REVIEW OF QUANTUM
MECHANICS
IMRANA ASHRAF ZAHID
DEPARTMENT OF PHYSICS
QUAID-I-AZAM UNIVERSITY
ISLAMABAD, PAKISTAN
Layout
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Quantum Mechanics: Revisited
Radiative Processes for isolated atoms
Transition rates: Semi-classical
Line-broadening Mechanisms
– Homogeneous broadening
– Inhomogeneous broadening
Quantum Mechanics:
Revisited
Wave Mechanics
Quantum 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
ô   d r r, t ô r, t .
3

Probability
 The probability of finding the system in volume
element d3r is.
y ( r, t ) y ( r, t ).
*
 If system exist, its probability of being somewhere
has to equal 1.
ò y (r, t ) y (r, t ) d r =1
*
3
ò y (r, t ) y (r, t ) d r = d
*
n
3
m
nm
The time development of wave function is determined by
Schrödinger equation,

i  r , t   H r , t .
t
H- the Hamiltonian of the system. The energy of the
unperturbed system- an atom not interacting with light
is
2
P
H 
 V r 
2m
Stationary States
Stationary states of Schrödinger equation are those for
which space and time dependence are separated
yn ( r,t ) = U n (r)e -iwt
Time independent equation,
Eigen functions having same eigen values are normal
®
ò U ( r )U ( r ) d r = d
*
n
m
nm
Completeness
*
åU n (r)U n (r ) =1
n
The wave function
 r , t    n r , t   Cn t U n r e
n
 in t
n
Cn  t  - Expansion coefficients
-constant for problems related with free Hamiltonian
-time dependent for interaction Hamiltonian
åC
2
n
=1
n
Gives the probability of finding the system in state n.
ˆ 
O
 C nC me

 i  n  m t
ˆ
O
nm
n ,m
Where
Onm   d rU
3
*
ˆ
r
oU


n
m r
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.
A vector V can be expanded as,



V  Vx x  Vy y
In Dirac Notation
V  Vx x  Vy y
X-component of a vector is obtained by


V . x  Vx
In Dirac Notation
x V  Vx and y V  Vy
Using these equations we can write
V  x xV  y yV


x
x  y
y
V
 x x  y y 1
For n dimensions
V   n nV
n

n
n
n I
^
The expectation value of the operator
O
ˆ   t  O
ˆ  t 
O
Hermitian
*
é
ù
†
y (t ) Ô y ( t ) = êë y (t ) Ô y (t ) úû = y (t ) Ô y (t )
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.
*

   C n xn
n 0
Eigen vectors are orthonormal
X n X m   nm
 nm  1 for n  m
 0 for n  m
Completeness relation for discrete case is

n
Xn Xn  I
State vectors obey the Schrödinger's equation

i 
H ,
   C n e  i
nt
n
n
Expectation value can be written as
⌢
y t Oy t
()
()
= åC nC me (
*
n,m
where
Ônm = m Ô n
)
-i w n -w m t
Ônm
Two-level Atomic System
Wave function
State vector
  Ca e
ia t
a  Cbe
ibt
b
Radiative Processes for
Isolated Atoms
Spontaneous Emission
• Atom is in state 2 of given material.
• E2 >E1 – atom will tend to decay to state 1
• The corresponding energy difference (E2-E1) is released
by the emission of a photon.
( E2  E1 )

h
Two possible ways for atom to decay
1.Radiative: Spontaneous emission
2.Non-radiative:
Energy difference
E2  E1
Is delivered in some other form than
electromagnetic radiation e.g. it may go into
kinetic energy of the surrounding molecules.
Let N2 number of atoms in level 2 per unit volume
The rate of decay of these atoms are
 dN 2 

  N2
 dt  sp
 dN 2 

   AN 2
 dt 
A- Transition probability for spontaneous
emission or Einstein’s A coefficient.
 sp
1

A
Radiative wave has no definite phase relationsh ips
to that emitted by another atom.
Any direction is possible.
N 2  N 0e
N 0 is the initial population .
 At
STIMULATED EMISSION
•Atom is initially in level 2
•An electromagnetic wave of frequency υ given by
E2  E1  h
incident on it.
Finite probability of emission of another in phase photon.
Rate of transition
æ dN 2 ö
ç
÷ = -W21 N 2
è dt øst.em
W21- stimulated transition probability or rate- (time)-1
W21- depends on intensity of electromagnetic wave.
STIMULATED ABSORPTION:
•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 Transition Rate
 dN1 
 W12 N1


 dt st .ab.
N1 -number of atoms at a given time lying in level 1
Transition Rates
TRANSITION RATES OF STIMULATED
EMISSION AND ABSORPTION
•Semi-classical Theory
•Atom is quantized
•Field is treated classically (using Maxwell’s equations)
System: Two-Level Atomic System
 r , t   a1 t u1 r e
 iE1t

 a2 t u 2 r e
 iE2t

u1(r) & u2(r) - eigen -function of unperturbed Hamiltonian(H0).
satisfies time-independent equation.
H 0 u i  Ei u i
a1(t) - probability amplitudes.
ai  t  - probability of finding atoms in state i.
2
0
- Transition frequency
E2  E1
0 
 21

•A monochromatic electromagnetic wave incident on atom.
•Atom acquire an additional energy H’ during interaction
Total Hamiltonian
H  H0  H 
TIME EVOLUTION OF THE SYSTEM
Schrodinger equation
 r , t 
i
 H r , t 
t
Putting value of ψ(r,t)
iE1t
iE2t
iE1t
iE2t








i  a1 t u1 r e   i  a2 t u2 r e   H 0  a1 t u1 r e  a2 t u2 r e  
t 
 t 
 

iE1t
iE2t



H  a1 t u1 r e  a2 t u2 r e 



u
Multiplying with 1 r  and integrating over whole space.
Using

3




u
r
u
r
d
r 1
1
 1
 u r  H u r  d r   E u r  u r  d r  E

i
3
0 i

i i
3
i

3

3








u
r
H
u
r
d
r

E
u
r
u
r
d
1 02
 2 1 2 r 0

3




u
r
u
r
d
r 0
2
 1

3






u
r
H
u
r
d
r  H11
1
 1

3






u
r
H
u
r
d
r  H 12
2
 1
i




1
 i 0 t
  a 2 t H 12
e
a1 t  
a1 t H 11
i
Similarly
1
i 0 t
 a1 t e

a 2 t  
H 21
 a 2 t H 22
i
INTERACTION HAMILTONIAN:
H I  H ED  H EQ  H MD  H NL
H ED
-dominates
H  -due to the Interaction of the electric dipole moment of
the atom with the electric field of the electromagnetic
wave called- Electric-dipole interaction.
DIPOLE APPROXIMATION
r
= vector indicating the electron’s position with respect to
the nucleus
p  er
ELECTRIC FIELD IN TERMS OF PLANE WAVE
Distance of electron from the nucleus is given by Bohr radius.
0
a 0  5  10 11 m  0.5 A
<< Optical wavelength
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.
 2E0 cos t
 E 0 cos t
Interaction Hamiltonian
H   2er  E0 cos t
Using this

3


H 12   u1 r H u 2 r d r

3
  u r er u 2 r   E 0 cos td r

1
 2 p12  E0 cos t
where
p12   u r eru 2 r d r

1
3
is the matrix element of electric dipole operator.
Similarly
H11  2 p11  E0 cos t
p11   u r eru1 r d r  0

1
3
A level does not have a dipole moment.
er- has odd parity
u1 r  and u1 r  - has even parity
  H 22
 0
 H11
Electric dipole transition only occurs between states of
opposite parity.
Using these we get
1
 e i0t a 2 t 
a1 t   H 12
i
1
 e i0t a1 t 
a 2 t  
H 21
i
Two differential equations- can be solved by using initial
conditions.
Let at t  0 , atom is in level 1
a1 0  1
a2 0  0
Assume transition probability is weak--- perturbation analysis
can be used.
According to Perturbation theory
a1  t   a
(0)
1
a
(2)
1
a
(4)
1
 
a2  t   a2  a2  a2  
(1)
(3)
As atom is initially in level 1
a1
(0)
1
(5)
First order corresponds to the probability that atom go to
level 2 from level 1.
t
2  p21  E0  i  0 t
( 0)
i  0 t



a
t
dt

e

e
a
1 dt
0 2

i 0
2

t
Putting value of a1
a2
(1)

(0)
 e i  0 t  1 e i  0 t  1
1
  p 21  E0 


i
 i  0  
 i  0 
ω ~ field frequency and
ω0 ~ atomic frequency
At resonance
  0
  0 
is rapidly oscillating term. Neglecting
e
i  0 t
This is called “Rotating wave approximation (RWA)”.
a1
The second order of
( 2)
t 
is obtained by substituting the
value of a 2 (1) t 
a1  t   1  p12  E0 
2
Then putting this in
a1 t 
a 2 t 
function of time
a 2 t  , we get the third order of
a 2 t 
= series is even powers of E0
= series is odd powers of E0
First order is enough for transition rates.
( 3)
Probability of finding the atom in state 2 is
a 2 t 
2

4 p 21  E0
2
Expand
sin t
2
sin t
2
 t


 sin t 

2







t
2
2  t 

2
3
2
3!
 
2

2
 
t 2
3!
3
 
For limit
  0
t 
 sin  t

2 






2
t2

4
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 the area under the curve equal to unity it can be
replaced by Dirac delta function with properties
 t  t   
 t  t   0

t  t
t  t
  0 t
 sin


2 







0




2
can be replaced by Dirac delta function
 sin t 

2







2

t
2
    0 
  
Using
a2 t  
2
2 p21  E0

2
2
t   0 
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,
p 21  E 0
2
 E 0 p 21 cos 2 
2
2
Take average over all the random orientations of dipole moment.
If all angles θ are equally probable, then
p21  E0
 E0 p21
2
2
2
cos2 

cos 2 

2
cos
 sin d

0

 sind

0
 sind   cos 

0
2
0

2
cos
 sin d  ?

0
Putting
 sin d  dx
cos  x

 cos
2
3 1
1
x
 sin d 
2 dx
1
0
x

3
1
2

3
2
1
3
cos  

2
3
2
 p21  E0
a 0 t 
2
2
1
2
 p21 E0 2
3
2
2
 2 p 21 E 0t    0 
3
The energy density of electromagnetic wave is
n 2 0 E0
 
2
4

2
W12  2 p 21 2   
3
n 0
At
  0
   0   
 W12  
  0
   0   0
 W12  0
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
W12
is valid provided the Dirac delta function - an infinitely
sharp function centered at
  0
and of unit area such that
     d  1
0
Is replaced by a new function
g t   0 
again centered at    0 , again of unit area.
 g    d  1
t
0
but with a finite spectral width.
The shape of function and value of its line width depends
Upon the particular broadening mechanism involved.
W12
4

p 21
2
3
2
g t  
Stimulated emission rate W21 is obtained by changing
initial conditions.
a1  0  0
and
p12  p21  p
As
a2 0  1
p 21  p12
W  W12  W21

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 Broadening
Broadening in Gases
Due to collision of an atom with other atoms, ions, free
electrons or the walls of the container.
Broadening in Solids
Due 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’d
How 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
Let
d   d 
Be the energy density of the wave in the
frequency interval between   and    d 
4
2
dW12  2 2 p12   (   0 )d 
3 n  0
The overall transitio n probabilit y is obtained by
intergrati ng over the entire frequency spectrum
of the radiation.
We assume the distribution of the values of  can
be described by a probability density
1  / T2
P  e
T2
P d is the probability that the time interval
between two successive collisions lies between
 and   d . T2 is the mean free path in which there
is no collision.

 c    P d  T2
0
If P( ) be the probabilit y that a collision occurs after
a time  , we get

P( )   P d  exp(  /  c )

The field
E(r,t)  2E 0 cos( t ) is a monochromatic field where
cos( t ) gives the phase or the mono chromalicity of the field
and this field leads to unphysical result
4
2
W12  2 2 p21  (  )
3 n 0
 FOURIER THEORY:
This method is used to handle the multiple frequency
The 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) 
- i t
F(

)e
d

-
1
F( ) 
2



F(t)e  i t 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 called
cross correlation for same function auto correlation
E t  E t   
1
 Lim
T  2T
T
 dtE  t  E  t   
T
When the functions are alinged we get
maximum of Auto -correlation function
 Lim
T 

T
T
 dtE  t 
T
2
Power spectrum W   can be obtained from the Parseval's
theorem

T
-
T
2


W

d


C
E
  
  t dt

it i t

E t    dd e e E  E  
2



2
E
 t dt  2  dE  E   



 2  d

E  
2
E  
   2
1
E    
2
to 

Eo e
 i t i t
2
dt 
to
Sin    / 2 
E     

 
   
2
2 E02 
2
2
As    Then we again get a  - function.
Times of flight are distributed according to P  
Total intensity is made up of large number of time segments
 
tot
2 E0

T2
2 
2 E0

T2
e
0
2
 / T2

Sin      
      

1

2

1










T

2
2



2 
2 

2
d
    g     
where




1 
1

g      

T2  

       2  1 2  


T 2 

Such that

'
g

    d  1

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  0 which is moving
with a velocity v in the propagation direction of the
wave. Atom will see a wave of frequency

V

    1  
 C
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
v

    1    0
c

Rewriting above equation

0
v

1  
c

This is equivalent to say that atom is not moving but has a
resonant frequency equal to
 0 
Or
0
v

1  
c

Incident field sees a shifted transition frequency of the atom.
Absorption will occur when the frequency
of e.m.wave is
equal to  0
If atom is moving away from the field ( same direction)

v

     1    0
c

 
0
v

1  
c

Incident field sees a shifted frequency of the atom and
And absorption occur when this   0 where
 0 
0
v

1  
c

 v  0  0 
c
0
We need to find the spectral function for Doppler broadening
g     g   0 
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

 Mv 
 M 
p d  
 exp 
 d
 2 KT 
 2 KT 
2
g    d
gives the probability that the transition frequency
lies between
and

The frequency function


  d
is
g related
  to
p d
g  0   0 d 0  p d
That is the number of atoms absorbing with in the frequency
interval from
to
is equal to the fraction of
Atoms moving with velocity between v and v+ dv as

  d
dv
d  0

c
0
  Mc   0  
c
M
 g    0  
exp 

2
0 2 KT
2
KT

0


2
The shape of curve corresponding to this equation is called
Gaussian. The maximum again occurs at   0
g   max
Find FWHM
c  M 



0  2 KT 
1
2
Spontaneous Emission Rate
SPONTANEOUS EMISSION TRANSITION RATE
 dN 2 
  AN 2


 dt  sp.em.
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

0
0 2
 0
 2 3 
 c e kT  1
0
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.
AN  W21 N  W21 N
e
2
e
2
e
1
Define
and
W21  B21  0
W12  B12  0
where B21 and B12 - called Einstein B co-efficient.
From Boltzmann statistics
e
2
e
1
N
e
N
0  E2  E1
  0


kT 

for
N 2  N1
 N 2e
N 2e
A e  B21  0  e
N1
 N1
 0 
Putting value of

  B12  0

N
A
N
e
2
e
1

 N 2e  
 B12  B21  e  
 N1  

N 2e
, we get
e
N1
 
0
Ae
 0
B12  B21e
kT
 0
A

kT
B12e
0
kT
 B21
For a medium,

0 n 3
 0

2 3

 c e kT  1
2
0
0
n- refractive index of the medium
Comparing two values we get,
A
B12 e
  0
kT
0 n
 0
 2 3 
 c e kT  1
2
 B21
3
0
B12  B21  B
Probability of absorption and stimulated emission due to black
body radiation are equal.
A
3


n
1
0
B

 0
2 3
0

c
kT
kT
e
1
e
1
3
A
0 3n 3


B
 2c3
 0 n
A
2
3
 c
3
3
B
Once the value of B, due to black body radiation is known
We can find the value of A as we know
W 
4
3n  0 
2
p   
2
2
This is true for monochromatic field.
For black body radiation, the elemental spectral energy density
of radiation whose frequency lies between
and
d

dW 

4
3n  0 
2
dW 

2
p
2
  d    
0
0
4
3n 2  0  2
4
B
3n 2 0
p  0
2
2
p
2

4 0
p n
A
3 c 3 0
3
2
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.
The order of magnitude of spontaneous emission at optical
frequency
0  1015 hz
  5  10 5 cm
where a is Bohr radius
p  ea
8
a  10 cm
Putting these values, we get
A  10 s
8
t sp
1
1
  10 8 sec
A
For magnetic dipole transitions, A is approximately
105 times small.
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 filtertransmitting 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
N
N
e
2
e
1
 e
   0



kT


The density of electromagnetic radiation

at any point inside the cavity will also be given by

0

 2 3 
 c e kT  1
2
0
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
N 2 A d
-Number of atoms per unit time which upon decay
emit a photon of frequency between ω and ω+dω.
A
 A d
Similarly,
NB  d
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
A N d  B   N d  B   N d
e
2
e
2
Using above equations we get,
A
B
A

B
e
1
where
B
can be obtained from
W 
4
3n  0 
2
p g t  
2
2
B  Bg  
A
A

Bg  
B
 A  Ag   
The spectrum of the radiated wave is the same as for absorption
or stimulated emission.
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