review of quantum mechanics imrana ashraf zahid department of physics quaid-i-azam university...
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Quantum Mechanics: RevisitedTRANSCRIPT
REVIEW OF QUANTUM MECHANICS
IMRANA ASHRAF ZAHIDDEPARTMENT OF PHYSICSQUAID-I-AZAM UNIVERSITY
ISLAMABAD, PAKISTAN
Layout• Quantum Mechanics: Revisited• Radiative Processes for isolated atoms• Transition rates: Semi-classical• Line-broadening Mechanisms–Homogeneous broadening – Inhomogeneous broadening
Quantum Mechanics: Revisited
Wave MechanicsQuantum mechanical systems ( such as atoms,
molecules, ions etc.) are given by wave function ψ(r, t).
Itself ψ(r, t) has no physical meaning- it allows to calculate the expectation values of all observables of interest.
Measurable quantities are called observable and are represented by hermition operator Ô. Expectation values is given by
.,o,o 3 trtrrd
ProbabilityThe probability of finding the system in volume
element d3r is.
If system exist, its probability of being somewhere has to equal 1.
.,, trHtrt
i
H- the Hamiltonian of the system. The energy of the unperturbed system- an atom not interacting with light is
rVmPH 2
2
The time development of wave function is determined by Schrödinger equation,
Stationary StatesStationary states of Schrödinger equation are those for which space and time dependence are separated
Time independent equation,
Eigen functions having same eigen values are normal
Completeness
The wave function
n
tinn
nn
nerUtCtrtr ,,
- Expansion coefficients nC t-constant for problems related with free Hamiltonian
-time dependent for interaction Hamiltonian
Gives the probability of finding the system in state n.
,
ˆ ˆn mi tnmn m
n m
O e OC C
3 *mˆr oUnm nO d rU r
Where
DIRAC’S NOTATION
• The wave function of wave mechanics corresponds to the state vector of the Dirac’s formulation of the quantum mechanics.
• The relation between state vector and wave function is analogous to using vectors instead of coordinates.
In Dirac Notation
yVxVV yx
A vector V can be expanded as,
yVxVV yx
X-component of a vector is obtained by
xVxV
.In Dirac Notation
yx VVyVVx and
Using these equations we can write
VyyVxxV
1x x y y
For n dimensions
VnnVn
x x y y V
n
n n I
The expectation value of the operator ^O
ˆ ˆO t O t
Hermitian
The set of eigen vectors of a hermitian operator is complete. This means that any arbitrary vector can be expressed as a sum of orthogonal eigen vectors.
nn
n xC
0
Eigen vectors are orthonormal
mnfor 0 mnfor 1
nm
nmmn XX
Completeness relation for discrete case is
n
nn IXX
State vectors obey the Schrödinger's equation
n
tin neC
Hi
n
,
Expectation value can be written as
Two-level Atomic SystemWave function
State vector
a bi t i ta bC e a C e b
Radiative Processes for Isolated Atoms
Spontaneous Emission• Atom is in state of given material.• E2 >E1 – atom will tend to decay to state • The corresponding energy difference (E2-E1) is released
by the emission of a photon.
hEE )( 12
1
2
1.Radiative: Spontaneous emission 2.Non-radiative:
12 EE
Is delivered in some other form than electromagnetic radiation e.g. it may go into kinetic energy of the surrounding molecules.
Two possible ways for atom to decay
Energy difference
22 N
dtdN
sp
22 AN
dtdN
The rate of decay of these atoms are
Let N2 number of atoms in level 2 per unit volume
Asp1
AteNN 02
atom.another by emitted that toipsrelationsh phase definite no has waveRadiative
possible. isdirection Any
.population initial theis N0
A- Transition probability for spontaneous emission or Einstein’s A coefficient.
STIMULATED EMISSION
hEE 12incident on it.Finite probability of emission of another in phase photon.Rate of transition
• Atom is initially in level 2• An electromagnetic wave of frequency υ given by
W21- stimulated transition probability or rate- (time)-1
W21- depends on intensity of electromagnetic wave.
• Atom is in level 1• Electromagnetic wave of frequency υ incident on the
material.• Finite probability that atom will be raised to level 2.• Energy of incident wave is absorbed by the atom
Process is called the stimulated absorption.
STIMULATED ABSORPTION:
N1 -number of atoms at a given time lying in level 1
112 1
. .st ab
dN W Ndt
Stimulated Transition Rate
Transition Rates
TRANSITION RATES OF STIMULATED EMISSION AND ABSORPTION
• Semi-classical Theory• Atom is quantized• Field is treated classically (using Maxwell’s equations)
tiEtiE
erutaerutatr21
2211,
System: Two-Level Atomic System
u1(r) & u2(r) - eigen -function of unperturbed Hamiltonian(H0).satisfies time-independent equation.
iii uEuH 0a1(t) - probability amplitudes.
2
ia t - probability of finding atoms in state i.
2112
0
EE
- Transition frequency0
• A monochromatic electromagnetic wave incident on atom.• Atom acquire an additional energy H’ during interaction Total Hamiltonian
HHH 0
TIME EVOLUTION OF THE SYSTEMSchrodinger equation
trHttri ,,
Putting value of ψ(r,t)
tiEtiE
tiEtiEtiEtiE
erutaerutaH
erutaerutaHerutat
ierutat
i
21
2121
2211
221102211
Multiplying with ru1 and integrating over whole space.
Using
1311 rdruru
3 30i i i i i iu r H u r d r E u r u r d r E
03212
3201 rdruruErdruHru
0321 rdruru
113
11 HrdruHru
123
21 HrdruHru
tieHtaHtai
ta 01221111
1
22212120
1 HtaetaHi
ta ti
INTERACTION HAMILTONIAN:
NLMDEQEDI HHHHH EDH -dominates
-due to the Interaction of the electric dipole moment of the atom with the electric field of the electromagnetic wave called- Electric-dipole interaction.
H
Similarly
DIPOLE APPROXIMATIONr = vector indicating the electron’s position with respect to
the nucleus
ELECTRIC FIELD IN TERMS OF PLANE WAVE
Distance of electron from the nucleus is given by Bohr radius.
011
0 5.0105 Ama
<< Optical wavelength
erp
The spatial variation of the electric field across the dimensions of the atom is very small - neglecting it for long wave-lengths - is called dipole approximation.
tE cos2 0
tE cos0
tEerH cos2 0
rdruHruH 32112
rtdErureru 3021 cos
tEp cos2 012
rdrerurup 32112
Interaction Hamiltonian
Using this
where
is the matrix element of electric dipole operator.
tEpH cos2 01111
031111 rdrerurup
ru 1 ru1
11 22 0H H
Similarly
A level does not have a dipole moment.er- has odd parity
and - has even parity
Electric dipole transition only occurs between states of opposite parity.
taeHi
ta ti2121
01
taeHi
ta ti1212
01
Two differential equations- can be solved by using initial conditions.
Let at 0t , atom is in level 1
101 a 002 a
Assume transition probability is weak--- perturbation analysis can be used.
Using these we get
(0) (2) (4)1 1 1 1a t a a a
(1) (3) (5)2 2 2 2a t a a a
1)0(1 a
As atom is initially in level 1
According to Perturbation theory
First order corresponds to the probability that atom go to level 2 from level 1.
dtaeeEp
idtta titi
tt)0(
10
021
02
00
22
Putting value of )0(1a
00021
)1(2
111 00
ie
ieEp
ia
titi
ω ~ field frequency and ω0 ~ atomic frequency
At resonance
0
0 is rapidly oscillating term. Neglecting tie 0
This is called “Rotating wave approximation (RWA)”.
ta )2(1
ta )1(2
The second order of is obtained by substituting the
21 12 01a t p E function of time
ta2 ta )3(2Then putting this in , we get the third order of
ta1
ta2
= series is even powers of E0
= series is odd powers of E0
First order is enough for transition rates.
value of
2
2
20212
22sin4
tEpta
!32
22sin
3ttt
!32
22sin
32 ttt
Expand
Probability of finding the atom in state 2 is
As t increases, the maximum in the curve moves upwards proportional to t2 and zeros of the function move in along the horizontal axis towards the origin.
For limit 0 t
22sin 24
t t
tt tt 0 tt tt
2
0
02sin
t
0
22sin
2
tt can be replaced by Dirac delta function
For the area under the curve equal to unity it can be replaced by Dirac delta function with properties
Using
02
20212
2
2
t
Epta
This is for a single atom.
Electromagnetic wave interact with an ensemble of atoms with randomly oriented dipole moment with respect to field.If θ is the angle between p and E0,
2221
20
2021 cospEEp
Take average over all the random orientations of dipole moment.
If all angles θ are equally probable, then
2 22 221 0 0 21 cosp E E p
d
d
0
0
2
2
sin
sincoscos
2cossin0
0
d
?sincos0
2
d
xcos dxd sinPutting
32
3sincos
1
1
31
1
2
0
2
xxd dx
31
23
2cos2
2 2 221 0 21 0
13
p E p E
002
212
20 3
2 tEpta
200
2 En
0
2
221212 3
4n
pW
The energy density of electromagnetic wave is
0 0
12W
At
0 00
12 0W This is physical unacceptable result
Reason:
• We have assumed that the interaction between the electromagnetic wave and system could continue coherently for an infinite time.
• There are of number of phenomena that prevent the interaction of atom with electromagnetic wave for long time. For example collision, Spontaneous emission.
• Above equations are valid only in time interval between one collision and the next.
• After each collision the relative phase between the atom’s wave-function and electric field of the wave under go a random jump.
• The problem is to find the interaction of atom with a broadband field
is valid provided the Dirac delta function - an infinitely sharp function centered at
0 and of unit area such that
10 d
0 tg
0
10 dgt
again centered at , again of unit area.
but with a finite spectral width.
12W
Is replaced by a new function
The shape of function and value of its line width depends Upon the particular broadening mechanism involved.
tgpW 2
21212 34
Stimulated emission rate is obtained by changing initial conditions.
1 0 0a 102 aand 1221 ppppp 2112 As
2112 WWW
21W
Line-broadening Mechanisms
•Line- Broadening Mechanisms
Broadening• The term is used to denote the finite
spectral width of the response of the atomic systems to the electromagnetic fields.
Two Types of Line Broadening 1. Homogeneous Broadening 2. Inhomogeneous Broadening
Homogeneous Broadening• Broadening mechanism is homogeneous
when it broadens the line of each individual atom and therefore of the whole system in the same way.
• Atoms are indistinguishable• All atoms have same transition frequency
and same energy spectrum• Examples: Collision and Natural
Broadening
Inhomogeneous broadening• A Broadening mechanism is said to be
inhomogeneous when it broadens the atomic lines by different amount for different atoms.
• In this case different atoms in an ensemble has different transition frequency and frequencies are distributed over a range.
• Example: Doppler Broadening
Collision BroadeningBroadening in Gases
Due to collision of an atom with other atoms, ions, free electrons or the walls of the container.
Broadening in SolidsDue to the interaction of with the phonons of the lattice
• It leads to the change of relative phase between atomic dipole moment and that of a incident wave
• Collision interrupt the process of coherent interaction between the atom and the incident wave
• Atom no longer sees a monochromatic wave instead a broadband field
Collision Broadening cont’dHow to deal with this?• Add all the frequencies of jump during
collision.• Use Fourier theory to handle multiple
frequencies.• Assume no collision for time interval T2 –
mean free path
dpn
dW
d
)(3
4 and between intervalfrequency in the wave theofdensity energy theBe
ddLet
02
120
2212
radiation. theofspectrumfrequency entire over the ngintergrati
by obtained isy probabilitn transitiooverall The
2/
2
We assume the distribution of the values of can be described by a probability density
1P
P is the probability that the time interval between two successive collisions lies between
and
TeT
d
2
c 20
d . T is the mean free path in which there is no collision.
Pd T
)/exp(P)P(
get we, timeaafter occurscollision ay that probabilit thebe )P( If
cd
0
212 212 2
0
The fieldE(r,t) 2E cos( ) is a monochromatic field wherecos( ) gives the phase or the mono chromalicity of the field and this field leads to unphysical result
4W ( )3
tt
pn
FOURIER THEORY:This method is used to handle the multiple frequencyThe Fourier theory allows the representation of a function in terms of its frequency or temporal characteristics and one can easily
-
-
move between the two representation
F(t) F( )e
1 F( ) F(t)e2
i t
i t
d
dt
The power spectrum W can be obtained as
the Fourier transform of the signal auto -correlation function Wiener- Kinchine theorem
CORRELATION:The method of calculating the similarity of two function is
called the correlation integral and result is correlation function.
AUTO CORRELATION FUNCTION:If the two functions are different ,the integral is calledcross correlation for same function auto correlation
1 2
T
TT
E t E t Lim dtE t E tT
2
When the functions are alinged we get maximum of Auto -correlation function
T
TT
Lim dtE tT
2
-
Power spectrum W can be obtained from the Parseval's
theoremT
T
W d C E t dt
EEeeddtE titi2
Ed
EEddttE
2
2
2
2
2
2202
12
2 / 2
o
o
ti t i t
ot
E E e dt
E SinE
function.- aget again Then we As
2
2 E
2
22
/02
2 0
20
2 2
2
2 2
12 2 1
tot
TSinE
e dT
ET
T
Times of flight are distributed according to P
Total intensity is made up of large number of time segments
22
2
2
where
1 1
1
g
gT
T
Such that
' 1g d
Function is maximum for 0
Doppler broadening• This is due to the random motions of the atoms.• It only occurs in gasses
Consider a field of frequency incident on an atom with transition frequency which is moving with a velocity v in the propagation direction of the wave. Atom will see a wave of frequency
0
1 VC
Due to Doppler effect
• The negative and positive sign applies whether the velocity is in the same or opposite direction to that of the wave. If the atom is moving in the opposite direction to that of wave the frequency observed by the atom is higher than the value observed in lab. frame.
• The absorption will occur only when the apparent frequency as seen from the atom is equal to the atomic transition frequency
0
01 vc
0
1 vc
Rewriting above equation
This is equivalent to say that atom is not moving but has a resonant frequency equal to
00
1 vc
Or
Incident field sees a shifted transition frequency of the atom. Absorption will occur when the frequency of e.m.wave is equal to If atom is moving away from the field ( same direction)
0
01 vc
0
1 vc
Incident field sees a shifted frequency of the atom andAnd absorption occur when this where0
00
1 vc
0 00
cv
We need to find the spectral function for Doppler broadening
0g g
As atoms are moving with different velocities, therefore field Sees different transition frequencies.
Using kinetic theory of gasses that an atom of mass M in a gas at temperature T has a velocity component between v and v+ dv is given by Maxwellian distribution
2 2
exp2 2M Mvp d dKT KT
gives the probability that the transition frequency lies between and
g d d
The frequency function is related to g p d
0 0 0g d p d
That is the number of atoms absorbing with in the frequency interval from to is equal to the fraction ofAtoms moving with velocity between v and v+ dv as
d
00
dv cd
20
0 20 0
exp2 2
Mcc MgKT KT
The shape of curve corresponding to this equation is calledGaussian. The maximum again occurs at 0
1
2
max0 2c Mg
KT
Find FWHM
Spontaneous Emission Rate
SPONTANEOUS EMISSION TRANSITION RATE
2..
2 ANdtdN
emsp
100
032
20
kTec
where A is called the Einstein A co-efficient or spontaneous emission transition rate.• Assume that the material is placed is blackbody cavity whose walls are kept at a constant temperature T. • Once thermodynamic equilibrium is reached, an
electromagnetic energy density with spectral distribution
Established
• Material will immersed in this radiation.
• Both stimulated emission and absorption processes will occur in addition to the spontaneous process.
• In thermal equilibrium, number of transitions per second from level 1 to level 2 must be equal to number of transition per second from level 2→1.
eee NWNWAN 1212212
02121 BW
01212 BW
kTe
e
eNN 0
1
2
120 EE 12 NN
Define
and
where B21 and B12 - called Einstein B co-efficient.From Boltzmann statistics
for
00 121
221
1
2 B
NN
BNN
A e
e
e
e
0
2
1
212 21
1
e
e
e
e
NAN
NB BN
e
e
NN
1
2Putting value of , we get
0
0 0 0
12 21 12 21
kT
kT kT
Ae A
B B e B e B
100
032
320
kTecn
For a medium,
n- refractive index of the medium
100
032
320
2112
kTkT ecn
BeB
A
BBB 2112
Comparing two values we get,
1
1
100 32
330
kTkT ecn
e
BA
3 302 3
A nB c
Probability of absorption and stimulated emission due to black body radiation are equal.
Bcn
A 32
330
Once the value of B, due to black body radiation is knownWe can find the value of A as we know
2
20
234 p
nW
This is true for monochromatic field.
d
00
2
20
234
dpn
dW
0
2
20
234
p
ndW
22 2
0
43
B pn
For black body radiation, the elemental spectral energy density of radiation whose frequency lies between and
This is same result as obtained from Quantum Electrodynamics approach because her we use Planck’s law (which is quantum electrodynamically correct). As A increases as the cube of the frequency, so the process of spontaneous emission increases rapidly with frequency.It is easy to produce infrared laser as compared to UV laser If frequency increases by a factor 1010, then A increases by 101000.
230
30
43
p nA
c
where a is Bohr radius cma 810
1810 sA
sec101 8A
t sp
Putting these values, we get
For magnetic dipole transitions, A is approximately 105 times small.
The order of magnitude of spontaneous emission at optical frequency
150 10 hz cm5105 eap
SPECTRUM OF THE SPONTANEOUS EMISSION
Line shape due to natural broadening• For any transition, the spectrum of the emitted
radiation is the same as that observed in absorption
• Assume that an ideal electromagnetic filter- transmitting only those frequencies between ω and ω + dω is placed between the material and the walls of the black body cavity.
• If the material, the filter, and the black body cavity are kept at the same temperature T, then ratio between the populations of the two levels will again be given by
kTe
e
eNN 0
1
2
The density of electromagnetic radiation
at any point inside the cavity will also be given by
1032
20
kTec
and the net exchange of the energy between the material and the cavity within the transmission bandwidth of the filter must be zero.
This means that the energy emitted by the material in the bandwidth dω around ω due to spontaneous and stimulated emission must be equal to energy absorbed.
Define a spectral co-efficient Aω such that
-Number of atoms per unit time which upon decay emit a photon of frequency between ω and ω+dω.
dAN 2
dAA
Similarly,
dNB
dNBdNBdNA eee122
represents the number of transitions per unit absorption or stimulated emission induced by black body radiation with frequency between ω and ω+dω.Equilibrium condition
Using above equations we get,
BA
BA
B
tgpn
W 2
20
234
BgB
BA
BgA
where can be obtained from
The spectrum of the radiated wave is the same as for absorption or stimulated emission.
A Ag
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