) has several components
(intrinsic degrees of freedom):
ψ(x) =
ψ1(x)...
ψN(x)
. (258)
The transformation characteristic of this (multi-component) wave function under Poincare
transformations is defined as
ψ′(x′) ≡ M(Λ)ψ(x) , (259)
where the matrix M(Λ) denotes how the Poincare transformations act on the space
spanned by the intrinsic degrees of freedom. As indicated earlier, this transformation
characteristic should follow from the demand of having the same wave equation for each
inertial observer:
if Dψ(x) = 0 for the linear differential operator D and wave function ψ(x) in
inertial system S, then it should follow automatically that also D′ψ′(x′) = 0
for the transformed differential operator D′ and wave function ψ′(x′) in inertial
system S ′.
The linear differential operator D is invariant under translations by a constant 4-vector aµ
since ∂/∂xµ = ∂/∂ (xµ + aµ). Therefore M(Λ) is independent of a as well as the coordi-
nates x and x′. Furthermore, M(Λ) is a linear operator (matrix) in the space of intrinsic
degrees of freedom, because the superposition principle should hold in both S and S ′.
Additionally we require the matrices M(Λ) to reflect the properties (group structure) of
the Lorentz transformations:
M(Λ1Λ2) = M(Λ1)M(Λ2) and M(Λ−11 ) = M−1(Λ1) (260)
for arbitrary Lorentz transformations Λ1 and Λ2 . This guarantees that in the absence of
a Lorentz transformation also nothing happens in the space of intrinsic degrees of freedom.
This fixes the normalization of the matrices M(Λ) in such a way that a (projective) matrix
representation of the Lorentz group is obtained.
2) Relativity principle and spin for the Klein–Gordon equation: next we specif-
ically consider the Klein–Gordon equation. We know that the d’Alembertian and speed
of light are scalar quantities, which do not change under the transition from one inertial
frame to the other. Consequently also the Klein–Gordon operator is invariant under a
change of inertial system:
D′KG =
(
′ +
m2c ′ 2
~2
)
=(
+m2c2
~2
)
= DKG .
105
This differential operator acts on each component of ψ(x) separately without mixing the
components, implying that each component should separately satisfy the Klein–Gordon
equation. As such, the transformation matrix M(Λ) in the Klein–Gordon theory is propor-
tional to the identity matrix. In order to guarantee the validity of the relativity principle it
hence suffices to demand that ψ(x) is a scalar field11, i.e. ψ′(x′) = ψ(x) = ψ(Λ−1[x′−a]
).
The Poincare transformations therefore act exclusively on x-space, leaving the intrinsic de-
grees of freedom of ψ(x) untouched. The particles described by ψ(x) are spin-0 particles,
as a spatial rotation merely affects the (spacetime) argument of the wave function without
mixing the intrinsic components. The generator of infinitesimal spatial rotations is in this
case exclusively given by the orbital angular momentum operator (see the lecture course
Quantum Mechanics 2).
The non-relativistic limit: before investigating the probabilistic interpretation of the
wave function, we first check whether the relativistic theory reproduces non-relativistic QM
at low velocities. To extract a non-relativistic wave function from the solution ψ(x) to the
Klein–Gordon equation, the rest-mass term has to be removed from the time evolution:
ψ(x) ≡ ψNR(x) exp(−imc2t/~) .
After multiplication by exp(imc2t/~), the Klein–Gordon equation (255) yields
(
− ~2 ∂
2
∂t2+ 2i~mc2
∂
∂t+ m2c4
)
ψNR(x) =(
m2c4 − ~2c2 ~
2)
ψNR(x)
⇒ i~∂
∂tψNR(x) − ~
2
2mc2∂2
∂t2ψNR(x) = − ~
2
2m~2ψNR(x) .
The wave function ψNR(x) oscillates (in time) due to the non-relativistic kinetic energy
ENR = E −mc2 ≪ mc2, so that
∣∣∣
~2
2mc2∂2
∂t2ψNR(x)
∣∣∣ ≪
∣∣∣~
∂
∂tψNR(x)
∣∣∣ .
In the non-relativistic limit the Klein–Gordon equation for ψ(x) changes into a Schrodinger
equation for ψNR(x).
Problems with the Klein–Gordon equation: last but not least we have a closer look
at the quantum mechanical interpretation of the solutions to the Klein–Gordon equation.
For those solutions we will try to find a continuity equation of the form ∂ρ/∂t+~·~j = 0, in
analogy with what can be done in the non-relativistic case for solutions to the Schrodinger
equation. This continuity equation will be interpreted as a manifestation of conservation
11A second possible solution is a so-called pseudoscalar field, for which ψ′(x′) = det(Λ)ψ(x).
106
of probability, with ρ being the quantum mechanical probability density and ~j the quan-
tum flux. For solutions to the Klein–Gordon equation we can derive in a trivial way that
0(256)==== ψ∗(x)ψ(x)− ψ(x)ψ∗(x)
(257)==== ∂µ
[ψ∗(x)∂µψ(x)− ψ(x)∂µψ∗(x)
]
(D.11)===⇒ ∂
∂tρ(x) + ~ ·~j(x) = 0 , (261)
with
ρ(x) ≡ i~
2mc2
[
ψ∗(x)∂
∂tψ(x)− ψ(x)
∂
∂tψ∗(x)
]
,
~j(x) ≡ − i~
2m
[
ψ∗(x)~ψ(x)− ψ(x)~ψ∗(x)]
. (262)
The prefactor of ρ(x) is chosen here in such a way that in the non-relativistic limit we
have ρ(x) ≈ |ψNR(x)|2 . For a proper quantum mechanical interpretation we would like to
interpret ρ(x) as relativistic probability density. To check this, we consider the plane-wave
solutions to the Klein–Gordon equation (256):
ψp(x) = exp(−ip ·x/~) , with p ·p = p2 = m2c2 ⇒ p0 =E
c= ±
√
m2c2 + ~p 2 .
(263)
E
0
mc2
−mc2
Based on these plane-wave solutions we find that
ρ(x) is not positive definite if p0 < 0 (see exer-
cise 22). As such, ρ(x) is not suitable as proba-
bility density. The next logical step is to simply re-
gard all negative-energy solutions as being unphysi-
cal and remove them from the theory. This approach
turns out to be not viable, since it will unavoidably
clash with the fundamental superposition principle
(see exercise 22). Apart from the problems with the
probabilistic interpretation, the negative-energy so-
lutions will result in a free-particle energy spectrum
that is unbounded from below (as sketched in the
figure on the right). This highly unstable scenario
would imply that an infinite amount of energy could
be extracted from the system if an external perturba-
tion would induce a transition from a positive-energy
eigenstate to a negative-energy eigenstate.
107
Two strategies present themselves at this point. On the one hand we could try to inter-
pret the negative-energy states in a different way. We will come back to this option in
chapter 5, when trying to link the Klein–Gordon theory to the quantum theory of the
electromagnetic field that is formulated in chapter 4. On the other hand we could follow
the rather logical strategy of trying to construct an alternative relativistic wave equation
that explicitly does not involve squaring the energy–momentum relation. After all, the
sign ambiguity for the energy was caused by this squaring action. In § 3.2 we will address
this latter (historical) approach, resulting in an entirely different wave equation that does
allow for a viable probability interpretation.
Static, radially symmetric solutions to the Klein–Gordon equation:
to round things off, we briefly discuss the physical applicability of the Klein–Gordon equa-
tion. To this end we consider time-independent (static) solutions φ(r) with radial sym-
metry. In that case the Klein–Gordon equation simplifies to
(~2
− m2c2
~2
)
φ(r) =1
r
( d2
dr2− m2c2
~2
)(rφ(r)
)= 0 .
This differential equation has exponential solutions, with the physically interesting branch
consisting of exponentially decreasing functions of the form
φYu(r) = Cexp(−r/a)
r, with a ≡ ~
mc> 0 . (264)
Solutions of this type are used to describe interactions that are transmitted by massive in-
termediary bosons, such as the weak nuclear forces (transmitted by the W and Z bosons)
and the interactions among nucleons (transmitted by pions), see the lecture course “Struc-
tuur der Materie: Subatomaire Fysica”. In the latter case φYu(r) represents an attractive
interaction potential of the Yukawa-type (i.e. with C < 0), where the constant length scale
a = ~/mc is a measure for the effective range of the potential.
If the mass would have been absent (i.e. for m = 0) we would have been dealing with the
well-known Laplace equation
~2φ(r) =
1
r
d2
dr2(rφ(r)
)= 0 ,
with solutions
φ(r) =A
r+ B .
For B=0 such a solution can be used in the description of infinite-range interactions that
are transmitted by massless intermediary bosons, such as the electromagnetic Coulomb-
interactions (transmitted by photons) and the strong nuclear forces (transmitted by gluons).
108
3.2 Second attempt: the Dirac equation (1928)
Based on the previous observations we now use the non-relativistic expression
i~∂
∂tψ(x) = Hψ(x) (265)
as starting point, with H independent of ∂/∂t. In this way the problematic differential
operator − ~2∂2/∂t2 is absent. The essential difference with the usual non-relativistic
Schrodinger equation resides in the requirement that space and time coordinates are to
be treated on equal footing. As a result, H should now be taken linear in ~ . Intrinsic
degrees of freedom are accounted for by writing the wave function as a column vector with
N components:
ψ(x) =
ψ1(x)...
ψN(x)
. (266)
For a free particle H will not depend on t or ~r . The simplest form of the Hamilton
operator then amounts to
H = c ~α · ~p + βmc2 = c3∑
j=1
αj pj + βmc2 . (267)
The operators ~α and β are independent of t, ~r , E , ~p and act as N×N matrices on
the N components of ψ(x), bearing in mind that H should be a linear operator in order
to guarantee the validity of the superposition principle. In this way we have obtained the
so-called Dirac equation for a free particle with rest mass m:
(
i~∂
∂t+ i~c ~α · ~ − βmc2
)
ψ(x) = 0 . (268)
In the Dirac equation the correct relativistic relation between energy and mo-
mentum should be contained. Or to phrase it differently, the matrices ~α and β
should be chosen in such a way that each component of a solution to the Dirac
equation should automatically be a solution to the Klein–Gordon equation.
This gives rise to the following set of matrix identities:
β2 = IN ,αj , αk
= 2δjk IN and
αj , β
= 0 (j , k = 1 , 2 , 3) . (269)
From the anticommutation relationsαj , β
= 0 we can readily deduce that ~α and β
should indeed be matrices. The symbol IN is used to denote an identity matrix of rank N .
109
Proof: 0(268)====
(
i~∂
∂t− i~c ~α · ~ + βmc2
)(
i~∂
∂t+ i~c ~α · ~ − βmc2
)
ψ(x)
=(
− ~2 ∂
2
∂t2+
~2c2
2
3∑
j,k=1
αj, αk
jk + i~mc33∑
j=1
αj, β
j − β2m2c4)
ψ(x) .
This yields the Klein–Gordon equation (255) if the matrix relations listed above are sat-
isfied. In order to guarantee conservation of probability, we additionally have to impose
that H = H†, which translates into the matrices ~α and β having to be hermitian:
αj = (αj)† and β = β† . (270)
Armed with this set of conditions we now try to find explicit solutions for ~α and β with
the lowest possible rank N .
1. The matrices ~α and β have eigenvalues ± 1.
Proof: the eigenvalues of ~α and β are real, since the matrices are hermitian. More-
over, from the matrix relations (269) it follows that β2 = (αj)2 = IN , causing the
square of the eigenvalues to be 1.
2. The matrices ~α and β are traceless.
Proof: from the matrix relations in equation (269) it can for instance be inferred that
Tr(αj) = Tr(β2αj)cyclic==== Tr(βαjβ)
αj ,β=0======= −Tr(αjβ2) = −Tr(αj) = 0.
3. Since the trace equals the sum of the eigenvalues, the results of steps 1 and 2 can
be combined into the requirement that the matrices ~α and β should have equal
numbers of +1 and −1 eigenvalues. The rank N thus has to be even.
4. Based on steps 1 to 3, the lowest possible value for the rank is N = 2. In that case we
are dealing with 2×2 matrices, spanned by the identity matrix I2 and the three Pauli
spin matrices ~σ . However, we are looking for four different, mutually anticommut-
ing matrices ~α and β . This is impossible for a representation of the Dirac equation
with N = 2 , since the identity matrix commutes with all 2×2 matrices.
From this we can draw the conclusion that the representation of the Dirac equation with
the lowest rank involves N = 4. The corresponding wave functions with four intrinsic
degrees of freedom are called Dirac spinors:
ψ(x) =
ψ1(x)...
ψ4(x)
. (271)
In fact there are infinitely many, physically equivalent representations for the matrices ~α
and β . After all, matrix relations are invariant under arbitrary unitary transformations.
110
In the remainder of this lecture course we will adopt the so-called
Dirac representation : β =
(
I2 Ø
Ø − I2
)
, ~α =
(
Ø ~σ
~σ Ø
)
, (272)
where the 4×4 matrices have been subdivided into 2×2 blocks. As will be proven in the
exercise class, these matrices indeed satisfy the matrix relations (269) and (270). The
corresponding proof will be based on the fundamental properties of the Pauli spin matrices
σ1 = σx , σ2 = σy and σ3 = σz . In this notation the commutator algebra for the Pauli
spin matrices reads[σj , σk
]= 2i
3∑
l=1
ǫjkl σl , (273)
in terms of the completely antisymmetric coefficient
ǫjkl =
+1 if (j, k, l) is an even permutation of (1, 2, 3)
−1 if (j, k, l) is an odd permutation of (1, 2, 3)
0 else
. (274)
3.2.1 The probability interpretation of the Dirac equation
As the Dirac equation is linear in both the spatial gradient and the time derivative, the
probability interpretation of the Dirac equation can be obtained in a straightforward way.
Define the hermitian conjugate row vector
ψ†(x) ≡(ψ∗1(x), · · · , ψ∗
4(x)). (275)
Then
∂
∂t
[ψ†(x)ψ(x)
]= ψ†(x)
∂ψ(x)
∂t+∂ψ†(x)
∂tψ(x)
(268)==== − c ψ†(x)
(~α · ~ +
imc
~β)ψ(x)
− c(~ψ†(x)
)· ~α†ψ(x) +
imc2
~ψ†(x)β†ψ(x)
(270)==== − ~ ·
[ψ†(x)c~α ψ(x)
],
from which the following continuity equation can be extracted:
∂
∂tρ(x) + ~ ·~j(x) = 0 , (276)
with
ρ(x) ≡ ψ†(x)ψ(x) =4∑
i=1
|ψi(x)|2 and ~j(x) ≡ ψ†(x)c~α ψ(x) . (277)
This looks really promising, since this time ρ(x) is guaranteed to be positive definite and
therefore allows an interpretation as probability density. In that interpretation the vector ~j
will play the role of quantum flux, which would imply that the matrix c~α can be interpreted
as a kind of velocity operator in Dirac spinor space. For a viable probability interpretation
it was essential that the matrices ~α and β (and as such also H) are hermitian.
111
3.2.2 Covariant formulation of the Dirac equation
We need one more ingredient before being able to use the Dirac equation as a relativistic
wave equation. We have to demand that ψ(x) transforms in such a way under Poincare
transformations that the Dirac equation has the same form for any inertial observer. As a
first preparatory step we rewrite the Dirac equation in a form that facilitates the transition
to other inertial frames. To this end we combine the four matrices β and ~α in the 4-vector-
like object
γµ : γ0 ≡ β and
γ1
γ2
γ3
≡ ~γ ≡ β ~α (278)
⇒ Dirac representation : γ0 =
(
I2 Ø
Ø − I2
)
, ~γ =
(
Ø ~σ
−~σ Ø
)
.
These are the so-called γ-matrices of Dirac. This notation is slightly misleading in the
sense that it gives the impression that γµ is a contravariant vector quantity. For that
to be true an appropriate contraction in spinor space is required, which also includes the
Dirac spinors in the vector quantity. The γ-matrices have the following properties (which
are derived in the exercise class):
γµ, γν
= 2gµν I4 (Clifford algebra) , (γµ)† = γ0γµγ0 =
γ0 if µ = 0
− γj if µ = j
⇒ Tr(γµ) = 0 , (γ0)2 = I4 , (γj)2 = − I4 (j = 1 , 2 , 3) . (279)
The Dirac equation can now be rewritten in a covariant form:
β
c
(
i~∂
∂t+ i~c ~α · ~ − βmc2
)
ψ(x) = 0(269),(D.11)======⇒
(i~γµ∂µ−mc
)ψ(x) = 0 . (280)
Expressions of the type γµaµ are represented in Feynman slash notation as
a/ ≡ γµaµ = γµaµ = γ0a0 − ~γ · ~a exercise 23
======⇒ a/ 2 = a2I4 . (281)
In this notation the Dirac equation obtains its final compact form
(i~ ∂/−mc
)ψ(x) = 0 . (282)
In exercise 23 it will eventually be shown that the Klein–Gordon operator (+m2c2/~2 )
can be derived straightforwardly from the Dirac operator (i~∂/−mc).
112
Basis of 4×4 matrices: as a last preparation we introduce an explicit basis of 4×4 ma-
trices:
I4 , γµ , σµν ≡ i
2
[γµ, γν
], γµγ5 , γ5 ≡ iγ0γ1γ2γ3 , (283)
that has the following form in the Dirac representation (for j , k = 1 , 2 , 3):
I4 =
(
I2 Ø
Ø I2
)
, γ0 =
(
I2 Ø
Ø − I2
)
, γj =
(
Ø σj
−σj Ø
)
,
σjk =
(
− i2
[σj, σk
]Ø
Ø − i2
[σj , σk
]
)
=
3∑
l=1
ǫjkl
(
σl Ø
Ø σl
)
, σ0j =
(
Ø iσj
iσj Ø
)
,
γ5 =
(
Ø I2
I2 Ø
)
, γ0γ5 =
(
Ø I2
− I2 Ø
)
, γjγ5 =
(
σj Ø
Ø −σj
)
. (284)
The reason why these matrices form a basis of 4×4 matrices is a trivial consequence of
the fact that each 2×2 subblock has the matrices I2 and ~σ as basis. Note that the five
groups of matrices in (283) contain an increasing number of γ-matrices, ranging from zero
to four.
3.2.3 Dirac spinors and Poincare transformations
The transformation characteristic of the Dirac spinors under Poincare transformations is
defined as follows [see equation (259)]:
ψ′(x′) ≡ S(Λ)ψ(x) , (285)
where S(Λ) denotes how the Poincare transformations act on the space spanned by the
intrinsic degrees of freedom (i.e. spinor space). This transformation characteristic should
follow from the demand that the Dirac equation has the same form for each inertial ob-
server: if (i~∂/−mc)ψ(x) = 0 in inertial system S, then it should follow automatically that
also (i~∂/ ′−mc)ψ′(x′) = 0 in inertial system S ′. The Dirac operator (i~∂/−mc) is invari-
ant under constant translations, since ∂/∂xµ = ∂/∂ (xµ+ aµ) for a constant 4-vector aµ.
Therefore S(Λ) is independent of the coordinates x and x′. Furthermore, S(Λ) is a
linear operator (matrix) in spinor space, because the superposition principle should hold
in both S and S ′.
If the Dirac equation (i~∂/−mc)ψ(x) = 0 should automatically imply that
(i~∂/ ′−mc
)ψ′(x′)
(285),(D.11)=======
(
i~ γµ∂ν (Λ−1)νµ −mc
)
S(Λ)ψ(x)
Dirac eqn.======= i~
(
γµ (Λ−1)νµS(Λ)− S(Λ)γν)
∂νψ(x) = 0 ,
113
then S(Λ) should satisfy
S−1(Λ)γµ (Λ−1)νµS(Λ) = γν∗Λρ
ν===⇒ S−1(Λ)γρS(Λ) = Λρνγ
ν . (286)
Next we try to find a matrix S(Λ) for each Lorentz transformation Λ such that
S(Λ1Λ2) = S(Λ1)S(Λ2) and S(Λ−11 ) = S−1(Λ1) (287)
for arbitrary Lorentz transformations Λ1,2 . In the absence of a Lorentz transformation
accordingly nothing happens in spinor space. This fixes the normalization of the matrices
S(Λ) in such a way that a projective matrix representation of the Lorentz group is obtained.
(I) Rotations and boosts: we start with the proper orthochronous Lorentz transforma-
tions. Since these Lorentz transformations (and corresponding spinor transformations) can
be connected continuously with the unit transformation, equation (286) only needs to be
solved for infinitesimal Lorentz transformations:
Λµν = gµν + δωµν , δωµν = − δωνµ ∈ IR infinitesimal . (288)
The reason why the tensor δωµν should be antisymmetric follows directly from the expres-
sion for the inverse of an infinitesimal Lorentz transformation:
(Λ−1)µν(288)==== gµν − δωµν (D.8)
==== Λνµ (288)==== gµν + δωνµ .
Writing the corresponding transformation matrix S(Λ) in infinitesimal form:
S(Λ) ≈ I4 + s(δω) ⇒ S−1(Λ) ≈ I4 − s(δω)S−1(Λ)=S(Λ−1)========= I4 + s(−δω) , (289)
we can derive from equation (286) that
(I4 − s(δω)
)γρ(I4 + s(δω)
)≈ γρ + δωρ
νγν first order
======⇒ ∀ρ
[γρ, s(δω)
]= δωρ
νγν .
(290)
The transformation matrix S(Λ) can now be decomposed in terms of the basis (283)
of 4×4 matrices. In exercise 24 it will be proven that all requirements are fulfilled if
s(δω) = − i4δωµν σ
µν . For an infinitesimal Lorentz transformation the corresponding spinor
transformation is then given by
S(Λ) ≈ I4 − i
4δωµνσ
µν , δωµν = − δωνµ ∈ IR infinitesimal (291)
⇒ the matrices σµν generate the Lorentz transformations in Dirac spinor space .
For a finite Lorentz transformation of this type we get
S(Λ) = limN→∞
(
I4 − i
4
ωµν
Nσµν)N
= exp(
− i
4ωµν σ
µν)
,
where the finite real coefficients ωµν are determined by the Lorentz transformation.
114
The spin operator and intrinsic spin: under a spatial rotation by an angle α about
the axis oriented along the unit vector ~en , the Dirac spinors transform as (see exercise 25):
ψ′(x′) = exp(−iα~en · ~S/~
)ψ(x) , ~S =
~
2
(
~σ Ø
Ø ~σ
)
. (292)
The hermitian operator ~S can be interpreted as the Dirac spin operator, since it generates
the rotations in the space spanned by the four intrinsic degrees of freedom of the Dirac
spinors. Because this spin operator satisfies
∀ψ
~S 2ψ(x) =~2
4
(
~σ 2 Ø
Ø ~σ 2
)
ψ(x)(B.4)====
3
4~2ψ(x) , (293)
we are led to conclude that the Dirac equation is a relativistic wave equation for spin-1/2
particles. In chapter 5 it will be explained why the number of degrees of freedom is twice as
large as expected from the non-relativistic case, where the spin space for spin-1/2 particles
is 2-dimensional.
Lorentz boosts, without proof (no exam material): consider an inertial system S ′ moving
at constant velocity v in the ~en-direction with respect to inertial system S. In spinor space
the corresponding Lorentz boost gives rise to the transformation
S(Λ) = cosh(η/2) I4 − sinh(η/2)
(
Ø ~σ · ~en~σ · ~en Ø
)
with η ≡ 1
2ln
(1 + v/c
1− v/c
)
.
(294)
This transformation is not unitary, since S†(Λ) = S(Λ) 6= S−1(Λ). This implies that
the normalization of the wave function in a given spacetime point is not conserved. This
change in normalization is needed to compensate for the Lorentz contraction caused by
the boost (see § 3.2.4).
(II) Spatial inversion and time reversal: the transformation characteristic for the
discrete homogeneous Lorentz transformations can be derived straight from equation (286)
by again decomposing S(Λ) in terms of the basis (283). For the parity transformation it
can be deduced that (see extra exercise)
ψ′(x′) = S(ΛP)ψ(x) , S(ΛP) = exp(iϕP)γ0 (ϕP ∈ IR) . (295)
Similarly we find for the time reversal transformation that
ψ′(x′) = S(ΛT)ψ∗(x) , S(ΛT) = exp(iϕT)σ
13 (ϕT ∈ IR) . (296)
(III) Spacetime translations: finally we consider constant spacetime translations of the
origin, i.e. x′µ = xµ+aµ with aµ a constant 4-vector. Since the Dirac operator (i~∂/−mc)is invariant under such translations, there is no need for a compensatory transformation
in spinor space.
115
3.2.4 Solutions to the Dirac equation
The Dirac equation for a free particle involves a time-independent Hamilton operator
that does not depend on ~x, i.e.[H, ~p
]= 0. As such, solutions can be found that are
simultaneous eigenfunctions of H and ~p :
ψp(x) = u(~p ) exp(−ip · x/~) , (297)
where the eigenvalues E of E = i~ ∂/∂t and ~p of ~p = − i~~ are combined into the
contravariant momentum vector pµ with p0 = E/c. These stationary plane-wave solutions
to the Dirac equation have to satisfy the additional equation
(i~∂/ −mc
)ψp(x) = 0
(297)===⇒ ( p/−mc)u(~p ) = 0 . (298)
Solutions in the rest system (zero-momentum solutions): we start by trying to
find the solutions in the so-called rest system, for which ~p = ~0 . In that case u(~0 ) should
satisfy the eigenvalue equation (Eγ0−mc2)u(~0 ) = 0, where the matrix γ0 =
(
I2 ØØ − I2
)
has eigenvalues ± 1. The desired plane-wave solutions thus read
E = E+ = mc2 : eigenvectors u(1)(~0 ) =
1000
, u(2)(~0 ) =
0100
,
(299)
E = E− = −mc2 : eigenvectors u(3)(~0 ) =
0010
, u(4)(~0 ) =
0001
.
Negative energies and the Dirac sea: we observe again that negative-energy states
occur. This time this is caused by the fact that H is linear in ~p rather than quadratic.
The Dirac spinors seem to describe two types of spin-1/2 particles simultaneously, one
type having positive energies and one having negative energies. Again the free-particle
energy spectrum is unbounded from below. However, since we are dealing with fermions
here, induced transitions between positive- and negative-energy states can be avoided
(Pauli-blocked). To this end Dirac assumed that all 1-fermion negative-energy states are
occupied in the vacuum state of the universe. In this way the vacuum consists of an infinite
sea of particles with negative energies (Dirac sea) and no particles with positive energies.
The exclusion principle then forbids the unwanted transitions to these occupied negative-
energy states. In § 3.2.6 we will come back to the implications of this specific assumption
and we will also have a critical look at potential problems linked to the existence of such
a Dirac sea.
116
Finite-momentum solutions: for completeness we have a brief look at how the plane-
wave solutions change for finite momenta. These solutions can be obtained by transforming
u(~0 ) to the correct inertial frame by means of an appropriate Lorentz boost S(Λ). How-
ever, it is much simpler to make use of the fact that any solution to the Dirac equation
is automatically a solution to the Klein–Gordon equation, which causes the desired plane-
wave solutions to satisfy the condition p2 = m2c2. The corresponding eigenvectors can thus
be obtained straight from the solutions (299) in the rest system:
u(r)(~p ) = N (p/ +mc)u(r)(~0 ) , E =
E+ =√
m2c4 + ~p 2c2 for r = 1, 2
E− = −√
m2c4 + ~p 2c2 for r = 3, 4,
since (p/−mc)u(r)(~p ) = N (p2 −m2c2) u(r)(~0 ) = 0 for p2 = m2c2.
While determining the normalization factor N we should properly take into account that
the normalization of ψ changes under Lorentz boosts. This is a direct consequence of
conservation of probability: the probability to observe the system as being localized in
a spatial volume d~x around ~x in the rest frame S should be identical to the proba-
bility to observe the system as being localized in the transformed spatial volume d~x ′
around ~x ′ in the boosted inertial frame S ′. Phrased differently, we have to demand that
|ψ(x)|2 d~x = |ψ′(x′)|2 d~x ′, with d~x ′ = (mc2/E+) d~x = d~x√
1− ~v 2/c2 as a result of the
Lorentz contraction in the direction of the boost. Consequently |ψ(x)|2 should transform
as the time component of a contravariant vector field, as expected from the fact that
|ψ(x)|2 = j0(x)/c. This translates into the normalization condition
u(r)(~p )† u(s)(~p ) ≡ E+
mc2δrs ⇒ |N | =
1√
2m(E+ +mc2). (300)
Example: consider ~p = p~ez . Then p/+mc = p0γ0 − pγ3 +mc I , so that
u(1)(p~ez) =N
c
E+ +mc2
0cp
0
, u(2)(p~ez) =N
c
0E+ +mc2
0− cp
,
u(3)(p~ez) =N
c
− cp
0E+ +mc2
0
, u(4)(p~ez) =N
c
0cp
0E+ +mc2
.
117
Remark: the twofold degeneracy of the energy eigenvalues for a given momentum eigen-
value ~p can be lifted by also specifying an appropriate spin quantum number. Such an
appropriate spin quantum number is given by the spin in the direction of motion (helicity),
with corresponding operator S(~p ) = ~S · ~ep (see exercise 26). In the example discussed
above, u(r)(p~ez) has helicity ~/2 if r = 1 , 3 and − ~/2 if r = 2 , 4. The fact that
H , ~p and S(~p ) are mutually commuting operators follows from a symmetry argument
that is valid for arbitrary relativistic free-particle theories. First of all, a free-particle the-
ory is invariant under translations, since empty space is completely isotropic. This implies
that H and ~p commute. The same in principle holds for rotational invariance. But
mind you, ~p actually introduces an explicit direction. As a result, rotational symmetry is
restricted to rotations about this direction, and H and ~p exclusively commute with the
angular momentum component ~p · ~J = ~p · ~S .
3.2.5 The success story of the Dirac theory
At this point we should have a closer look at the implications of the Dirac theory. We start
with its successes. To this end we assume that we are dealing with spin-1/2 particles with
charge q and rest mass m. Subsequently we consider the interaction with classical electro-
magnetic fields described by a real vector potential ~A(x) and a real scalar potential φ(x).
The associated interaction can be obtained in the usual way through minimal substitution:
E → E − qφ(x) and ~p → ~p − q ~A(x) . (301)
In terms of the electromagnetic 4-vector potential
Aµ(x) : A0(x) ≡ φ(x)/c and
A1(x)A2(x)A3(x)
≡ ~A(x) (302)
minimal substitution can be written in covariant form:
pµ → pµ − qAµ(x)position repr.
=========⇒ i~∂µ → i~∂µ − qAµ(x) . (303)
In this way we obtain the Dirac equation for spin-1/2 particles with charge q and rest
mass m that are subjected to a classical electromagnetic field:
(i~∂/ −mc− qA/(x)
)ψ(x) = 0 . (304)
To see what has changed with respect to the non-relativistic case, three special scenarios
will be considered of which only the first one will be worked out in full detail.
118
(I) Non-relativistic limit: intrinsic magnetic dipole moment.
As a first example we consider the interaction with static electromagnetic fields, such as
constant magnetic fields or the Coulomb potential generated by an atomic nucleus. A static
electromagnetic field is described classically by a time-independent vector potential ~A(~x )
and a time-independent scalar potential φ(~x ). The corresponding Hamilton operator
Hposition repr.========= c~α ·
[− i~~− q ~A(~r )
]+ qφ(~r ) + βmc2 (305)
does not depend on t ≡ x0/c but it does depend on ~r ≡ ~x . Therefore it makes sense to
look for stationary solutions of the type
ψ(~r, t) = ψ(~r ) exp(−iEt/~) , H ψ(~r, t) = i~∂
∂tψ(~r, t) = Eψ(~r, t) . (306)
Next we select a positive-energy solution, we write E = ENR +mc2 in order to split off
the rest-mass term, and we decompose the Dirac spinor in terms of 2-dimensional spinors
ψU
and ψD:
ψ(~r ) ≡(
ψU(~r )
ψD(~r )
)
. (307)
From the eigenvalue equation Hψ(~r, t) = Eψ(~r, t) we obtain in the Dirac representation
[see equation (272)]
(ENR +mc2)ψU
= (qφ+mc2)ψU+ c~σ · (− i~~− q ~A )ψ
D,
(ENR +mc2)ψD
= c~σ · (− i~~− q ~A )ψU+ (qφ−mc2)ψ
D. (308)
The mc2 terms drop out from the first equation, but not from the second equation. As a
result we obtain
ψD
=c~σ · (− i~~− q ~A )ψ
U
ENR + 2mc2 − qφ
NR limit======⇒ ψ
D≈ ~σ ·
( − i~~− q ~A
2mc
)
ψU, (309)
bearing in mind that in the non-relativistic limit the total and potential non-relativistic
energies are substantially smaller than the rest-mass energy, i.e. |ENR| , |qφ| ≪ mc2.
This means that the 2-dimensional (“small”) component ψD
is suppressed by a factor
O(p/mc) = O(v/c) with respect to the other 2-dimensional (“large”) component ψU.
Inserting the limit (309) in the first equation of (308) yields
ENRψU= qφ ψ
U+
1
2m
[
~σ · (− i~~− q ~A )]2
ψU
=
(
qφ +1
2m(− i~~− q ~A )2 +
i
2m~σ ·[
(− i~~− q ~A )× (− i~~− q ~A )])
ψU,
119
which gives rise to the so-called Pauli equation for the 2-dimensional spinor ψU:
ENRψU=[
qφ +1
2m(− i~~− q ~A )2 − q~
2m~σ · ~B
]
ψU. (310)
To arrive at this expression we made use of the operator identity (B.6) and the relation
(− i~~− q ~A )× (− i~~− q ~A ) = i~q (~× ~A ) = i~q ~B .
The first two terms on the right-hand side of the Pauli equation we recognize as the
interaction between a charged spin-0 particle and a classical electromagnetic field. The
last term is one of the big successes of the Dirac theory. The Dirac theory predicts the
experimentally observed interaction between the magnetic field ~B and the spin-induced
intrinsic magnetic dipole moment ~MS :
H spinB
= − ~MS · ~B = − q~
2m~σ · ~B ≡ |e|~
4mg ~σ · ~B . (311)
Consequence: an elementary particle with charge q = −|e| that is described by the Dirac
theory (such as an electron or muon) automatically has a lowest-order g-factor of 2. This
g-factor agrees with experimental observations and could not be explained in the non-
relativistic theory! It should be noted though that non-elementary particles (such as
antiprotons), which can be considered as consisting of even smaller particles, in general do
not require g = 2 at lowest order.
(II) Relativistic corrections: fine structure for central potentials.
Consider an electrostatic central potential, for which ~A(~r ) = ~0 and qφ(~r ) = V (r). By
taking the afore-mentioned non-relativistic approximation one step further, we can obtain
the first-order relativistic corrections to the Pauli equation. This results in three relativistic
corrections to the non-relativistic Hamilton operator (without proof):
∆HFS = − ~p 4
8m3c2+
~2
8m2c2V2(~r ) +
~
4m2c2V3(r) ~L · ~σ ,
with V2(~r ) ≡ ~2V (r) and V3(r) ≡ 1
r
dV (r)
dr, (312)
which coincides with the three perturbative corrections that are for instance responsible for
the fine structure of 1-electron atoms (see the lecture course Quantum Mechanics 1). The
first term is a relativistic correction to the kinetic energy. The second term is a relativistic
correction to the potential energy (called Darwin term). The last term is the well-known
spin–orbit interaction, which describes the interaction between the spin-induced intrinsic
magnetic dipole moment of the spin-1/2 particle and the magnetic field that is caused by
the orbital motion (orbital angular momentum) of the particle.
120
(III) Discrete energy levels of the relativistic 1-electron atom.
Again consider the Hamilton operator for an electrostatic central potential
H = c~α · ~p + βmc2 + V (r) . (313)
In the non-relativistic case the operators H , ~L 2, Lz , ~S2, Sz and P formed a complete set
of commuting observables. However, ~L and ~S no longer individually commute with the
Hamilton operator due to the presence of relativistic spin interactions in the Dirac equation
(such as the spin–orbit interaction). The total angular momentum operator ~J = ~L+ ~S
does commute with the Hamilton operator, since V (r) respects the rotational symmetry
of empty space. The energy levels of the relativistic 1-electron atom can therefore be
specified in terms of the quantum number j belonging to ~J 2, which is typical for the rel-
ativistic QM. The orbital angular momentum quantum number ℓ is not a “good quantum
number” anymore, but the total angular momentum quantum number j is. In the absence
of a preferred direction, the energy levels will be independent of the magnetic quantum
number mj belonging to Jz . Note also that the non-relativistic parity operator P no
longer commutes with the Hamilton operator, since H is linear in ~p andP, ~p
= 0.
The relativistic parity operator β P does commute with the Hamilton operator becauseβ, ~α
= 0, with β representing the action of the parity transformation in spinor space
[see S(ΛP ) in equation (295)].
For a 1-electron atom with V (r) = −Zα~c/r the energy spectrum consists of two contin-
uous branches with |E | ≥ mec2 and a discrete set of positive-energy levels corresponding
to bound states (see § 15.5 of Bransden & Joachain for more details):
Enj = mec2
1 +(Zα)2
(
n−j−1/2 +√
(j+1/2)2 − (Zα)2)2
−1/2
,
with n = 1 , 2 , · · · and j = 1/2 , 3/2 , · · · , n−1/2 . (314)
These energy levels are (2j+1)-fold degenerate as a result of the magnetic quantum number
mj = −j , · · · , j . Moreover, each level has an additional twofold degeneracy due to the
parity quantum number (−1)ℓ with ℓ = j ± 1/2, with the exception of j = n−1/2 for
which only ℓ = j − 1/2 = n − 1 is allowed. This so-called fine structure is confirmed
experimentally at high precision. In the non-relativistic limit one can expand in powers of
v/c = O(Zα/n):
Enj −mec2 ≈ − 1
2mec
2 (Zα)2
n2
[
1 +(Zα)2
n2
( n
j + 1/2− 3
4
)
+ · · ·]
, (315)
which agrees with the result derived in the lecture course Quantum Mechanics 1.
121
3.2.6 Problems with the 1-particle Dirac theory
As a relativistic wave equation, the Dirac equation gives rise to a few aspects that are
tough to interpret. As we have seen, there is the problem with the negative-energy states
and the ensuing energy spectrum that is unbounded from below. Transitions from positive-
to negative-energy states can be avoided by invoking the exclusion principle and assuming
that all negative-energy states are occupied. This is rather unsatisfactory from a 1-particle
perspective, since the relevant number of particles is in fact infinite and we actually should
take into account the interactions among these particles. On top of that, an infinite neg-
ative vacuum energy may not pose a problem at the level of quantum mechanics, but it
may have cosmological implications.
Question: When do we expect to see observable manifestations of the presence of the Dirac
sea, which would force us to let go of the Dirac theory as a 1-particle theory?
In order to excite a particle from the Dirac sea a minimum 2mc2 of energy is required. The
excitation would leave behind a hole in the sea and the corresponding negative-energy parti-
cle would be promoted to a positive-energy particle. This pair creation of a particle and a
hole is the relativistic analogue of the electron–hole excitation in hole theory (see § 1.7.1).
A quantum mechanical limitation to the 1-particle theory: consider a particle with
positive energy that is localized in a box of size L. According to Heisenberg’s uncertainty
principle, the quantum mechanical uncertainty ∆p on the momentum of the particle is
in that case at least O(~/L). The uncertainty on the relativistic energy of the particle
would roughly amount to c∆p = O(~c/L). If this uncertainty exceeds the 2mc2 bar-
rier pair creation can occur and the Dirac sea will manifest itself. Hence, if the particle
is localized in a box of size L = O(~/mc), which is referred to as the Compton wavelength
of the particle, we are in fact not dealing with a 1-particle theory anymore.
Quantum mechanics thus involves two characteristic length scales for a parti-
cle with mass m and momentum p: on O(~/p) length scales the quantum-
mechanical wave-like nature of the particle manifests itself and on even smaller
length scales of O(~/mc) we have to let go of the concept of an individual
particle.
In the latter case it makes no sense anymore to talk about the concept of conservation
of probability. After all, additional particles can be created. We will therefore have to
assign a different meaning to the conservation law that was responsible for the continuity
equation (276). Note: in the non-relativistic limit many-particle aspects do not play a
prominent role yet. After all, for non-relativistic momenta |~p | ≪ mc also |∆~p | ≪ mc
and therefore the uncertainty on the position of the particle is at least |∆~r | ≫ O(~/mc),
which largely exceeds the Compton wavelength.
122
4 Quantization of the electromagnetic field
We are in a bit of a pickle now. We have no clue how to interpret the successes and
shortcomings of the relativistic 1-particle quantum theories. To be able to make any
conclusive statements, we turn our attention now to a theory for which we already know
the correct relativistic wave equation . . . electromagnetism.
In this chapter the electromagnetic field will be quantized by applying the usual
canonical quantization conditions to the classical electromagnetic Hamiltonian.
In the end we will see that the quantization of the electromagnetic field will
turn out to be a very natural application of second quantization, as described
in §1.5. The free electromagnetic theory actually is a many-particle theory for
massless spin-1 quanta, called photons.
Similar material can be found in Schwabl (Ch. 14) and Merzbacher (Ch. 23).
4.1 Classical electromagnetic fields: covariant formulation
We start from the Maxwell equations for an electromagnetic field in vacuum with charge
density ρc(~r, t) and current density ~jc(~r, t):
~ · ~B(~r, t) = 0 , ~× ~E(~r, t) = − ∂
∂t~B(~r, t) , (316)
~ · ~E(~r, t) =1
ǫ0ρc(~r, t) , (317)
~× ~B(~r, t) =1
c2∂
∂t~E(~r, t) + µ0
~jc(~r, t) . (318)
Here ~E(~r, t) and ~B(~r, t) are the real electric and magnetic fields. In addition, the con-
stants ǫ0 and µ0 are the permittivity and permeability of the vacuum, which are linked
to the speed of light c in vacuum through the relation
ǫ0µ0 = 1/c2 . (319)
Next we introduce the real scalar potential φ′(~r, t) and the real vector potential ~A ′(~r, t),
such that
~E(~r, t) = − ~φ′(~r, t) − ∂
∂t~A ′(~r, t) , ~B(~r, t) = ~× ~A ′(~r, t) . (320)
123
In this way the first two Maxwell equations (316) are satisfied automatically, since on
general grounds ~ ·( ~ × ~A ′(~r, t)
)= 0 and ~ ×
( ~φ′(~r, t))
= ~0 . The Maxwell
equation (317) can be rewritten now as
1
ǫ0ρc(~r, t) = − ∂
∂t
( ~ · ~A ′(~r, t))− ~ ·
( ~φ′(~r, t))
=( 1
c2∂2
∂t2− ~ 2)
φ′(~r, t) − ∂
∂t
(~ · ~A ′(~r, t) +
1
c2∂
∂tφ′(~r, t)
)
.
By employing the identity
~×( ~× ~A ′(~r, t)
) general======= ~( ~ · ~A ′(~r, t)
)− ~ 2
~A ′(~r, t)
the Maxwell equation (318) finally becomes
µ0~jc(~r, t) = ~×
( ~× ~A ′(~r, t))+
1
c2∂2
∂t2~A ′(~r, t) +
1
c2∂
∂t~φ′(~r, t)
=( 1
c2∂2
∂t2− ~ 2)
~A ′(~r, t) + ~(~ · ~A ′(~r, t) +
1
c2∂
∂tφ′(~r, t)
)
.
Next we define the electromagnetic 4-vector potential
A′µ(x) : A′ 0(x) ≡ φ′(~r, t)/c and
A′ 1(x)A′ 2(x)A′ 3(x)
≡ ~A′ (~r, t) (321)
and the electromagnetic 4-current
jµc (x) : j0c (x) ≡ cρc(~r, t) and
j1c (x)j2c (x)j3c (x)
≡ ~jc(~r, t) . (322)
By inserting the relation 1/ǫ0 = µ0c2 the Maxwell equations take their final covariant
form
A′ µ(x)− ∂µ(∂νA
′ ν(x))
= µ0 jµc (x) . (323)
By taking the 4-divergence of this wave equation, we readily derive that the electromagnetic
4-current has to satisfy the continuity equation ∂µ jµc (x) = 0. This reflects the conservation
of total electric charge, as opposed to the quantum mechanical probability interpretation
that we would have assigned to it in the previous chapter. As a result of the relativity
principle, A′ µ(x) should transform as a contravariant vector field (see exercise 28). If
regarded as a quantum mechanical wave function, the electromagnetic 4-vector potential
therefore should describe spin-1 particles.
124
For the quantum mechanical interpretation of A′µ(x) we restrict the discussion to a free
electromagnetic field without charges or charge currents. In that case the wave equation
reduces to
A′ µ(x)− ∂µ(∂νA
′ ν(x))
= 0 . (324)
Note that the potentials φ′(~r, t) and ~A ′(~r, t) are not fixed completely by the condition
(320). For arbitrary real differentiable functions χ(~r, t), the transformed potentials
~A(~r, t) = ~A ′(~r, t) − ~χ(~r, t) and φ(~r, t) = φ′(~r, t) +∂
∂tχ(~r, t)
⇒ Aµ(x) = A′ µ(x) + ∂µχ(x)
(325)
give rise to exactly the same electric/magnetic fields and therefore describe the same
physics. The associated freedom to choose the potentials is referred to as gauge freedom.
This gauge freedom can be exploited to transform the potentials in such a way that the
Coulomb conditions
~ · ~A(~r, t) = 0 and φ(~r, t) = 0 , (326)
are satisfied, which implies that
~E(~r, t) = − ∂
∂t~A(~r, t) and ~B(~r, t) = ~× ~A(~r, t) . (327)
This choice of gauge is called Coulomb gauge. We prefer this non-covariant gauge con-
dition to a covariant condition such as ∂µAµ(x) = 0, since the former allows us to more
easily identify the two physical degrees of freedom of the theory. In a covariant gauge the
two remaining, unphysical (read: physically non-relevant) degrees of freedom have to be
taken into account as well. In the Coulomb gauge the Maxwell equations simplify to the
free electromagnetic wave equation
~A(~r, t) =( 1
c2∂2
∂t2− ~ 2)
~A(~r, t) = 0 , ~ · ~A(~r, t) = 0 , (328)
implying that each component of ~A(~r, t) satisfies the Klein–Gordon equation with m = 0.
Also the electric and magnetic fields satisfy this simple massless Klein–Gordon equation.
4.1.1 Solutions to the free electromagnetic wave equation
Next we will solve the wave equation (328) for ~A(~r, t) inside a finite cubic enclosure with
edges L and periodic boundary conditions. These restrictions can be lifted later on by
taking the continuum limit L→ ∞ (see App.A.2). The periodicity condition allows us to
125
express the solutions in terms of plane waves with quantized values for the wave vectors
(propagation vectors) ~k = ~p/~ featuring in the spatial Fourier modes ∝ exp(i~k · ~r ):
~k ≡ k ~ek ≡ ωk
c~ek
§ 2.5.2====
2π
L~ν , with νx,y,z = 0 ,±1 ,±2 , · · · ∧ ~ν 6= ~0 . (329)
The Fourier mode corresponding to ~ν = ~0 is left out here, since it does not refer to an
electromagnetic wave. The general solution to the free electromagnetic wave equation then
reads
~A(~r, t) =1√V ǫ0
∑
~k
2∑
λ=1
√
~
2ωk
~ǫλ(~ek)[positive frequency sol.︷ ︸︸ ︷
a~k,λ exp(i~k · ~r − iωkt) +
negative frequency sol.︷ ︸︸ ︷
a∗~k,λ exp(iωkt− i~k · ~r )]
=1√ǫ0
∑
~k
2∑
λ=1
√
~
2ωk~u~k,λ(~r )
[
a~k,λ(t) + ηλ a∗−~k,λ
(t)]
(330)
in terms of the Fourier components
~u~k,λ(~r ) ≡
Fourier mode︷ ︸︸ ︷
exp(i~k · ~r )√V
∗unit vector︷ ︸︸ ︷
~ǫλ(~ek) . (331)
The factor√
~/2ωk has been introduced to render the Fourier coefficients
a~k,λ(t) ≡ a~k,λ exp(−iωkt) (332)
dimensionless, which is crucial for performing the quantization. As a result of the Coulomb
condition (326), one of the three degrees of freedom of the vector potential ~A(~r, t) is not
physical, since it can be gauged away. This aspect is manifested in the transversality of
the polarization vectors ~ǫλ(~ek) ∈ IR3 :
~ǫλ(~ek) · ~k~· ~A=0
======= 0 , with ~ǫ 2λ (~ek) = 1 and ~ǫ1(~ek) · ~ǫ2(~ek) = 0 . (333)
These polarization vectors are chosen in such a way that the unit vectors ~ǫ1(~ek) , ~ǫ2(~ek)
and ~ek form a right-handed coordinate system:
~ǫ1(~ek)×~ǫ2(~ek) = ~ek , ~ǫ2(~ek)× ~ek = ~ǫ1(~ek) , ~ek ×~ǫ1(~ek) = ~ǫ2(~ek) ,
2∑
λ=1
ǫλ,j(~ek)ǫλ, l(~ek) +kjk
klk
completeness========= δjl (j , l = x, y , z ) , (334)
and that this right-handedness is preserved under spatial inversion ~k → −~k :
~ǫλ(~ek)~k→−~k−−−−→ ~ǫλ(−~ek) ≡ ηλ~ǫλ(~ek) , with η1 = +1 and η2 = − 1 . (335)
126
Some more remarks about equation (330).
• In the second line of equation (330) we replaced ~k by −~k in the second term.
Subsequently we exploited the inversion symmetry of the ~k sum and used the trans-
formation property (335) of the polarization vector. It will depend on the explicit
application which of the two expressions for the vector potential will turn out to be
the most useful one. For example, in order to read off that ~A(~r, t) is real the first ex-
pression is more suitable, whereas the second expression will be ideal for performing
spatial integrals (as we will see).
• In the previous chapter we would at this point have interpreted the positive-frequency
terms ∝ exp(−iωkt) as positive-energy solutions and the negative-frequency terms
∝ exp(iωkt) as negative-energy solutions. That is precisely what we will not do in
this case, since we actually know what the classical energy of the system should be.
4.1.2 Electromagnetic waves in terms of harmonic oscillations
The next step involves determining the Hamiltonian (energy) of the free electromagnetic
field for a given solution (330). From classical electrodynamics we know that this Hamil-
tonian is given by
H =ǫ02
∫
V
d~r[
~E 2(~r, t) + c2 ~B 2(~r, t)]
(327)====
ǫ02
∫
V
d~r
[( ∂ ~A(~r, t)
∂t
)2
+ c2(~× ~A(~r, t)
)2]
,
(336)
where ρH(~r, t) = ǫ0
[~E 2(~r, t) + c2 ~B 2(~r, t)
]/2 is the energy density of the classical elec-
tromagnetic field. By means of the orthonormality relations for the Fourier components,
which are given in appendix E, this energy density can be expressed in terms of the inde-
pendent coefficients a~k,λ(t) and a∗~k,λ(t). For the electric-field contribution to the energy
density this proceeds as follows:
HE(336)====
ǫ02
∫
V
d~r( ∂ ~A(~r, t)
∂t
)2 ~A∈ IR3
====ǫ02
∫
V
d~r( ∂ ~A(~r, t)
∂t
)
·( ∂ ~A∗(~r, t)
∂t
)
(330)====
1
2
∑
~k,~k′
2∑
λ,λ′=1
~
2√ωkωk′
[a~k,λ(t) + ηλ a
∗−~k,λ
(t)][a∗~k′,λ′
(t) + ηλ′ a−~k′,λ′(t)]∗
∗∫
V
d~r ~u~k,λ(~r ) · ~u ∗~k′,λ′
(~r )(E.5)====
1
4
∑
~k
2∑
λ=1
~
ωk| a~k,λ(t) + ηλ a
∗−~k,λ
(t)|2
(332)====
1
4
∑
~k
2∑
λ=1
~ωk |a~k,λ(t)− ηλ a∗−~k,λ
(t)|2 .
127
The magnetic-field contribution to the energy density can be obtained in a similar way:
HB(336)====
c2ǫ02
∫
V
d~r(~× ~A(~r, t)
)2 ~A∈ IR3
====c2ǫ02
∫
V
d~r(~× ~A(~r, t)
)
·(~× ~A∗(~r, t)
)
(330)====
1
2c2∑
~k,~k′
2∑
λ,λ′=1
~
2√ωkωk′
[a~k,λ(t) + ηλ a
∗−~k,λ
(t)][a∗~k′,λ′
(t) + ηλ′ a−~k′,λ′(t)]∗
∗∫
V
d~r(
~k × ~u~k,λ(~r ))
·(
~k′ × ~u ∗~k′,λ′
(~r ))
(E.6),(329)=======
1
4
∑
~k
2∑
λ=1
~ωk |a~k,λ(t) + ηλ a∗−~k,λ
(t)|2 .
This can be combined into the compact expression
H =1
2
∑
~k
2∑
λ=1
~ωk
(
|a~k,λ(t)|2 + |a−~k,λ(t)|2)
=∑
~k
2∑
λ=1
~ωk |a~k,λ(t)|2 . (337)
Because |a~k,λ(t)|2 = |a~k,λ|2 we readily obtain that the Hamiltonian is properly time inde-
pendent, as expected for an isolated system. Finally, we rewrite the result in terms of the
real quantities
Q~k,λ(t) ≡[
a~k,λ(t) + a∗~k,λ(t)]√
~
2ωk
and P~k,λ(t) ≡ Q~k,λ(t)(332)==== − iωk
[
a~k,λ(t)− a∗~k,λ(t)]√
~
2ωk. (338)
The Hamiltonian becomes much more accessible in this way:
H =1
2
∑
~k
2∑
λ=1
(
P 2~k,λ
(t) + ω2kQ
2~k,λ
(t))
= H(Q~k,λ , P~k,λ
). (339)
The quantities Q~k,λ and P~k,λ satisfy the Hamilton equations
∂H∂P~k,λ
= P~k,λ
(338)==== Q~k,λ and
∂H∂Q~k,λ
= ω2k Q~k,λ
(332)==== − Q~k,λ
(338)==== − P~k,λ .
As such they form conjugate pairs of coordinates and momenta.
An electromagnetic wave in vacuum is the result of harmonic oscillations of the electro-
magnetic field. The Hamiltonian consists of a countably infinite set of independent linear
harmonic oscillators, characterized by the direction of propagation ~ek of the wave, its
polarization vector ~ǫλ(~ek) and its frequency ωk = ck .
128
4.2 Quantization and photons
Since the Hamiltonian simply corresponds to a set of linear harmonic oscillators, we can
quantize along the lines of § 1.6.1 and § 1.5. The Hamiltonian is then promoted to a
Hamilton operator in terms of the generalized position operators Q~k,λ(t) in the Heisenberg
picture and the corresponding conjugate momentum operators P~k,λ(t). These operators
satisfy the fundamental (canonical) commutation relations
[Q~k,λ(t), P~k′,λ′(t)
]= i~ δλ,λ′ δ~k,~k′ 1 ,
[Q~k,λ(t), Q~k′,λ′(t)
]=[P~k,λ(t), P~k′,λ′(t)
]= 0 . (340)
Subsequently we define creation, annihilation and number operators in the Heisenberg
picture according to
creation operators : a†~k,λ(t) ≡√ωk
2~Q~k,λ(t) − i√
2~ωk
P~k,λ(t) ≡ a†~k,λ exp(iωkt) ,
annihilation operators : a~k,λ(t) ≡ a~k,λ exp(−iωkt) ,
number operators : n~k,λ(t) ≡ a†~k,λ(t) a~k,λ(t) = a†~k,λ a~k,λ ≡ n~k,λ . (341)
These creation and annihilation operators satisfy the bosonic commutation relations
[a~k,λ(t), a
†~k′,λ′
(t)]= δλ,λ′ δ~k,~k′ 1 ,
[a~k,λ(t), a~k′,λ′(t)
]=[a†~k,λ(t), a
†~k′,λ′
(t)]= 0 . (342)
In analogy with the Hamilton operator (76) for a single linear harmonic oscillator, we ob-
tain the following Hamilton operator for the quantized electromagnetic field:
H =1
2
∑
~k
2∑
λ=1
~ωk
(a†~k,λ a~k,λ + a~k,λ a
†~k,λ
)=∑
~k
2∑
λ=1
~ωk
(n~k,λ +
121). (343)
Such observables belonging to collective quantities of the electromagnetic field can be ob-
tained in an alternative way by simply replacing the fields in the classical quantities by the
corresponding field operators in the Heisenberg picture. The real vector potential of equa-
tion (330) is then changed into the corresponding hermitian electromagnetic field operator
~A(~r, t) =1√V ǫ0
∑
~k
2∑
λ=1
√
~
2ωk~ǫλ(~ek)
[
a~k,λ exp(i~k · ~r − iωkt) + a†~k,λ exp(iωkt− i~k · ~r )
]
=1√ǫ0
∑
~k
2∑
λ=1
√
~
2ωk~u~k,λ(~r )
[
a~k,λ(t) + ηλ a†−~k,λ
(t)]
(344)
in the Heisenberg picture. This field operator satisfies the same free electromagnetic wave
129
equation as the classical vector potential. The quantized versions of the electric and mag-
netic fields follow from the usual relations
~E(~r, t) = − ∂
∂t~A(~r, t) and ~B(~r, t) = ~× ~A(~r, t) . (345)
Using appendix E as well as the commutation relations (342) the following can be derived
(see exercise 29 for an example).
• The total Hamilton operator for the quantized electromagnetic field reads
H ≡ ǫ02
∫
V
d~r[
~E(~r, t) · ~E †(~r, t) + c2 ~B(~r, t) · ~B †(~r, t)]
=∑
~k
2∑
λ=1
~ωk
(n~k,λ +
121).
(346)
• The total momentum operator for the quantized electromagnetic field follows from
the Poynting vector and reads
~P ≡ ǫ0
∫
V
d~r[
~E(~r, t)× ~B †(~r, t)]
=∑
~k
2∑
λ=1
~~k n~k,λ . (347)
• The total spin operator for the quantized electromagnetic field reads
~S ≡ ǫ0
∫
V
d~r[
~E(~r, t)× ~A †(~r, t)]
=∑
~k
~~ek(n~k,R − n~k,L
), (348)
where the number operators n~k,R = a†~k,R a~k,R and n~k,L = a†~k,L a~k,L are defined
according to
a~k,R = − 1√2( a~k,1 − ia~k,2 ) and a~k,L =
1√2( a~k,1 + ia~k,2 ) . (349)
4.2.1 Particle interpretation belonging to the quantized electromagnetic field
From the above-given observables for collective electromagnetic quantities we can directly
read off the particle interpretation of the quantized free electromagnetic theory. Ow-
ing to the bosonic commutation relations, the electromagnetic field can be regarded as
being composed of bosonic quanta, called photons. The operator a†~k,λ creates such a
photon with momentum ~~k , energy ~ωk = ~ck and polarization λ , whereas a~k,λ an-
nihilates such a photon. The number operator n~k,λ counts such photons. From the
additive spin operator (348) it can be concluded that photons are spin-1 particles, as
already predicted in exercise 28 by employing the relativity principle. However, the
helicity of the photon, i.e. the spin component along the direction of propagation ~ek ,
130
can exclusively take on the values + ~ (right-handed circularly polarized light) and − ~
(left-handed circularly polarized light). The absence of the helicity eigenvalue 0 is a di-
rect consequence of the fact that one of the three degrees of freedom of the vector field can
be gauged away and is therefore unphysical. This statement only makes sense as being
independent of the frame of reference if the photons are massless , which indeed follows
from the relativistic relation between mass, energy and momentum:
m2c4 = E2 − ~p 2c2 = ~2ω2
k − ~2k2c2
(329)==== 0 .
If the photons would have had a finite mass, then a rest system would exist with respect to
which the photons would be at rest. In that case no direction of propagation and therefore
no helicity could have been defined. Massless particles move at the speed of light in any
inertial frame, excluding the existence of a rest frame.
4.2.2 Photon states and density of states for photons
Based on § 1.2.1, the multi-photon states can be described in terms of the bosonic eigen-
states of the number operator:
|n1, n2, · · ·〉 =(a†1)
n1
√n1!
(a†2)n2
√n2!
· · · |0〉 ≡∏
j
(a†j)nj
√nj!
|0〉 ,
with |0〉 = the normalized photon vacuum
. (350)
Just like we did in chapter 1, the label j = 1 , 2 , · · · numbers the complete set of quantum
numbers ~k and λ . The multi-photon states have the following bosonic properties [see
equation (30)]:
nj | · · · , nj , · · ·〉 = nj | · · · , nj, · · ·〉 ,
aj | · · · , nj , · · ·〉 =√nj | · · · , nj−1, · · ·〉 ,
a†j | · · · , nj , · · ·〉 =√nj+1 | · · · , nj+1, · · ·〉 . (351)
The state function of the radiation field can be described in terms of this complete basis,
with the size of a particular occupation number representing the strength of the corre-
sponding monochromatic component for the given radiation field.
For later use we also derive the density of states for photons with wave vector ~k and polar-
ization λ . In complete analogy to the steps presented in § 2.5.2, we determine the number
of 1-photon states with energy on the interval [E,E+dE ], polarization λ , and ~ek within
131
the solid angle dΩ around the fixed (θ, φ)-direction ~eΩ :
ρλ(E,Ω) dΩdE =V
8π3
(k2dk dΩ
) k = E/(~c)=======
V E2
(2π~c)3dΩdE
⇒ ρλ(E,Ω) =V E2
(2π~c)3(λ = 1 , 2) . (352)
4.2.3 Zero-point energy and the Casimir effect
For the total Hamilton operator in equation (346) we have
H |n1, n2, · · ·〉(346)====
(∑
j
~ωjnj + E0
)
|n1, n2, · · ·〉 , (353)
where ~ωj is the energy of the photon mode labeled by j . As in the case of the linear
harmonic oscillator, here too we encounter a zero-point energy E0 :
H |0〉 = E0 |0〉 =1
2
∑
~k
2∑
λ=1
~ωk |0〉 =∑
~k
~ωk |0〉 , (354)
which originates from the operator combinations 12( a†~k,λ a~k,λ + a~k,λ a
†~k,λ
). The zero-point
energy of the photon vacuum represents an absolute infinite energy scale with respect to
which the photon states have energies that are higher by multiples of the photon energies
~ωj , j = 1 , 2 , · · · . In this respect, two scenarios can be distinguished.
• E0 does not depend on external parameters such as the dimensions of the box. This
would for instance apply to a free electromagnetic field without any restrictions. From
a quantum mechanical perspective only transitions between energy levels and energy
differences are measurable. In that case merely the energy differences between the
individual multi-photon energy eigenstates are relevant and the zero-point energy
can simply be subtracted by defining that H |0〉 ≡ 0 |0〉 . This can be accomplished
by normal ordering of the operator H (see § 1.2.2):
a†~k,λ a~k,λ → N(a†~k,λ a~k,λ) ≡ a†~k,λ a~k,λ ,
a~k,λ a†~k,λ
→ N(a~k,λ a†~k,λ
) ≡ a†~k,λ a~k,λ ,
which brings H in additive form:
N(H) =∑
~k
2∑
λ=1
~ωk n~k,λ .
132
• E0 does depend on external parameters such as the edge size L of the cubic box. A
variation of these parameters can result in physically measurable energy shifts and
forces (see the example below). In such cases we are at best allowed to subtract a
universal (parameter-independent) energy, such as the zero-point energy belonging
to an electromagnetic field without any restrictions.
free fieldfree field
perfectly conducting plates
aρE0(free)
ρE0(a)ρE0(free)
Casimir effect (Hendrik Casimir, 1948):
consider two parallel, perfectly conducting
plates. These plates are positioned at a
very short distance a of each other, such
that the plates can effectively be considered
as infinitely large on O(a) length scales.
Since the zero-point energy between the
plates differs from the one on the outside,
a net attractive force between the plates is
generated (see exercise 30). This is referred
to as the Casimir effect. Experimentally
it proved very difficult to align two plates
with uniform micrometre precision. Even-
tually it took 50 years to successfully verify
the Casimir effect experimentally. This was
done by switching from a two-plate config-
uration to a sphere-and-plate one: S.K. Lamoreaux covered in this way the a > 0.6µm
range in 1997, and U. Mohideen and A. Roy covered the a > 0.1µm range in 1998. The
Casimir effect is actually one of those rare purely quantum mechanical phenomena that
extend beyond the molecular range.
By the way: the concept of zero-point energy might actually contain more pro-
found physics than we anticipate at this moment. Not only energy differences
have physical relevance, but also the absolute value of the energy. For instance,
zero-point energy can have cosmological implications in view of the relation be-
tween energy and mass in general relativity. It is very well possible that the
construction of a consistent fundamental theory of Nature requires the exact
cancellation of the infinities caused by the zero-point energies that originate
from the various fundamental particles that are present in the theory.
4.2.4 Continuum limit
For completeness we list now the changes that would result from taking the size of the
enclosure to infinity (continuum limit). For our cube this continuum limit would imply
133
taking the limit L → ∞ (see App.A.2). The ~k -sum extends over all integer numbers
νx,y,z = (L/2π)kx,y,z , which causes the ~k -spectrum to become very closely spaced (i.e. ef-
fectively continuous) for L→ ∞ and changes the sum into an integral:
∑
~k
actually=======
∑
~ν
L→∞−−−−→ V
(2π)3
∫
d~k . (355)
In addition, for L→ ∞ the commutation relations for creation and annihilation operators
should satisfy the expressions for a continuous representation:
[aλ(~k ), a
†λ′(~k
′)]= δλ,λ′ δ(~k − ~k′) 1 ⇒ a~k,λ
L→∞−−−−→
√
(2π)3
Vaλ(~k ) . (356)
The field operator ~A(~r, t) in equation (344) thus changes into
~A(~r, t) =2∑
λ=1
∫
d~k
√
~
16π3ǫ0ωk
~ǫλ(~ek)[
aλ(~k ) exp(i~k ·~r−iωkt) + a†λ(~k ) exp(iωkt−i~k ·~r )
]
.
(357)
4.3 A new perspective on particle –wave duality
The field operator ~A(~r, t) contains the wave aspects that belong to the photon
states. This reflects particle–wave duality in the sense that each creation and
annihilation operator corresponds to a specific term in the plane-wave expansion
of the vector potential ~A(~r, t).
Let’s see how this differs from the approach that we adopted in the previous chapter.
• The positive-frequency terms: when attempting to set up a 1-particle QM we would
not have been dealing with the operator a~k,λ exp(i~k · ~r − iωkt), but rather with the
plane wave ∝ exp(i~k · ~r − iωkt) as the corresponding stationary solution to the rel-
ativistic wave equation. By acting with the operators E = i~ ∂/∂t and ~p = − i~ ~on this plane wave we would have reached the conclusion that we were dealing with
a state with energy ~ωk and momentum ~~k . However, to the quantized electromag-
netic theory a many-particle interpretation should be assigned, with the annihilation
operator a~k,λ exp(i~k · ~r − iωkt) acting on multi-photon states. Applied to a multi-
photon state with n~k,λ photons in the 1-photon state with quantum numbers ~k and
λ , this operator gives rise to a similar type of plane wave:
a~k,λ exp(i~k · ~r − iωkt) → √
n~k,λ exp(i~k · ~r − iωkt) ,
where the size of the proportionality factor depends on the occupation number of
the given 1-photon state. In this context particle –wave duality refers to the fact
that the annihilation of a photon with positive energy ~ωk and momentum ~~k will
always be accompanied by a plane-wave factor ∝ exp(i~k · ~r − iωkt).
134
• The negative-frequency terms: the main difference between the 1-particle and many-
particle theory follows from the operator a†~k,λ exp(iωkt − i~k · ~r ). In the 1-particle
theory this would have corresponded to a plane wave ∝ exp(iωkt − i~k · ~r ), whichwe would have interpreted as a state with negative energy −~ωk and momentum
−~~k in the previous chapter. In the many-particle theory, however, the signs of
these quantum numbers are effectively reversed, since we are dealing with pho-
ton creation rather than annihilation! The operator a†~k,λ exp(iωkt − i~k · ~r ) cor-
responds to the creation of a photon with positive energy ~ωk and momentum ~~k ,
which will always be accompanied by a plane-wave factor ∝ exp(iωkt− i~k · ~r ).
Second quantization: the prescription for quantizing the electromagnetic field that we
have just discussed is an explicit example of second quantization, as described in § 1.5 for
the Schrodinger field. Starting from a classical wave equation we switched directly to a cor-
responding many-particle theory formulated in the Heisenberg picture. The classical vector
potential ~A(~r, t) was a classical, three-component function defined in each spacetime point.
The quantized vector potential ~A(~r, t) is a field operator that acts on the state functions
in the Fock space for photons. In this field operator ~r and t are not quantum mechanical
variables. They are simply parameters that parametrize the plane waves. In this approach
~r and t are treated on equal footing, in contrast to the non-relativistic QM where t is
a parameter and ~r an eigenvalue belonging to the observable ~r . This aspect is precisely
what is needed for constructing the relativistic QM (see Ch. 5).
4.4 Interactions with non-relativistic quantum systems
In classical electrodynamics the influence of a classical electromagnetic field on a classical
particle with mass m and charge q follows from minimal substitution. In the Coulomb
gauge this implies that the momentum ~p of the classical particle should be replaced by
~p−q ~A(~r, t), where ~r is the coordinate of the particle. After this substitution ~p represents
the canonical momentum of the particle in the presence of the electromagnetic field.
The quantum mechanical Hamilton operator of the complete system is given by
H = H0 + He.m. + V (t) , with V (t) = Vint(t) + Vspin(t) , (358)
where He.m. is the Hamilton-operator (346) of the free electromagnetic field inside an
enclosure with volume V and periodic boundary conditions. The term H0 represents the
Hamilton operator of the unperturbed non-relativistic particle, which can be a free particle
or a particle that is bound by a potential. The interactions between the particle and the
135
electromagnetic field are given by
Vint(t) = − q
m~A(~r, t) · ~p +
q2
2m~A 2(~r, t) , (359)
Vspin(t) = − ~MS · ~B(~r, t) , ~MS = − |e|~2m
g~S
~. (360)
The first interaction is obtained by applying minimal substitution to the kinetic term
~p 2/(2m) and by subsequently inserting the Coulomb condition ~ · ~A(~r, t) = 0. The sec-
ond interaction is the extra interaction between the magnetic field and the spin-induced
intrinsic magnetic dipole moment of the particle that we have encountered in chapter 3.
The interaction V (t) acts both on the charged-particle states, through ~r and ~p , and on
the multi-photon states, through a~k,λ and a†~k,λ . The field operator ~A(~r, t) provides the
contact between the charged particle and the radiation field, with ~r being the position at
which the particle experiences the interaction and with the interaction corresponding to
the creation (emission) and annihilation (absorption) of photons. To phrase it differently:
the field operator ~A(~r, t) is the universal force carrier (mediator) of the electromagnetic
interactions. Stronger radiation fields will result in stronger emission and absorption ef-
fects in view of the annihilation and creation factors√nj and
√nj+1 in equation (351)
being larger in that case. The difference between these two factors will play an important
role in the quantum mechanical phenomenon of spontaneous photon emission.
Vint(t) in first-order perturbation theory: absorption/emission of one photon.
In order to assess the implications of the quantization of the electromagnetic field, we
consider electromagnetic processes that involve the absorption/emission of precisely one
photon with quantum numbers ~k and λ . To further simplify the problem we assume for
convenience that the interaction Vint(t) can be treated as a weak perturbation and that
the magnetic interaction Vspin(t) can be neglected with repsect to Vint(t) (as is true for the
dipole approximation in § 4.4.2). As a result, the spin of the particle can be effectively left
out in the analysis. Owing to the periodic time dependence of the electromagnetic field,
we can apply periodic perturbation theory (see the lecture course Quantum Mechanics 2).
At first order in perturbation theory, only the linear term in ~A contributes to 1-photon
processes and effectively
V (t) →∑
~k
2∑
λ=1
[
C†~k,λ
exp(−iωkt) + C~k,λ exp(iωkt)]
,
with C~k,λ
(344),(359)======= − q
m
√
~
2Vǫ0ωk
a†~k,λ exp(−i~k · ~r )~ǫλ(~ek) · ~p . (361)
136
After all, at first order in perturbation theory the number of photons changes by ± 1 due
to the linear term in ~A and by 0 , ± 2 due to the quadratic term.12 Take the system prior
to the interaction to be in the following unperturbed initial state:
|φ(0)i 〉 = |ψA〉| · · · , n~k,λ, · · ·〉 , (362)
|ψA〉 = unperturbed initial state of the charged particle with energy EA ,
| · · · , n~k,λ, · · ·〉 = photon state with n~k,λ photons of the type (~k, λ) .
Subsequently we wait a sufficiently long time and ask ourselves the question what the
transition probability will be to the final state
|φ(0)f 〉 = |ψB〉| · · · , n~k,λ ± 1, · · ·〉 , (363)
|ψB〉 = unperturbed final state of the charged particle with energy EB ,
| · · · , n~k,λ ± 1, · · ·〉 = the same photon state with one (~k, λ)-photon more or less .
The relevant matrix elements that determine this transition probability are
Absorption : (C†~k,λ
)fi(361)==== − q
m
√
~
2Vǫ0ωk〈φ(0)
f | a~k,λ exp(i~k · ~r )~ǫλ(~ek) · ~p |φ(0)i 〉
(351)==== − q
m
√
~n~k,λ2Vǫ0ωk
〈ψB|exp(i~k · ~r ) ~p |ψA〉 · ~ǫλ(~ek) (364)
if the photonic final state is given by | · · · , n~k,λ−1, · · ·〉 , and
Emission : (C~k,λ )fi(361)==== − q
m
√
~
2Vǫ0ωk
〈φ(0)f | a†~k,λ exp(−i~k · ~r )~ǫλ(~ek) · ~p |φ
(0)i 〉
(351)==== − q
m
√
~(n~k,λ+1)
2Vǫ0ωk〈ψB|exp(−i~k · ~r ) ~p |ψA〉 · ~ǫλ(~ek) (365)
if the photonic final state is given by | · · · , n~k,λ+1, · · ·〉 . By employing Fermi’s Golden
Rule for periodic perturbation theory (see the lecture course Quantum Mechanics 2) this
gives rise to the following transition rates (i.e. probabilities per unit of time) per photon
state:
wabs~k,λ
=2π
~|(C†
~k,λ)fi|2 δ(EB −EA−~ωk) , wem
~k,λ=
2π
~|(C~k,λ )fi|2 δ(EB −EA+~ωk) .
(366)
Depending on the size of n~k,λ a few different scenarios occur.
12At first order in perturbation theory the quadratic term contributes to 2-photon processes and to
photon scattering off atoms. Note, though, that second-order perturbative corrections due to the linear
term contribute to these processes as well.
137
• If n~k,λ is large enough that we may take√
n~k,λ+1 ≈√
n~k,λ , then we are dealing
with a semi-classical scenario. In that case the absorption/emission of a single photon
has a negligible impact on the radiation field, which effectively can be regarded as a
classical field with a classical field strength |A0(ωk)| ≡√
2~n~k,λ/(Vǫ0ωk) .
• For decreasing n~k,λ the scenario becomes more and more quantum mechanical.
Absorption will still be approximated well by the semi-classical approach, since
|A0(ωk)|2 ∝ n~k,λ . However, this time the absorption of a photon does have an
impact on the strength of the remaining radiation field. On the other hand, emission
will not be predicted adequately by the semi-classical approach, since n~k,λ+1 will
differ noticeably from n~k,λ .
• The most extreme differences with the semi-classical approach occur in the absence
of a radiation field , i.e. for n~k,λ = 0. In that case no transition is predicted to occur
classically, whereas quantum mechanically there actually is the possibility of photon
emission from an excited state! This purely quantum mechanical phenomenon is
called spontaneous emission.
We observe that both types of emission are treated simultaneously in the quantum me-
chanical approach: spontaneous emssion if the right frequency component happens to be
absent in the radiation field (n~k,λ = 0) and stimulated emission if a radiation field is sup-
plied that does contain the right frequency component (n~k,λ > 0). In the latter case it
follows directly from equation (365) that the transition probability increases for increasing
n~k,λ . Hence the name stimulated emission.
4.4.1 Classical limit
Consider a monochromatic radiation field |n~k,λ〉 . The classical electromagnetic description
applies if the quantization effects due to the commutation relations (342) can be negelected.
This occurs when adding/removing a photon has no impact on the field strength, i.e. when
n~k,λ is large. Phrased more precisely: the number of (~k, λ) -photons per characteristic
volume ~3/|~pγ|3 = 1/k3 should be very large. When acting on such a strong radiation
field, the creation and annihilation operators a†~k,λ and a~k,λ can be represented by complex
numbers e− iδk√n~k,λ respectively eiδk
√n~k,λ (see § 1.6.4), so that
~A(~r, t)|n~k,λ〉 ≈√
~n~k,λ2Vǫ0ωk
~ǫλ(~ek)[
eiδk exp(i~k · ~r − iωkt) + e− iδk exp(iωkt− i~k · ~r )]
|n~k,λ〉
≈ |A0(ωk)| ~ǫλ(~ek) cos(~k · ~r − ωkt+ δk )|n~k,λ〉 , |A0(ωk)| ≡√
2~n~k,λVǫ0ωk
.
The action of a radiation field |n~k,λ〉 on a charge carrier is in that case equivalent with
the classical action of a monochromatic plane wave.
138
We thus observe a clear distinction between the classical theories of radiation and particles
(such as electrons).
• The classical limit of the Schrodinger wave mechanics for an electron describes the
behaviour of that particle as governed by Newtonian mechanics, whereas a classical
gas of such particles is obtained at low densities.
• The classical limit of the quantum theory of radiation describes the behaviour of
very many photons at large densities by means of a classical electromagnetic wave
that satisfies the Maxwell equations.
This also explains why first the wave aspects of light and the particle aspects of electrons
were observed, causing both phenomena to appear fundamentally different.
4.4.2 Spontaneous photon emission (no exam material)
Using the density of states for photons from § 4.2.2, the transition rate for spontaneous
emission of a photon with polarization λ and ~ek within a solid angle dΩ around ~eΩ is
given by
W spont.em.A→B+γλ, dΩ
=2π
~dΩ
∫
d(~ωk) ρλ(~ωk,Ω) |(C~k,λ )fi|2 δ(EB − EA + ~ωk)
(365),(352)=======
q2ωAB
8π2~c3m2ǫ0
∣∣∣〈ψB|exp(−iωAB
~eΩ · ~r/c) ~p |ψA〉 · ~ǫλ(~eΩ)∣∣∣
2
dΩ ,
with the usual definition ωAB
≡ (EA −EB)/~
. (367)
Application 1: can a free particle with mass m 6= 0 and well-defined momentum ~pA ≡ ~~kA
spontaneously emit a photon? Answer: no!
Proof: the unperturbed spatial wave function of the particle prior to the interaction is
ψA(~r ) = (2π)−3/2 exp(i~r · ~kA) .
As unperturbed final state of the particle we consider an arbitrary eigenfunction of the
momentum operator belonging to the momentum eigenvalue ~pB ≡ ~~kB :
ψB(~r ) = (2π)−3/2 exp(i~r · ~kB) .
The relevant matrix element for the transition is then
〈ψB|exp(−i~k · ~r ) ~p |ψA〉 = ~~kA (2π)−3
∫
d~r exp(i~r · [~kA−~kB−~k ]
)= ~~kA δ(~kA−~kB−~k ) .
139
Based on equation (367) this implies that, for the transition to be possible, two conditions
should be satisfied simultaneously . . . conservation of energy as well as momentum:
EB =√
~p 2B c
2 +m2c4 = EA−~ωk =√
~p 2A c
2 +m2c4 − ~ck and ~pB = ~pA−~~k .
Both conditions can be merged to arrive at
~p 2B +m2c2
momentum======= (~pA − ~~k)2 +m2c2
energy==== ~p 2
A +m2c2 + ~2k2 − 2~k
√
~p 2A +m2c2
k 6=0===⇒
√
~p 2A +m2c2 = ~pA ·
~k
k≤ |~pA| ,
which is impossible if m 6= 0. This energy–momentum conservation argument holds just
as well for photon absorption and for 1-photon processes caused by Vspin(t).
Application 2: spontaneous transitions of 1-electron atoms in the dippole approximation.
For spontaneous-emission transitions of relatively light 1-electron atoms (such as hydro-
gen) the wavelength of the emitted photon is much larger than the typical size of the
emitting atom. For instance, the transition from the n = 2 to the n = 1 state in-
volves an energy difference ∆E = 3mec2 (Zα)2/8. This corresponds to a wavelength
λγ = 2π/k = 2π~c/∆E ≈ (1.2/Z2) × 10−7m, which is much larger than the typical
< ratoom >= O(10−10m) size of the charge distribution of the 1-electron atom, provided
that Z is not too large. In the electric dipole approximation we exploit this fact by replac-
ing the Fourier mode exp(−i~k · ~r ) by 1, since kr ≪ 1 across the full extent of the atom,
and by leaving out the magnetic interaction. The justification for the latter approximation
lies in the fact that the factor ~ǫλ(~ek) · ~p in Vint(t) gives rise to O(~/<ratoom>) effects,
whereas the factor ~~k ×~ǫλ(~ek) in Vspin(t) leads to much weaker O(~k) effects.
By means of the commutation relation
H0 =~p 2
2me+ V (~r ) ⇒
[H0, ~r
]= − i~
∂H0
∂ ~p= − i~
~p
me
the relevant matrix element for spontaneous photon emission in the dipole approximation
is given by
〈ψB|~p |ψA〉 =ime
~〈ψB|
[H0, ~r
]|ψA〉 = − iω
ABme 〈ψB|~r |ψA〉 ≡ − iω
ABme~rBA
.
From equation (367) it follows that in dipole approximation
W spont.em.A→B+γλ, dΩ
≈ e2ω3AB
8π2~c3ǫ0|~r
BA· ~ǫλ(~eΩ)|2 dΩ . (368)
140
Subsequently we sum over all photon polarizations, using the completeness relation
2∑
λ=1
ǫλ,j(~eΩ)ǫ∗λ, l(~eΩ)
~ǫλ(~eΩ) ∈ IR3
=======2∑
λ=1
ǫλ,j(~eΩ)ǫλ, l(~eΩ)(334)==== δjl − eΩ, j eΩ, l ,
and integrate over all directions ~eΩ while defining the z-axis along ~rBA
:
W spont.em.A→B+γ ≈ e2ω3
AB
8π2~c3ǫ0
∫
dΩ(|~r
BA|2 − |~r
BA· ~eΩ|2
)=
e2ω3AB
|~rBA
|28π2~c3ǫ0
2π∫
0
dφ
1∫
−1
dcos θ sin2 θ
=e2ω3
AB
3π~c3ǫ0|~r
BA|2 =
4
3αem
ω3AB
c2|~r
BA|2 ≈ W spont.em.
A→B+γ . (369)
In this expression αem is the electromagnetic fine-structure constant, which is defined in
the usual way as
αem ≡ e2
4πǫ0~c≈ 1/137 . (370)
The average lifetime τA and decay width ΓA of the state A can be derived from
τA =1
ΓA=
1∑
j
W spont.em.A→Bj +γ
, (371)
which is summed over all allowed final states Bj for electric dipole transitions. These
allowed final states have to satisfy the so-called dipole selection rules |∆ℓ| = 1 and
|∆mℓ| = 0 or 1, which is related to the spin of the emitted photon. Sometimes ~rBA
= ~0
for all states with EB < EA . For instance this is the case for the 2s state of the hy-
drogen atom, for which no dipole transition to the 1s ground state is possible as both
states have ℓ = 0. To determine the (longer) lifetime of such states one would have
to take along the next term in the multipole expansion. This implies that the magnetic
interaction cannot be discarded (consequence: magnetic dipole transitions) and that in
Vint(t) the Fourier mode exp(−i~k · ~r ) should be replaced by 1 − i~k · ~r (consequence:
electric quadrupole transitions). For the 2s state of the hydrogen atom even these tran-
sitions are not possible: the 1-photon transition 2s → 1s + γ is forbidden by angular
momentum conservation. The first allowed transition is the much weaker 2-photon transi-
tion 2s → 1s + γ + γ , which explains why the lifetime of the 2s state (122ms) is so much
larger than the one for the 2p state (1.6 ns).
4.4.3 The photon gas: quantum statistics for photons
As final example of the interaction between quantum systems and quantized electromag-
netic fields we consider the quantum statistics for photons. From equations (364)–(366) a
141
condition can be derived for thermal equilibrium between an atomic heat bath at temper-
ature T and electromagnetic radiation with quantum numbers ~k and λ (see exercise 31).
From this equilibrium condition it follows that the average occupation number of the cor-
responding 1-photon state at temperature T = 1/(kBβ) is given by
n~k,λ =1
exp(β~ωk)− 1, (372)
which we recognize as a Bose–Einstein distribution with α = 0. This is exactly the same
distribution as derived in exercise 14 for a canonical ensemble of systems that each consist
of a single linear harmonic oscillator. The fact that photons in general do not allow for
a grand canonical treatment has to do with the absence of a photon conservation law in
most quantum mechanical settings.13 As a result of absorption by the atoms in the heat
bath, the photons do not migrate to the heat bath as particles and exclusively contribute
to the thermodynamic energy balance. The entropy of the heat bath is in fact independent
of the number of photons. The absence of a photon conservation law also implies that no
Bose–Einstein condensation of low-energy photons can occur in such situations. It will
be entropically favourable for the atomic heat bath to simply absorb such photons: the
condensate only involves a single quantum state whereas the number of heat-bath states
increases upon energy absorption.
Let’s now enclose the electromagnetic field inside a black body with walls that can ab-
sorb/emit photons of all energies. In this idealized situation we are dealing with an ideal
photon gas. The energy distribution of the radiation field per unit of energy and volume
then reads
U(E)(372)====
1
V
E
exp(βE)− 1
∫
dΩ
2∑
λ=1
ρλ(E,Ω)(352)====
8π
h3c3E3
exp(βE)− 1. (373)
From this expression we can derive Planck’s law for the energy distribution per unit of
frequency ν = E/h and volume:
U(ν) = U(E)dE(ν)
dν
(373)====
8πhν3
c31
exp(hν/kBT )− 1, (374)
as well as the law of Stefan–Boltzmann for the total energy density:
Utot =
∞∫
0
dE U(E)(373), x=βE=======
k4BT4
π2~3c3
∞∫
0
dxx3
exp(x)− 1=
π2k4B15~3c3
T 4 . (375)
13We used to think that a photon gas with a conservation law for the total number of photons was not
realizable in the lab. However, in 2010 the situation changed. The photon gas inside the microresonator
described in exercise 21 effectively satisfies a photon conservation law and as such allows the formation of
a Bose–Einstein condensate.
142
Experimental realization: use a large cavity (furnace) with a tiny hole in it. If the walls
of the cavity are sufficiently rough and opaque, any light entering the cavity will be reflected
very many times before it will be able to escape the cavity through the tiny hole. The light
then has a high probability of being absorbed by the walls before it manages to escape. In
good approximation the tiny hole acts as a black body. By varying the temperature of the
cavity, Planck’s law of black-body radiation can be verified experimentally.
Another almost perfect example of Planck’s spectrum for black-body radiation is provided
by the present-day cosmic microwave background (CMB), which corresponds to a temper-
ature of T = 2.725K. The present-day CMB radiation has evolved from a snapshot of our
universe roughly 380,000 years after the Big Bang, when the universe became transparent
to radiation. Due to the expansion of the universe, the earlier (distant) universe was of
course warmer than the universe today. However, the corresponding radiation is redshifted
due to that same expansion, causing it to appear as having the same temperature as the
universe today!
143
5 Many-particle interpretation of the relativistic QM
Motivated by the quantization of the electromagnetic field, this chapter will be
devoted to assigning a many-particle interpretation to the solutions of the rela-
tivistic wave equations that were constructed in chapter 3.
More extended versions of the material covered in this chapter can be found in
Schwabl (Ch. 13) and Merzbacher (Ch. 24).
Negative-energy states and the 1-particle relativistic QM: in chapter 3 the at-
tempts to construct a relativistic 1-particle quantum theory always resulted in a set of
hard to interpret negative-energy states. In the bosonic case, such as in the Klein–Gordon
theory, this resulted in unsurmountable problems that basically disqualified the theory as
a viable 1-particle quantum theory. In the fermionic case, such as in the free Dirac theory,
Pauli’s exclusion principle provided a way out by assuming that all 1-fermion negative-
energy states are occupied in the vacuum state of the universe. These states in the Dirac
sea behaved dynamically as if we were dealing with objects with opposite charge. This
can be highlighted by looking at the relativistic 1-electron atom, which has bound states
with positive energy. If we would invert the nuclear charge (Z → −Z), only bound states
with negative energy occur. This is consistent with the picture that the positively charged
nucleus effectively repels the negative-energy electrons and that upon inverting the nuclear
charge the negative-energy electrons are effectively attracted. Another way to read off this
opposite charge is by selecting negative-energy states in the Dirac equation (308) by means
of the substitution E = ENR −mc2. In that case we obtain the same eigenvalue equation
as in § 3.2.5 (I) with ψU↔ ψ
D, ENR → −ENR , ~p → − ~p and q → − q . It was shown in
exercise 27 how a negative-energy state can indeed be turned into a positive-energy state
with opposite charge by performing charge conjugation.
Hidden inside the free Dirac theory apparently resides the description of a second type of
particle with opposite charge. By employing hole theory (see the discussion in § 1.7.1),these particles with opposite charge could be made manifest as holes in the Dirac sea.
However, hole theory will not be able to describe all observed phenomena involving the
oppositely charged particles.
A conflict with hole theory: the occurrence of the β+ decay process p→ n+e++νe in
the decay of proton-rich nuclei actually contradicts this hole-theory interpretation. Within
hole theory we would have liked to interpret the produced e+ as a hole in the Dirac sea.
However, the creation of a hole in the Dirac sea should automatically be accompanied by
the simultaneous creation of a positive-energy electron (see the discussion on electron–hole
excitations in § 1.7.1), which evidently is not the case in the observed β+ decay process!
144
Many-particle interpretation of the relativistic QM: all this seems to suggest that
it may be better to not set up the relativistic QM as a 1-particle theory, but rather as a
many-particle theory in analogy to the treatment of the electromagnetic field. This idea
does not contradict with the non-relativistic QM. After all, as noted earlier, photons are
massless and should therefore always be treated relativistically. These photons can (in
most situations) be created and annihilated without energy threshold. Massive particles,
however, do have an energy threshold as a direct consequence of their non-zero rest mass:
• for kinetic energies ≪ mc2 such systems behave non-relativistically, i.e. they corre-
spond to a fixed number of particles;
• for kinetic energies = O(mc2) the many-particle aspects manifest themselves .
The many-particle theory will be set up along the lines worked out for the electromagnetic
field: the field equation will be solved by performing a Fourier decomposition (plane-
wave expansion) and the Hamiltonian belonging to the field equation will be expressed
as an infinite set of independent linear harmonic oscillators. Finally, this Hamiltonian
will be quantized in the Heisenberg picture and the fundamental energy quanta will be
interpreted as particles. Particle–wave duality then follows automatically from the fact
that the creation/annihilation of a particle is always accompanied by a corresponding
plane-wave factor, as can be read off from equation (344) in the electromagnetic case.
From this point onward the 1-particle quantum mechanical approach makes way for a
many-particle formulation based on second quantization. This many-particle formulation
is referred to as relativistic quantum field theory and is usually described in terms of path
integrals or Lagrangian densities for continuous systems. In this lecture series we will
content ourselves with deriving the salient aspects of quantization while avoiding field-
theoretic bells and whistles as much as possible.
5.1 Quantization of the Klein–Gordon field
Again we start off by considering the Klein–Gordon equation for free spin-0 particles with
rest mass m and charge q . For the derivation of the many-particle formulation we ignore
the problems with the 1-particle quantum mechanical interpretation and treat the Klein–
Gordon equation as a relativistic classical wave equation.
The classical energy density: in order to highlight the analogy with the electromagnetic
case we will rescale the Klein–Gordon field according to ψ(x) =√2mc2 φ(x)/~ . This
rescaled Klein–Gordon field φ(x) satisfies the field equation
(
+m2c2
~2
)
φ(x) =( 1
c2∂2
∂t2− ~2
+m2c2
~2
)
φ(x) = 0 (376)
145
and has a corresponding classical energy density14 (without proof)
ρH(x) =
( ∂
∂tφ∗(x)
)( ∂
∂tφ(x)
)
+ c2(~φ∗(x)
)
·(~φ(x)
)
+m2c4
~2φ∗(x)φ(x) . (377)
In this form the similarity with the classical electromagnetic energy density is apparent.
The first two terms are the analogues of the electric and magnetic contributions to the
energy density. The last term is a rest-mass term, which is absent for the massless pho-
tons. The other differences between both energy densities reflect that photons have spin 1,
whereas the particles belonging to the Klein–Gordon theory have spin 0 (see § 3.1).Remark: in the relativistic quantum-field-theory approach all this is handled more sys-
tematically by working with Lagrangian densities for continuous systems, using φ(x) and
φ∗(x) as generalized coordinates. Everything else follows from this Lagrangian density.
The Klein–Gordon equation simply follows from the Euler–Lagrange equation (equation
of motion) that belongs to this Lagrangian density. Moreover, if we go from the Lagrangian
density to the corresponding Hamiltonian density of the system we obtain an expression
that coincides with the energy density given above.
Plane-wave expansion and quantization: in analogy to the electromagnetic case we
also solve the Klein–Gordon equation inside a finite cubic enclosure with periodic boundary
conditions:
φ(x) =∑
~p
~√2E~p
u~p (~x )[
a~p (t) + c~p (t)]
, u~p (~x ) =1√V
exp(i~p · ~x/~) . (378)
The positive- and negative-frequency Fourier coefficients a~p (t) and c~p (t) satisfy
a~p (t) = a~p exp(−iE~p t/~) and c~p (t) = c~p exp(iE~p t/~) ,
with ~p =2π~
L~ν , E~p ≡
√
m2c4 + ~p 2c2 > 0 (νx,y,z = 0 ,±1 ,±2 , · · ·) . (379)
The Fourier components u~p (~x ) are normalized in the usual way:∫
V
d~x u~p (~x )u∗~p ′(~x )
(E.4)==== δ~p, ~p ′ . (380)
The total energy inside the volume V then simply reads
H(t) =
∫
V
d~x ρH(x)
(377),(380)=======
∑
~p
E~p
2
(
|a~p (t)− c~p (t)|2 + |a~p (t) + c~p (t)|2)
=∑
~p
E~p
(
|a~p (t)|2 + |c~p (t)|2)
=∑
~p
E~p
(
|a~p |2 + |c~p |2)
= H . (381)
14For φ(x) ∈ IR this energy density should be multiplied by a factor 1/2, since the number of degrees
of freedom is halved in that case.
146
Next this superposition of vibrational modes is quantized in the Heisenberg picture in
analogy with equations (337)–(341) for the electromagnetic field:
a~p (t) → a~p (t) ≡ a~p exp(−iE~p t/~) ,
c~p (t) → b†−~p (t) ≡ b†−~p exp(iE~p t/~) . (382)
In order to be able to give a correct particle interpretation to the negative-energy solutions,
it is essential that the second operator is a creation operator if the first one is an annihilation
operator. The scalar Klein–Gordon field then becomes a scalar field operator
φ(x) =∑
~p
~√2E~p
u~p (~x )[
a~p (t) + b†−~p (t)]
=∑
~p
~√2VE~p
[
a~p exp(−ip · x/~) + b†~p exp(ip · x/~)] (383)
with
p0 ≡ E~p/c =√
m2c2 + ~p 2 , (384)
satisfying the same wave equation as the classical Klein–Gordon field.
Particle interpretation: the following particle interpretation can be assigned to the
quantized Klein–Gordon theory (see the many-particle operators listed on the next page).
The operators a~p and a†~p annihilate and create spin-0 particles with momentum ~p and
positive energy E~p . The operators b~p and b†~p annihilate and create the corresponding
antiparticles with the same spin, momentum and positive energy, but with opposite charge.
The field operator φ(x) acts on the state functions in the Fock space belonging to both the
spin-0 particles and antiparticles.15 For a sensible treatment of negative energy solutions
to the Klein–Gordon equation, particles and antiparticles are thus described in a joint
way in the many-particle formulation of the relativistic QM. If a positive-energy solution
corresponds to the annihilation/creation of a positive-energy particle, a negative-energy so-
lution corresponds to the creation/annihilation of a positive-energy antiparticle with oppo-
site charge. Particles and antiparticles can be regarded as identical particles with opposite
values for the charge quantum number, whereas the field operator φ(x) corresponds to
one specific value of this quantum number. This quantum number indicates how the par-
ticles interact with the outside world. We distinguish three types of (conserved) charge:
electric charge, weak charge (weak isospin) and strong charge (colour charge), belonging
to the three fundamental non-gravitational forces that are known to us.
15If φ(x) ∈ IR, then b†~p = a†~p . This lifts the distinction between particles and antiparticles, which
therefore should carry zero charge.
147
Motivation for this particle interpretation: just like in the electromagnetic case, the
creation and annihilation operators satisfy bosonic commutation relations:
[a~p (t), a
†~p ′(t)
]=[b~p (t), b
†~p ′(t)
]= δ~p, ~p ′ 1 , all other commutators vanish . (385)
Now we introduce the number operators
n(+)~p ≡ a†~p (t) a~p (t) = a†~p a~p and n
(−)~p ≡ b†~p (t) b~p (t) = b†~p b~p (386)
for counting particles and antiparticles, respectively. In analogy to equation (343), this
allows us to write the total Hamilton operator of the free non-interacting many-particle
system in a compact way as
H =∑
~p
E~p
(
a†~p(t) a~p(t) + b−~p(t) b†−~p(t)
)
=∑
~p
E~p
(
n(+)~p + n
(−)~p + 1
)
. (387)
The energy spectrum is indeed nicely bounded from below. Again we obtain a zero-point
vacuum energy that is infinite and positive. The rejected “probability density” of § 3.1 is
given a completely different meaning in the many-particle formulation. It is related to the
total charge operator of the many-particle system, since
Qtot ≡ iqc
~
∫
V
d~x(
φ†(x)∂0 φ(x)− ∂0 φ†(x)φ(x)
)
(383), 1st
====q
2
∑
~p
([a†~p (t) + b−~p (t)
][a~p (t)− b†−~p (t)
]+[a†~p (t)− b−~p (t)
][a~p (t) + b†−~p (t)
])
(385)==== q
∑
~p
(
n(+)~p − n
(−)−~p − 1
)
= q∑
~p
(
n(+)~p − n
(−)~p − 1
)
= Qtot (388)
indeed counts particles with charge q and antiparticles with charge − q . The Lorentz
invariant continuity equation that was derived in § 3.1 in fact does not represent conser-
vation of probability, but rather conservation of total charge of the many-particle system.
The total momentum operator of the free non-interacting many-particle system is given by
the additive expression (without proof)
~P =∑
~p
~p(
n(+)~p + n
(−)~p
)
. (389)
The particle interpretation that was given on the previous page can now be read off directly
from the three many-particle operators (387)–(389).
148
Locality: physical observables at different spatial positions should mutually commute and
as such be measurable simultaneously. After all, the results of a repeated quantum me-
chanical measurement in a certain spatial point should not be affected by simultaneously
performing a repeated quantum mechanical measurement in another spatial point, irrespec-
tive of how far the two points are apart. This is called locality or microcausality. If these
measurements would influence each other, it would constitute a transfer of information
that exceeds the speed of light, which is forbidden by the theory of relativity. During one
measurement we would be able to observe the fact that the other measurement is taking
place. In the coordinate representation the physical observables can be written in terms
of the field operators (383). Hence, locality should follow directly from the commutation
relations that pertain to these field operators. The relevant commutators are
[φ(~x, t), φ(~x ′, t)
]=
[φ†(~x, t), φ†(~x ′, t)
] (383),(385)======= 0 , (390)
[φ(~x, t), φ†(~x ′, t)
]=
∑
~p,~p ′
~2u~p (~x )u∗~p ′ (~x ′)
2√E~pE~p ′
[a~p (t) + b†−~p (t) , a
†~p ′(t) + b−~p ′(t)
] (385)==== 0 .
These commutators nicely vanish in all required situations. By the way, this would not
have been the case if the creation and annihilation operators would have satisfied fermionic
anticommutation relations.
5.2 Quantization of the Dirac field
The classical energy density: consider a Dirac field ψ(x) that satisfies the classical
Dirac equation for free spin-1/2 particles with rest mass m and charge q :
(i~ ∂/−mc
)ψ(x) = 0 ⇒
(
i~∂
∂t+ i~c ~α · ~ − βmc2
)
ψ(x) = 0 . (391)
The corresponding classical energy density is given by
ρH(x) = ψ†(x) i~
∂
∂tψ(x) = ψ†(x)
(
− i~c~α · ~ + βmc2)
ψ(x) . (392)
Plane-wave expansion and quantization: subsequently the Dirac equation is solved
inside a finite cubic enclosure with periodic boundary conditions. This results in the
following plane-wave decomposition:
ψ(x) =∑
~p
∑
λ
√
mc2
E~pu~p (~x )
[
u+λ (~p )a~p ,λ(t) + u−λ (~p )c~p ,λ(t)]
(393)
≡∑
~p
∑
λ
exp(i~p · ~x/~)√
VE~p /mc2
[
u+λ (~p )a~p ,λ exp(−iE~p t/~) + u−λ (~p )c~p ,λ exp(iE~p t/~)]
,
149
with ~p and E~p the same as for the Klein–Gordon field [see equation (379)]. The Dirac
spinors u±λ (~p ) satisfy
~S · ~p u±λ (~p ) = λ~ ~p u±λ (~p ) and ( p/± −mc)u±λ (~p ) = 0 , (394)
where λ denotes the helicity eigenvalue (being +1/2 or −1/2), p0± = ±E~p/c the energy
eigenvalue and ~p± = ~p the momentum eigenvalue. For free spin-1/2 particles it is derived
in exercise 26 that H, ~p and ~S · ~p indeed form a commuting set of operators. These Dirac
spinors are normalized according to the normalization condition in equation (300):
u±λ (~p )†u±λ′(~p ) =
E~p
mc2δλ,λ′ and u±λ (~p )
†u∓λ′(~p ) = 0 . (395)
By means of the normalization condition (380) for the Fourier components u~p (~x ), the
total energy inside the volume V reads
H(t) =
∫
V
d~x ρH(x)
(392),(393)=======
∑
~p
∑
λ,λ′
mc2[
u+λ (~p )†a∗~p ,λ(t) + u−λ (~p )
†c∗~p ,λ(t)]
×
×[
u+λ′(~p )a~p ,λ′(t) − u−λ′(~p )c~p ,λ′(t)]
(395)====
∑
~p
∑
λ
E~p
(
|a~p ,λ(t)|2 − |c~p,λ(t)|2)
,
which results in the additive expression
H =∑
~p
∑
λ
E~p
(
|a~p ,λ|2 − |c~p ,λ|2)
. (396)
Looking at this minus sign, it is clear that it makes no sense to perform the quanti-
zation on the basis of bosonic creation and annihilation operators. The classical total
energy is not bounded from below and can therefore not be quantized as in the Klein–
Gordon theory. To compensate this minus sign we are forced to resort to the alterna-
tive many-particle formulation based on creation and annihilation operators that satisfy
fermionic anticommutation relations.
In this case the quantization in the Heisenberg picture goes as follows:
a~p ,λ (t) → a~p ,λ (t) ≡ a~p ,λ exp(−iE~p t/~) ,
c~p ,λ (t) → b†−~p ,λ (t) ≡ b†−~p ,λ exp(iE~p t/~) . (397)
The Dirac field becomes a field operator that satisfies the Dirac equation:
ψ(x) =∑
~p
∑
λ
√
mc2
E~p
u~p (~x )[
u+λ (~p ) a~p ,λ(t) + u−λ (~p ) b†−~p ,λ(t)
]
=∑
~p
∑
λ
√
mc2
VE~p
[
u+λ (~p ) a~p ,λ exp(−ip · x/~) + u−λ (−~p ) b†~p ,λ exp(ip · x/~)], (398)
150
with as before
p0 ≡ E~p/c =√
m2c2 + ~p 2 . (399)
Particle interpretation and its motivation: the operators a~p ,λ and a†~p ,λ annihilate
and create spin-1/2 particles with momentum ~p , positive energy E~p and helicity λ . The
operators b~p ,λ and b†~p ,λ annihilate and create the corresponding antiparticles with the same
spin, momentum, positive energy and helicity, but with opposite charge. As indicated, in
order to have an energy spectrum that is bounded from below we require the creation and
annihilation operators to satisfy fermionic anticommutation relations:
a~p ,λ(t), a
†~p ′,λ′(t)
=b~p ,λ(t), b
†~p ′,λ′(t)
= δ~p, ~p ′ δλ,λ′ 1 , all other anticommutators vanish .
(400)
Again this implies an identical-particle approach in which particles and antiparticles are
described in a joint way in the relativistic QM. Next we introduce the number operators
n(+)~p ,λ ≡ a†~p ,λ(t) a~p ,λ(t) = a†~p ,λ a~p ,λ and n
(−)~p ,λ ≡ b†~p ,λ(t) b~p ,λ(t) = b†~p ,λ b~p ,λ (401)
to count the number of particles and antiparticles. The total Hamilton operator of the free
non-interacting many-particle system can be written compactly as
∫
V
d~x ψ†(x) i~∂
∂tψ(x)
(380),(395),(398)=========
∑
~p
∑
λ
E~p
(
a†~p ,λ(t) a~p ,λ(t) − b−~p ,λ(t) b†−~p ,λ(t)
)
=∑
~p
∑
λ
E~p
(
n(+)~p ,λ + n
(−)−~p ,λ − 1
)
=∑
~p
∑
λ
E~p
(
n(+)~p ,λ + n
(−)~p ,λ − 1
)
= H . (402)
The energy spectrum is indeed nicely bounded from below thanks to the anticommutation
relations for b†~p ,λ(t) and b~p ,λ(t). In the many-particle formulation, the probability density
of § 3.2.1 receives the meaning of total charge operator of the many-particle system, since
Qtot ≡ q
∫
V
d~x ψ†(x)ψ(x)(380),(395),(398)========= q
∑
~p
∑
λ
(
a†~p ,λ(t) a~p ,λ(t) + b−~p ,λ(t) b†−~p ,λ(t)
)
(400),(401)======= q
∑
~p
∑
λ
(
n(+)~p ,λ − n
(−)~p ,λ + 1
)
= Qtot (403)
indeed counts particles with charge q and antiparticles with charge −q . Finally, the
total momentum operator of the free non-interacting many-particle system is given by the
additive expression
~P =∑
~p
∑
λ
~p(
n(+)~p ,λ + n
(−)~p ,λ
)
. (404)
151
Vacuum energy: in the expression for the Hamilton operator given above the vacuum
has an infinite negative energy, but it does define a lowest-energy state (ground state).
As a result of the fermionic quantization conditions the vacuum has a zero-point energy
per vibration mode that is opposite to the corresponding zero-point energy for bosonic
quantization conditions. It is very well possible that nature has equal amounts of bosonic
and fermionic degrees of freedom, allowing for an automatic cancellation of the infinities
in the zero-point energies. This idea is for instance used in supersymmetry , which links
bosonic to fermionic degrees of freedom by means of a symmetry. Quantum mechanically
speaking this cancellation of infinities is not essential, but cosmologically it is since an
energy density curves spacetime.
Locality: the requirement of locality forces us to impose the full set of anticommutation
relations (and not just the one for the b-operators). In exercise 32 it will be shown that in
that case
ψi(~x, t), ψi′(~x
′, t)
=ψ†i (~x, t), ψ
†i′(~x
′, t)
= 0 , (405)
ψi(~x, t), ψ
†i′(~x
′, t)
= δii′ δ(~x− ~x ′) 1 (i, i′ = 1 , · · · , 4) ,
causing the relevant anticommutators of the Dirac fields to vanish in all required situa-
tions. This would not have been the case if the creation and annihilation operators would
have satisfied bosonic commutation relations. In this context, an important thing to note
is that physical observables should consist of even numbers of Dirac fields, since physical
observations should not change under 2π rotations of space while Dirac spinors flip sign
(see exercixe 25). Products of even numbers of Dirac fields give rise to commutators of
observables that can be written in terms of anticommutators of Dirac fields (see the trick
on p. 10), allowing locality to be realized by means of either anticommutation relations or
commutation relations.
Spin – statistics theorem: spin and statistics are related in the relativistic QM. The quan-
tization of relativistic wave equations belonging to arbitrary spins will result in the generic
statement that bosons should have integer spin and fermions half integer spin.
In order to explicitly prove this theorem one should first of all demand the
energy spectrum to be bounded from below, This will determine whether the
creation and annihilation operators b†~p ,λ and b~p ,λ satisfy commutation relations
or anticommutation relations. From the requirement of locality then follows that
a†~p ,λ and a~p ,λ should satisfy the same operator algebra and that both sets of
operators should also commute or anticommute mutually.
152
5.2.1 Extra: the 1-(quasi)particle approximation for electrons
Starting from the many-particle theory for electrons and positrons (anti-electrons) we will
now try to understand the success of the 1-particle Dirac theory. Define to this end the
electron vacuum |0e〉 , with ψ(x)|0e〉 ≡ 0 and 〈0e|0e〉 ≡ 1 . (406)
This implies that ∀~p ,λ
a~p ,λ |0e〉 = b†~p ,λ |0e〉 = 0, i.e. this vacuum contains no electrons.
However, all possible 1-positron states are occupied in this vacuum, causing it to contain an
infinite number of positrons. We are dealing here with a quasi-particle vacuum belonging
to the quasi-particle annihilation operator ψ(x) in the Heisenberg picture. In view of
the exclusion principle, this way of constructing the electron vacuum exclusively works for
fermions and not for bosons.
Both adding an electron to the electron vacuum and removing a positron from the electron
vacuum can be regarded as the creation of a kind of 1-electron state, where in the latter
case we are dealing with a negative effective energy but electron-like effective charge − |e|with respect to the vacuum. In this quasi-particle picture an arbitrary 1-electron state can
be defined as a
1-charge state |1e〉 , with |1e〉 ≡∫
V
d~x ψ†(x)ψe(x)|0e〉 . (407)
This 1-charge state is defined in the Heisenberg picture, just as the field operator ψ(x).
This automatically implies that the time dependence of ψ†(x) should be compensated
by the time dependence of the spinor coefficient ψe(x). The coefficient ψe(x) will be
identified with the spinor wave function in the 1-electron Dirac theory. In exercise 32 it
will be proven that the states ψ†(x)|0e〉 have the following orthonormality property:
〈0e|ψi(~x, t)ψ†i′(~x
′, t)|0e〉 = δii′ δ(~x− ~x ′) 1 (i, i′ = 1 , · · · , 4) , (408)
from which we can deduce that
〈1e|1e〉 =
∫
V
d~x
∫
V
d~x ′ ψ†e(~x, t)〈0e|ψ(~x, t)ψ†(~x ′, t)|0e〉ψe(~x
′, t) =
∫
V
d~x ψ†e(~x, t)ψe(~x, t) ,
ψe(~x, t) =
∫
V
d~x ′ 〈0e|ψ(~x, t)ψ†(~x ′, t)|0e〉ψe(~x′, t)
(407)==== 〈0e|ψ(~x, t)|1e〉 . (409)
In this way the normalization of the spinor wave function ψe(x) follows automatically from
the normalization of the 1-charge state |1e〉 .
153
The charges of the above-mentioned many-particle states are given by
Qtot |0e〉(403)==== − |e|
∑
~p
∑
λ
(
n(+)~p ,λ − n
(−)~p ,λ + 1
)
|0e〉 = 0 |0e〉 , (410)
Qtot |1e〉(407)====
∫
V
d~x Qtot ψ†(x)|0e〉ψe(x)
(410)====
∫
V
d~x[Qtot, ψ
†(x)]|0e〉ψe(x)
(403)==== − |e|
∫
V
d~x ψ†(x)ψe(x)|0e〉(407)==== − |e| |1e〉 , (411)
using that n(+)~p ,λ |0e〉 = 0 |0e〉 and n
(−)~p ,λ |0e〉 = |0e〉 , as well as the relations
∑
~p
∑
λ
[n(+)~p ,λ − n
(−)~p ,λ + 1 , a†~p ′,λ′(t)
] (400),(401)======= a†~p ′,λ′(t)
and∑
~p
∑
λ
[n(+)~p ,λ − n
(−)~p ,λ + 1 , b~p ′,λ′(t)
] (400),(401)======= b~p ′,λ′(t) .
The terminology 1-charge state for |1e〉 thus makes good sense just like the designation
“vacuum” for |0e〉 , which finds further support in the expression for the total energy:
H |0e〉(402)====
∑
~p
∑
λ
E~p
(
n(+)~p ,λ + n
(−)~p ,λ − 1
)
|0e〉 = 0 |0e〉 . (412)
Finally we have
(
− i~c~α · ~ + βmc2)
ψe(x)(409)
======= 〈0e|(
− i~c~α · ~ + βmc2)
ψ(x)|1e〉
Dirac eqn.======= 〈0e|i~
∂
∂tψ(x)|1e〉
(409)==== i~
∂
∂tψe(x) . (413)
Hence, ψe(x) satisfies the Dirac equation, it is normalizable and it contains negative-
energy components. As such, ψe(x) can be identified with the spinor wave function in the
1-electron Dirac theory.
The 1-particle Dirac theory is therefore related to a specific 1-(quasi)particle approxima-
tion of the many-particle theory. This approximation can be applied successfully as long as
no fluctuations of order mec2 occur in the energy on time scales of order ∆t = O(~/mec
2)
or on length scales of order ∆r = O(~/mec). If such fluctuations would occur, effects from
the creation/annihilation of electrons and positrons should be taken into account. For
instance, electron–positron annihilation can occur, i.e. the emission of a photon caused
by the annihilation of an electron with a positron from the electron vacuum. The in-
verse of this photon-emission process is called pair creation. For calculating such effects
154
quantum field theory is the tool of choice, which describes all particles (electrons, positrons,
photons, . . . ) by means of field operators and represents the field equations including in-
teractions by means of Lagrangians. These interactions have their origin in so-called
gauge symmetries and the associated charge-conservation laws, such as conservation of
total electric charge in the case of electromagnetic interactions.
155
A Fourier series and Fourier integrals
Fourier decompositions are used quite frequently in these lecture notes. We briefly list the
relevant aspects of these decompositions in this appendix.
A.1 Fourier series
Systems with a ring-shaped structure and systems that are enclosed spatially can be de-
scribed in terms of state functions with a certain degree of periodicity. Consider to this
end a 1-dimensional function f(x), for which the function as well as the spatial derivative
are stepwise continuous on the interval x ∈(−L/2 , L/2
)and which additionally satisfies
the spatial periodicity condition f(x) = f(x+L). From Fourier analysis it follows (see the
lecture course “Trillingen en Golven”) that such a function can be decomposed in terms of
a basis of periodic plane waves (Fourier modes):
fn(x) =1√L
exp(2πinx/L) ≡ 1√L
exp(ikx) (n = 0 ,±1 ,±2 , · · ·) . (A.1)
We readily recognize these functions as being momentum eigenstates belonging to the
quantized momentum eigenvalues ~k . This basis satisfies the orthonormality relation
1
L
x0+L∫
x0
dx exp(ix[k − k′]
)= δkk′ , (A.2)
where x0 can be chosen arbitrarily. Since x is usually restricted to the interval(−L/2 , L/2
),
a handy choice is x0 = −L/2. By employing the completeness relation for periodic Fourier
modes, the periodic function f(x) can be written as a Fourier series:
f(x) =∞∑
n=−∞
An√L
exp(2πinx/L) (A.3)
=A0√L
+∞∑
n=1
[ An + A−n√L
cos(2πnx/L) + iAn − A−n√
Lsin(2πnx/L)
]
≡ B0√L
+∞∑
n=1
√
2
L
[
Bn cos(2πnx/L) + Cn sin(2πnx/L)]
, (A.4)
with
An =1√L
L/2∫
−L/2
dx f(x) exp(−2πinx/L) . (A.5)
Periodic systems vs enclosed systems: the close relation between periodic systems and sys-
tems that are contained inside an enclosure can be seen as follows. If we assume the im-
penetrable walls of the enclosure to be positioned at x = 0 and x = L, the state functions
i
have to satisfy the continuity condition ψ(0) = ψ(L) = 0 (see § 2.5). Inside the enclo-
sure this function coincides with a periodic function f(x) = f(x + 2L) with odd parity,
so that f(x) = − f(−x). As a result of this extra condition, only the standing waves
∝ sin(πnx/L) for n = 1 , 2 , · · · contribute to the Fourier series (A.4), which indeed guar-
antees that f(0) = f(L) = 0. The periodicity length is 2L now and at the same time n is
allowed to take on positive values only. The number of allowed quantized k values is in this
case identical to the number of k values for a system with periodicity length L, with the ex-
ception of the absence of the quantum number k = 0. The presence of the edge of the box
manifests itself in the absence of the constant Fourier mode with quantum number k = 0.
The influence of the edge of the enclosure is only felt when the quantum states with the
lowest absolute values for the momentum (and thus the largest de Broglie wavelengths)
contribute significantly to the description of the system. This happens for instance in
bosonic gases at low temperatures (see Ch. 2).
A.2 Fourier integrals
The Fourier decomposition of non-periodic quadratically integrable functions can be ob-
tained by taking the continuum limit L → ∞ of the periodic case given above. During
this transition the resulting basis of plane waves should be normalized to a δ function.
This is based on the fact that k = 2πn/L, causing the k-spectrum to become very closely
spaced for L→ ∞ and the summation over n to become an integral over k :
∞∑
n=−∞
L→∞−−→ L
2π
∞∫
−∞
dk . (A.6)
The relevant basis functions are the momentum eigenfunctions belonging to the momentum
eigenvalues ~k :
fk(x) = fn(x)
√
L
2π
(A.1)====
1√2π
exp(ikx) (k ∈ IR) , (A.7)
with orthonormality relation
1
2π
∞∫
−∞
dx exp(ix[k − k′]
)= δ(k − k′) . (A.8)
The Fourier series now becomes a Fourier transformation (Fourier integral):
f(x) =1√2π
∞∫
−∞
dk A(k) exp(ikx) and A(k) =1√2π
∞∫
−∞
dx f(x) exp(−ikx) .
(A.9)
ii
A.2.1 Definition of the δ function
The δ function in equation (A.8) is defined on an arbitrary continuous test function g(x):
∀g
g(x) =
∞∫
−∞
dx′ g(x′)δ(x− x′) . (A.10)
Since g(x) is arbitrary, δ(x − x′) should vanish for x 6= x′ . The peak of δ(x − x′) at
x = x′ should be such that the integral over δ(x− x′) around x = x′ always yields one.
The δ function therefore only exists as a limit of a series of functions that progressively
become more peaked around x = x′. The result of such a limiting procedure is called a
distribution. An example of this is given by the integral
δǫ(x) =1
2π
∞∫
−∞
dk exp(ikx− ǫ|k|) =1
2π
( 1
ix+ ǫ− 1
ix− ǫ
)
=1
π
ǫ
x2 + ǫ2(ǫ > 0) .
For decreasing ǫ the function δǫ(x) becomes smaller for all x 6= 0, whereas simultaneously
x2>0∫
x1<0
dx δǫ(x)v=x/ǫ====
1
π
x2/ǫ∫
x1/ǫ
dv
v2 + 1=
1
π
(
arctan(x2/ǫ)− arctan(x1/ǫ)) ǫ↓0
−−→ 1 .
The following series representations of the δ function will be used in these lecture notes:
δ(x) = limǫ↓0
1
π
ǫ
x2 + ǫ2=
1
2π
∞∫
−∞
dk exp(ikx) , (A.11)
δ(x) = limǫ↓0
θ(x+ ǫ)− θ(x)
ǫ= θ′(x) , (A.12)
where θ(x) is the usual Heaviside step function.
The following handy identities hold for the δ function:
g(x)δ(x− c) = g(c)δ(x− c) , (A.13)
as well as
δ(h(x)
)=∑
j
1
|h′(xj)|δ(x− xj) for h(xj) = 0 and h′(xj) 6= 0 , (A.14)
provided that h(x) is sufficiently well-behaved close to the roots of the equation h(x) = 0.
In three dimensions, finally, the following compact notation can be used:
δ(~r ) ≡ δ(x)δ(y)δ(z) ⇒ 1
(2π)3
∞∫
−∞
dkx
∞∫
−∞
dky
∞∫
−∞
dkz exp(i~k ·~r ) (A.11)==== δ(~r ) . (A.15)
iii
B Properties of the Pauli spin matrices
Since the world of QM is littered with spin-1/2 particles (such as electrons, nucleons, etc.),
we give here a short summary of the properties of the Pauli spin matrices
σx =
(
0 1
1 0
)
, σy =
(
0 − i
i 0
)
and σz =
(
1 0
0 −1
)
. (B.1)
These 2×2 matrices are linked directly to the spin operator ~S for spin-1/2 particles:
~σ ≡ 2
~
~S . (B.2)
From the fundamental commutator algebra for the spin operator and the fact that Sz has
eigenvalues ± ~/2, the following properties of the Pauli spin matrices can be derived:
σk = σ†k , Tr(σk) = 0 , det(σk) = −1 (k = x, y, z)
and[σx, σy
]= 2iσz (cyclic) . (B.3)
Denoting the identity matrix in spin space by I2, the following additional identities hold
for the Pauli spin matrices:
σ2k = I2 (k = x, y, z) and σxσy = −σyσx = iσz (cyclic)
⇒σj , σk
= 2δjk I2 (j , k = x, y, z ) . (B.4)
This allows us to complete the relevant trace identities in spin space to
Tr(I2) = 2 , Tr(σj)(B.3)==== 0 and Tr(σjσk)
(B.4)==== 2δjk (j , k = x, y, z ) .
(B.5)
Moreover, for arbitrary 3-dimensional vectors ~A and ~B the following holds:
(~σ · ~A )(~σ · ~B ) =1
2
∑
j,k
AjBk
( [σj , σk
]+σj , σk
) (B.3),(B.4)======= ( ~A · ~B )I2 + i~σ · ( ~A× ~B ) ,
(B.6)using that
∑
j,k
AjBk
2
[σj , σk
] (B.3)==== iσz (AxBy −AyBx)+ iσx (AyBz −AzBy)+ iσy (AzBx−AxBz) .
Combined with the identity matrix the Pauli spin matrices form a basis of 2×2 matrices.
As such, an arbitrary 2×2 matrix A can be decomposed as
A = A0 I2 + ~A · ~σ (A0 ∈ C and Ax,y,z ∈ C) . (B.7)
iv
C Lagrange-multiplier method
Suppose we want to determine the extrema of a quantity F (q1, · · · , qN) while simultane-
ously trying to satisfy the boundary condition f(q1, · · · , qN) = C . In that case it makes
good sense to employ the so-called Lagrange-multiplier method. This consists in looking
for solutions to the variational equation
δ(
F (q1, · · · , qN) − λ(q1, · · · , qN)f(q1, · · · , qN)− C
)
= 0 , (C.1)
where δ denotes the action of independently varying the coordinates q1 , · · · , qN as well as
the Lagrange multiplier λ . This results in
N∑
j=1
( ∂F
∂qj− (f − C)
∂λ
∂qj− λ
∂f
∂qj
)
δqj − (f − C) δλ = 0 ∀δλ, δqj
⇒ f(q1, · · · , qN) = C and∂F
∂qj= λ
∂f
∂qj(j = 1 , · · · , N ) . (C.2)
In this way the boundary condition is indeed satisfied automatically. By substituting
the obtained solution(s) for q1 , · · · , qN in the boundary condition f(q1, · · · , qN) = C , the
Lagrange multiplier λ can be related to the constant C . The Lagrange-multiplier method
is used in these lecture notes to maximize the entropy of ensembles in thermal equilibrium
while simultaneously imposing certain additional constraints.
D Special relativity: conventions and definitions
In this appendix we specify a few conventions that will allow us to deal with flat spacetime
(Minkowski space). The time coordinate t and spatial coordinates ~r of a particle are
combined into a contravariant position 4-vector
xµ : x0 ≡ ct and
x1
x2
x3
≡ ~x ≡ ~r , (D.1)
with µ = 0 , 1 , 2 , 3 the Minkowski index. Also the energy E and momentum ~p are
combined into a contravariant momentum 4-vector
pµ : p0 ≡ E/c =mc
√
1−~v 2/c2and
p1
p2
p3
≡ ~p =m~v
√
1−~v 2/c2, (D.2)
corresponding to a particle with rest mass m and velocity ~v = d~r/dt.
v
The metric tensor in Minkowski space is given by
gµν = gµν =
1 Ø−1
−1Ø −1
. (D.3)
From the contravariant vector aµ the corresponding covariant vector aµ can be derived as
aµ ≡ gµν aν . (D.4)
Here we made use of the Einstein summation convention, which states that a repeated
Minkowski index automatically implies summation of that index. Finally we introduce the
inner product (scalar product)
x · y ≡ xµyµ = xµyµ = gµν x
µyν = x0y0 − ~x · ~y , (D.5)
which is not positive definite as a result of the spacetime metric gµν in Minkowski space.
Inertial systems: reference frames in which force-free particles travel along straight lines
are referred to as inertial systems.
Relativity principle: all inertial systems are physically equivalent and the speed of light
is the same in each inertial system.
The relativity principle is reflected in the transformation characteristic of position vectors
while going from inertial system S with origin O to inertial system S ′ with origin O′.
Assume the origins to coincide (i.e. O=O′ ) at time t = 0 and indicate the corresponding
coordinate transformation as follows:
S : xµ → S ′ : x′µ = Λµν x
ν . (D.6)
The coordinate transformation (Lorentz transformation) Λ has to be linear to be able
to map linear motions on linear motions in the absence of forces. It can be defined in
accordance with the relativity principle by demanding x2 ≡ x ·x to be invariant under the
transformation. After all, in that case a particle that moves at the speed of light starting
from the origin O at t = 0, which implies c2 = ~v 2 = ~x 2/t2 ⇒ x2 = 0, automatically
travels with the speed of light in any other inertial system S ′, since x′ 2 = x2 = 0 implies
~v ′ 2 = ~x ′ 2/t′2 = c2. From this condition we can derive that
∀xx2 = x′ 2
(D.5),(D.6)======⇒ ∀
xgρσ x
ρxσ = ΛµρΛ
νσ gµν x
ρxσ ⇒ ΛµρΛµσ = gρσ , (D.7)
which implies that (Λ00)
2 = 1 + (Λ10)
2 + (Λ20)
2 + (Λ30)
2 ≥ 1. The inverse of the Lorentz
transformation then becomes
ΛµρΛµσ = gρσ =
1 Ø1
1Ø 1
⇒ Λµρ = (Λ−1)ρµ , det(Λ) = ± 1 . (D.8)
vi
The Lorentz transformations can be subdivided as follows. First of all there are the so-
called proper orthochronous Lorentz transformations (or short: “the Lorentz transforma-
tions”), for which det(Λ) = +1 and Λ00 ≥ 1. These transformations comprise:
• spatial rotations, about three independent spatial axes.
• Lorentz boosts, in three independent spatial directions. Example: S ′ moves relative
to S at constant velocity v = βc in the x1-direction. In that case
Λµν =
1/√
1−β2 −β/√
1−β2 0 0
−β/√
1−β2 1/√
1− β2 0 0
0 0 1 0
0 0 0 1
≡
cosh η − sinh η 0 0
− sinh η cosh η 0 0
0 0 1 0
0 0 0 1
(D.9)
β≪ 1, x0=ct−−−−−−→ non-relativistic Galilei transformation :
1 0 0 0− v/c 1 0 00 0 1 00 0 0 1
.
Two more classes of Lorentz transformations can be added to this: transformations with
det(Λ) = − 1 and Λ00 ≥ 1, such as the parity transformation that describes spatial inver-
sion, and transformations with Λ00 ≤ − 1, such as the time-reversal transformation. All
transformations can be combined into the so-called homogeneous Lorentz transformations,
for which an arbitrary transformation can be written as a product of rotations, boosts, spa-
tial inversion and time reversal. Spatial inversion and time reversal have a special status:
• free-particle systems are invariant under these transformations, but this invariance
gets lost once the weak (nuclear) force enters the game;
• these transformations cannot be connected continuously with the identity transfor-
mation, as opposed to the proper orthochronous Lorentz transformations which form
a continuous group and for which a finite transformation can be written as an in-
finite series of infinitesimal transformations. Spatial inversion and time reversal in
fact incorporate the possible discrete Lorentz transformations.
Finally, we also have to consider the possibility that the origin O′ is shifted with respect
to the origin O :
x′µ = Λµν x
ν + aµ (short: x′ = Λx+ a) , with aµ a constant 4-vector
⇒ xµ = (Λ−1)µν (x′ − a)ν (short: x = Λ−1[x′ − a] ) . (D.10)
All transformations together form the so-called inhomogeneous Lorentz transformations,
which are also known as the Poincare transformations. They are defined by demanding
the invariance (x− y)2 = (x′ − y′)2 .
vii
For the construction of a relativistic wave equation, which should have the same form in
all inertial systems, we need several types of quantities (independent of space and time)
and fields (dependent on space and time).
• Scalar quantities: quantities that remain the same in each inertial system, such as
p2 = p · p and the speed of light c.
• Scalar fields: fields with the following transformation characteristic
S : φ(x) → S ′ : φ′(x′) = φ(x)(D.10)==== φ
(Λ−1[x′ − a]
).
• Contravariant vector quantities: 4-vectors that transform as A′ µ = Λµν A
ν , such as
the momentum 4-vector pµ.
• Contravariant vector fields: fields with the following transformation characteristic
S : Aµ(x) → S ′ : A′µ(x′) = Λµν A
ν(x)(D.10)==== Λµ
ν Aν(Λ−1[x′ − a]
).
• Covariant vector quantities: 4-vectors that transform as
A′µ = gµρA
′ ρ = gµρΛρσA
σ = gµρΛρσ g
σνAν = Λ νµ Aν
(D.8)==== (Λ−1)νµAν .
• Covariant vector fields : fields with the following transformation characteristic
S : Aµ(x) → S ′ : A′µ(x
′) = gµρA′ ρ(x′) = gµρΛ
ρσA
σ(x) = gµρΛρσ g
σνAν(x)
(D.10)==== Λ ν
µ Aν
(Λ−1[x′ − a]
) (D.8)==== (Λ−1)νµAν
(Λ−1[x′ − a]
).
Examples: the gradient in Minkowski space,
∂µ ≡ ∂
∂xµ⇒ ∂0 =
1
c
∂
∂tand
∂1∂2∂3
= ~ , (D.11)
is a covariant vector quantity since
∂ ′µ =
∂
∂x′µ=( ∂xν
∂x′µ
) ∂
∂xν(D.10)==== (Λ−1)νµ
∂
∂xν(D.8)==== Λ ν
µ
∂
∂xν= Λ ν
µ ∂ν .
Consequently, ∂µ = gµν ∂ν is automatically a contravariant vector quantity. Furthermore
• Aµ(x) ≡ ∂µφ(x) is a covariant vector field if φ(x) is a scalar field.
Proof: A′µ(x
′) = ∂ ′µφ
′(x′) = Λ νµ ∂ν φ(x) = Λ ν
µ Aν(x).
• φ(x) ≡ Aµ(x)Bµ(x) is a scalar field if A and B are vector fields.
Proof: φ′(x′) = A′µ(x
′)B′µ(x′) = Aν(x) (Λ−1)νµΛ
µρB
ρ(x) = Aν(x)Bν(x) = φ(x).
• φ(x) ≡ ∂µAµ(x) is a scalar field if Aµ(x) is a contravariant vector field (see previous
example).
viii
E Electromagnetic orthonormality relations
Consider the periodic electromagnetic Fourier components
~u~k,λ(~r ) =1√V~ǫλ(~ek) exp(i~k · ~r ) , (E.1)
where the wave vector ~k is allowed to take on the following quantized values:
~k ≡ k ~ek ≡ ωk
c~ek =
2π
L~ν , with νx,y,z = 0 ,±1 ,±2 , · · · ∧ ~ν 6= ~0 . (E.2)
Together with the propagation direction ~ek the polarisation vectors ~ǫ1(~ek) and ~ǫ2(~ek)
form an orthonormal right-handed coordinate system:
~ǫ1(~ek)×~ǫ2(~ek) = ~ek , ~ǫ2(~ek)× ~ek = ~ǫ1(~ek) , ~ek ×~ǫ1(~ek) = ~ǫ2(~ek) ,
~ǫλ(~ek) · ~ǫλ′(~ek) = δλ,λ′ . (E.3)
By means of the orthonormality relation for periodic Fourier modes:
1
V
∫
V
d~r exp(i~r · [~k − ~k′ ]
)= δ~k,~k′ , (E.4)
the following identities can be derived for the Fourier components:
∫
V
d~r ~u~k,λ(~r ) · ~u ∗~k′,λ′
(~r )(E.4)==== ~ǫλ(~ek) · ~ǫλ′(~ek) δ~k,~k′
(E.3)==== δλ,λ′ δ~k,~k′ , (E.5)
∫
V
d~r(
~ek × ~u~k,λ(~r ))
·(
~ek′ × ~u ∗~k′,λ′
(~r ))
(E.4)====
(
~ek ×~ǫλ(~ek))
·(
~ek ×~ǫλ′(~ek))
δ~k,~k′
(E.3)==== δλ,λ′ δ~k,~k′ , (E.6)
∫
V
d~r ~u~k,λ(~r )×(
~ek′ × ~u ∗~k′,λ′
(~r ))
(E.4)==== ~ǫλ(~ek)×
(
~ek ×~ǫλ′(~ek))
δ~k,~k′
(E.3)==== ~ek δλ,λ′ δ~k,~k′ , (E.7)
∫
V
d~r ~u~k,λ(~r )× ~u ∗~k′,λ′
(~r )(E.4)==== ~ǫλ(~ek)×~ǫλ′(~ek) δ~k,~k′
(E.3)==== ~ek (δλ,1δλ′,2 − δλ,2δλ′,1 ) δ~k,~k′ . (E.8)
ix
Index
γ-matrices of Dirac, 112
absorption, 137
additive many-particle quantities
1-particle quantities, 16
2-particle quantities, 18
annihilation operator, 7
anticommutation relations, 10, 13
anticommutator, 10
atomic heat bath, 142
average occupation numbers, 88
binding energy, 75
for nuclei, 82
black body, 142
Bogolyubov transformation, 38
bosonic, 39
fermionic, 47
Boltzmann constant, 60
Bose gas, 93
degenerate, 94
Bose–Einstein condensation, 90, 94
critical temperature, 93
Bose–Einstein statistics, 2, 87
bosons, 2, 10
box, 69
Casimir effect, 133
Chandrasekhar limit, 86
charge conjugation, 144
charge number, 80
chemical potential, 68, 99
classical gas, 92
coherent state, 27
commutation relations, 10, 13
Compton wavelength, 122
conduction electrons, 77
continuity equation, 106, 111
continuum limit, 71, 76
for the e.m. field, 134
contravariant vector fields, viii
contravariant vector quantities, viii
Cooper instability, 46
Coulomb conditions, 125
Coulomb gauge, 125
covariant vector fields, viii
covariant vector quantities, viii
creation operator, 6
Curie’s law, 65
d’Alembertian, 104
Darwin term, 120
de Broglie hypothesis, 103
density matrix, 50, 53
density of states, 71
for photons, 132
density operator, 53
equation of motion, 59
dipole selection rules, 141
Dirac equation, 109
anticommutation relations, 150
Dirac spinors, 110
energy density, 149
field operator, 150
spin-1/2 antiparticles, 151
spin-1/2 particles, 115, 151
total charge operator, 151
Dirac representation, 111
Dirac sea, 116
Dirac spin operator, 115
distinguishable, 1
electric dipole approximation, 140
electromagnetism
4-current, 124
4-vector potential, 118, 124
x
classical limit, 138
commutation relations, 129
field operator, 129
spin-1 particles, 130
electron gas, 77
electron–hole excitation, 45
emission, 137
spontaneous, 138
lifetime, 141
stimulated, 138
energy quanta, 23, 25
photon, 130
ensemble average, 53
time evolution, 59
ensembles
canonical, 62
completely random, 58, 59
grand canonical, 67
microcanonical, 65
mixed, 51, 57
pure, 49, 57, 59
unpolarized, see completely random
entropy, 60
equipartition of energy, 64
Fermi energy, 45, 74, 76, 89
Fermi gas, 69, 98
completely degenerate, 74, 99
degenerate, 98
Fermi momentum, 76
Fermi sea, 45, 74
Fermi sphere, 76
Fermi surface, 76
Fermi temperature, 75
Fermi–Dirac statistics, 2, 87
fermions, 2, 10
field operator, 15
fine structure, 121
Fock space, 5
fontain effect, 43
Fourier integral, ii
Fourier modes, i
Fourier series, i
Fourier transformation, ii
freezing out degrees of freedom, 90
fugacity, 68
g-factor, 120
gauge freedom, 125
heat bath, 62
heat capacity, 92
Heisenberg equation of motion, 21
Heisenberg picture, 21
helicity, 118, 130
helium film, 44
helium I, 42
helium II, 42
hole, 47
hole theory, 45
ideal gas, 3, 90
identical particles, 1
incoherent mixture, 51
indistinguishable, 1
inertial system, vi
intrinsic degrees of freedom, 105
intrinsic magnetic dipole moment, 120
isolated system, 62
Klein–Gordon equation, 104
commutation relations, 148
energy density, 146
scalar field, 106
scalar field operator, 147
spin-0 antiparticles, 147
spin-0 particles, 106, 147
total charge operator, 148
Lagrange multiplier, v
Lagrange-multiplier method, v
xi
law of Stefan–Boltzmann, 142
left-handed circularly polarized light, 131
Liouville equation, 59
locality, 148
long-range correlations, 96
long-range order, 96
Lorentz transformations, vi
homogeneous, vii
Lorentz boosts, vii
orthochronous, vii
proper Lorentz transformations, vii
spatial rotations, vii
lowering operator, 24
macroscopically occupied, 94
magnetization, 65
magnetization density, 100
mass number, 80
Maxwell equations, 123
Maxwell–Boltzmann statistics, 87
microcausality, 149
minimal substitution, 118, 135
Minkowski space, v
contravariant vector, v
covariant vector, vi
Einstein summation convention, vi
inner product, vi
metric tensor, v
Minkowski index, v
neutron gas, 87
neutron star, 87
non-interacting many-particle systems, 3
normal ordering, 19, 132
nucleon distribution, 80
number density operator, 14
number operators, 6, 7
observables for identical particles, 1
occupation numbers, 6
distributions, 88
occupation-number representation, 6
order parameter, 96
oscillators
bosonic quanta, 25, 27
forced, 26
free, 23
overlapping wave functions, 1
pair number operator, 20
paramagnetic susceptibility, 65, 101
particle interpretation, 23
particle–wave duality, 103, 145
partition function
canonical, 63
grand canonical, 68
Pauli equation, 120
Pauli exclusion principle, 5, 11
Pauli spin matrices, iv, 56
periodic plane waves, i
permutation degeneracy, 1
permutation operator, 2
persistent current, 43
photon gas, 142
photon states, 131
photon statistics, 142
photons, 130
Planck’s law, 142
plasma, 77
Poincare transformations, vii
polarization for photons
polarization vector, 126
polarization for spin-1/2 particles
degree of polarization, 57
direction of polarization, 57
polarization vector, 56
pressure ionization, 84
pure state, 49, 54
quasi particles, 26, 27, 38, 41
quasi-particle vacuum, 27
xii
raising operator, 24
relativistic quantum field theory, 145
relativity principle, vi, 103
reservoir, 66
right-handed circularly polarized light, 131
Rollin film, 44
saturation, 83
scalar fields, viii
scalar potential, 123
scalar quantities, viii
Schrodinger field, 22
Schrodinger picture, 21
second quantization, 18, 22, 135, 144–152
slash notation, 112
Slater determinant, 4
spin–orbit interaction, 120
spin–statistics theorem, 152
state functions
product functions, 4
totally antisymmetric, 2, 14
totally symmetric, 2, 14
statistical weight, 53
super leak, 43
superfluidity, 31
critical velocity, 32
Landau criterion, 32
symmetrization postulate, 1
temperature, 63
thermal de Broglie wavelength, 91
thermal equilibrium, 61
Thomas–Fermi model, 80
total number operator, 8, 14
transversality, 126
two-fluid model, 42
vacuum state, 6
vector potential, 123
wave vector, 71, 76, 126
white dwarf, 84
zero-point energy, 25, 132
xiii