Physics 551, Problem Set 8due: Wednesday, November 14 at 10am

grader: M. YorkPlease give your completed problem sets to me at the beginning of class or place them in the “Physics551” box in the physics department mailroom (Rutherford 103b) before the due date. You areencouraged to discuss these problems with your colleagues, but you must write up your own solutions;the solutions you hand in should reflect your own work and understanding. Late problem sets willbe penalized 10% per day late, unless an extension has been obtained from me or the TA before thedue date. Late problem sets will not be accepted after solutions are handed out.

Announcement: The take-home exam will be distributed by email on Sunday, December 2 at Noon.It will be due on Monday, December 3 at Noon. If this scheduling presents a problem for you, pleasecontact me as soon as possible.

Reading: Chapters 1, 2, 3 & 4 of Sakurai & Napolitano.

1. Consider a pair of uncoupled harmonic oscillators with equal frequency, with raising/loweringoperators a± and a†±. In an earlier problem set you showed that T± = h̄a†±a∓ and T3 =h̄2(a†+a+ − a†−a−) obey the usual SU(2) algebra and that the casimir T 2 is given by

T 2 =h̄2

2N(

N

2+ 1)

where N = a†+a+ + a†−a− is the total number operator. This SU(2) symmetry is extremelyuseful for understanding the spectrum of the theory, and this also gives a nice way of visualizingthe representations of SU(2).

(a) Consider the two states of the system with N = 1, a†+|0〉 and a†−|0〉. Compute the matrixelements of T3 and T± between these states. How do these matrix elements compare tothe usual form for SU(2) generators in a spin 1/2 representation? Conclude that thesetwo states transform in the usual spin 1/2 representation of SU(2).

(b) The energy eigenstates of the theory take the form

|n+, n−〉 = Cn+,n−(a†+)n+(a†−)n−|0〉

where n± is the number of excitations of the ± oscillator and Cn+,n− is some normalizationconstant. What are the allowed energies of the system? How many states are there in eachenergy level? Show that T± commutes with N . Show that the states in each energy leveltransform in an irreducible representation of SU(2). Which representation is it? Concludethat the Hilbert space of the system is the direct sum

H = H0 ⊕H1/2 ⊕H1 ⊕ . . .

where Hj is the Hilbert space of the spin j representation of SU(2).

(c) So far we have only discussed infinitesimal SU(2) rotations, but it is easy to understandfinite SU(2) transformations. Consider the transformation(

a+

a−

)→ U

(a+

a−

)

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for some complex matrix U . How do the a†± transform? Show that the Hamiltonian isunchanged if U is an element of SU(2).

2. SU(2) arises in nature in many ways, not all of which involve spatial rotations. It turns out thatthe two lightest quarks (called up and down) are related by an approximate SU(2) symmetrycalled isospin; this is an approximate symmetry which is valid when we ignore the fact thatthe two quarks actually have different mass, but it can be used to understand a lot of physics!Explicitly, this means that if we denote by |+〉 the state of an up quark, and by |−〉 the stateof a down quark we can define the operators

T+ = |+〉〈−|, T− = |−〉〈+|, T3 =1

2(|+〉〈+| − |−〉〈−|)

which obey the SU(2) algebra. This means that the up and down quarks live in the twodimensional (i.e. spin 1/2) representation of SU(2). We will call this SU(2) symmetry “isospin”to distinguish it from the normal spin of a particle associated with spatial rotations.

(a) These quarks undergo complicated interactions (which we will not try to explain) whichcause them to form bound states. However, we will assume that the Hamlitonian describingthese interactions commutes with the SU(2) operators we defined above; this assumptionturns out to be a good approximation in many cases. This means that as far as theisospin transformation properties are concerned we can ignore the interactions betweenthe quarks. In particular, we can organize the quark bound states into representations ofSU(2) just as if they were non-interacting spin 1/2 particles.

i. Show that there are four possible bound states of two quarks, three of which live inan isospin 1 representation and the other of which has isospin 0. The triplet of isospin1 particles are referred to as pions, and the isospin 0 particle as an eta meson.

ii. Show that there are eight possible bound states of three quarks, four of which live intwo isospin 1/2 representations and four of which live in a isospin 3/2 representation.One of the isospin 1/2 representations describes the nucleons – i.e. the proton andthe neutron. The isospin 3/2 representation describes the Delta particles. The otherisospin 1/2 representation involves an unbound quark and is not observed in naturebecause of quark confinement.

(b) Argue that this model explains why the proton and neutron have nearly the same mass.Use this model to predict that the masses of the three pion states, as well as the massesof the four Delta states, are roughly the same. Look up these particle masses and checkyour predictions.

3. Consider a particle moving in three dimensions in a harmonic oscillator potential

H =1

2m~p2 +

ω

2~x2

(a) Write the Hamiltonian in terms of the three raising and lowering operators ai and a†i ,where i = 1, 2, 3.

(b) Show that the Hamiltonian of this system is invariant under the transformation a1

a2

a3

→ U

a1

a2

a3

2

Where U is a unitary 3 by 3 matrix with determinant 1.1 This means that the system hasan SU(3) symmetry.

(c) In fact, this SU(3) symmetry also appears in particle physics just as SU(2) isospin did. Itis an approximate symmetry where the three lightest quarks (up, down and strange) aretreated as having equal mass. This means that particle states organize into representationsof what is known as SU(3) flavour symmetry; the organization of particle states intoSU(3) representations is known as “the eight-fold way.” To understand the origin of thissomewhat melodramatic terminology, compute the dimension of the group SU(3), i.e. thenumber of continuous parameters required to specify an SU(3) matrix.

(d) What are the allowed energies of the system? How many states are there in each energylevel? Argue that there must be a representation of SU(3) with dimension 1

2(n+ 1)(n+ 2)

for any integer n. In fact, the representation theory of SU(3) is much more complicatedthan that of SU(2).

(e) How does the usual SO(3) rotation symmetry act on the raising and lowering operators?Show that this SU(3) symmetry includes the usual SO(3) rotation symmetry. You canthinks of this SU(3) as an enhanced version of the rotation symmetry that arises becauseof the linear force law for harmonic oscillators.

(f) Consider a particle moving in three dimensions under a more general potential

H =1

2m~p2 +

ω

2~x2 + εf(~x2)

In general, rotation symmetry implies that the states can be organized into representationswith integer angular momentum l (i.e. into spherical harmonics). Argue that when ε issmall, many different angular momentum states must be approximately degenerate withenergy splittings that are linear in ε. It is possible, using group theory, to understandexactly which different angular momentum states will become degenerate, but you neednot do so.

4. In this problem we will investigate a different kind of continuous symmetry, known as super-symmetry.

(a) Consider the one dimensional Hamiltonian

H1 =1

2mp2 + V1(x)

We will assume that this Hamiltonian has a ground state with energy 0 (if not, we couldalways just add a constant to V1). Define the operator A = W (x) + i 1√

2mp where W is

some (real) function of x. We want to ask whether it is possible to write the Hamiltonianas

H1 = A†A

How must V1(x) be related to W (x) in order for the Hamiltonian to take this form? Findan expression for the ground state wave function in terms of W (x).

1Actually, the assumption that the determinant equals one is not necessary, but the case where U has determinantnot equal to one is not relevant for this problem.

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(b) Consider a second Hamiltonian H2 = AA†. Show that this Hamiltonian takes the form

H2 =1

2mp2 + V2(x)

where V2(x) is a function you should compute in terms of W (x).

(c) How are the eigenstates and eigenvalues of H1 related to those of H2? Show that if youknow all of the eigenstates of H1 you can determine those of H2. (Hint: If |ψ〉 is aneigenstate of H1, consider A|ψ〉).

(d) Assume that W (x) and W ′(x) go to zero at x→ ±∞. Then we can consider the scatteringof an incident wave off of this potential. This means we should look for a solution ψ(x) tothe time independent Schrodinger equation which describes an incident plane wave eikx.Such a solution will look like

ψ(x) ∼{eikx +Re−ikx, x→ −∞

Teikx x→∞

where R(k) and T (k) are the transmission and reflection coefficients, which depend on thefrequency k. How are the reflection and transmission coefficients of H1 related to thoseof H2? Show that if you know the reflection and transmission coefficients of H1 you candetermine those of H2.

(e) We can define one “super-Hamiltonian” which combines both of these Hamiltonians, andis given by the two by two matrix

H =(H1 0

0 H2

)Define the operators

Q =(

0 0

A 0

), Q† =

(0 A†

0 0

),

Compute [H,Q] and [H,Q†]. Your answer shows that these operators generate a symme-try of the Hamiltonian. This symmetry is known as supersymmetry, and is responsiblefor the results you derived earlier. Compute {Q,Q†} and Q2. Note that because Q2 = 0,and because the algebra of charges is formed using anti-commutators rather than com-mutators, it is natural to interpret Q as a fermionic rather than bosonic operator. Thuswe can interpret the symmetry described above as relating bosonic and fermionic degreesof freedom. There are some hints that the Hamiltonian of nature is supersymmetric athigh energies; there is a real possibility that supersymmetry will be discovered at the LHCwithin the next few years.

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