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QUANTUM MECHANICS FORMULA SHEET
Contents
1. Wave Particle Duality
1.1 De Broglie Wavelength
1.2 Heisenberg’s Uncertainty Principle
1.3 Group velocity and Phase velocity
1.4 Experimental evidence of wave particle duality
1.4.1 Wave nature of particle (Davisson-German experiment)
1.4.2 Particle nature of wave (Compton and Photoelectric Effect)
2. Mathematical Tools for Quantum Mechanics
2.1 Dimension and Basis of a Vector Space
2.2 Operators
2.3 Postulates of Quantum Mechanics
2.4 Commutator
2.5 Eigen value problem in Quantum Mechanics
2.6 Time evaluation of the expectation of A
2.7 Uncertainty relation related to operator
2.8 Change in basis in quantum mechanics
2.9 Expectation value and uncertainty principle
3. Schrödinger wave equation and Potential problems
3.1 Schrödinger wave equation
3.2 Property of bound state
3.3 Current density
3.4 The free particle in one dimension
3.5 The Step Potential
3.7 Potential Barrier
3.7.1 Tunnel Effect
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3.8 The Infinite Square Well Potential
3.7.1 Symmetric Potential
3.9 Finite Square Well Potential
3.10 One dimensional Harmonic Oscillator
4. Angular Momentum Problem
4.1 Angular Momentum
4.1.1 Eigen Values and Eigen Function
4.1.2 Ladder Operator
4.2 Spin Angular Momentum
4.2.1 Stern Gerlach experiment
4.2.2 Spin Algebra
4.2.3 Pauli Spin Matrices
4.3 Total Angular Momentum
5. Two Dimensional Problems in Quantum Mechanics
5.1 Free Particle
5.2 Square Well Potential
5.3 Harmonic oscillator
6. Three Dimensional Problems in Quantum Mechanics
6.1 Free Particle
6.2 Particle in Rectangular Box
6.2.1 Particle in Cubical Box
6.3 Harmonic Oscillator
6.3.1 An Anistropic Oscillator
6.3.2 The Isotropic Oscillator
6.4 Potential in Spherical Coordinate (Central Potential)
6.4.1 Hydrogen Atom Problem
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7. Perturbation Theory
7.1 Time Independent Perturbation Theory
7.1.1 Non-degenerate Theory
7.1.2 Degenerate Theory
7.2 Time Dependent Perturbation Theory
8. Variational Method
9. The Wentzel-Kramer-Brillouin (WKB) method
9.1 The WKB Method
9.1.1 Quantization of the Energy Level of Bound state
9.1.2 Transmission probability from WKB
10. Identical Particles
10.1 Exchange Operator
10.2 Particle with Integral Spins
10.3 Particle with Half-integral Spins
11. Scattering in Quantum Mechanics
11.1 Born Approximation
11.2 Partial Wave Analysis
12. Relativistic Quantum Mechanics
12.1 Klein Gordon equation
12.2 Dirac Equation
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1.Wave Particle Duality
1.1 De Broglie Wavelengths
The wavelength of the wave associated with a particle is given by the de Broglie relation

h
h

where h is Plank’s constant
p mv
For relativistic case, the mass becomes m 
m0
v2
1 2
c
where m0 is rest mass and v is
velocity of body.
1.2 Heisenberg’s Uncertainty Principle
“It is impossible to determine two canonical variables simultaneously for microscopic
particle”. If q and pq are two canonical variable then
qp q 

2
where ∆q is the error in measurement of q and ∆pq is error in measurement of pq and h is
Planck’s constant (  h / 2 ) .
Important uncertainty relations

x  Px 

(x is position and px is momentum in x direction )
2

E   t 

( E is energy and t is time).
2

L   

(L is angular momentum, θ is angle measured)
2
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1.3 Group Velocity and Phase Velocity
According to de Broglie, matter waves are associated with every moving body. These
matter waves moves in a group of different waves having slightly different wavelengths.
The formation of group is due to superposition of individual wave.
Let If  1 x, t  and  2  x, t  are two waves of slightly different wavelength and frequency.
 1  A sin kx   t ,
 2  A sink  dk x    d t 
vg
 dk d t 
   1   2  2 A cos 
 sin kx   t 
2 
 2
v ph
The velocity of individual wave is known as Phase
t

velocity which is given as v p  . The velocity of
k
amplitude is given by group velocity vg i.e. v g 
The
vg 
relationship
between
group
d
dk
and
phase
velocity
is
given
by
dv p
dv p
d
 vp  k
; vg  v p  
dk
dk
d
Due to superposition of different wave of slightly different wavelength resultant wave
moves like a wave packet with velocity equal to group velocity.
1.4 Experimental evidence of wave particle duality
1.4.1 Wave nature of particle (Davisson-German experiment)
Electron strikes the crystals surface at an
D
S
angle  . The detector, symmetrically located
from the source measure the number of

electrons scattered at angle θ where θ is the
angle between incident and scattered electron


beam.
n  2d sin 
The Maxima condition is given by
 
or n  2d cos 
2
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where  
h
p
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1.4.2 Particle nature of wave (Compton and Photoelectric Effect)
Compton Effect
The Compton Effect is the result of scattering of a photon by an electron. Energy and
momentum are conserved in such an event and as a result the scattered photon has less
energy (longer wavelength) then the incident photon.
If λ is incoming wavelength and λ' is scattered
wavelength and  is the angle made by
scattered wave to the incident wave then
h
 ' 
1  cos 
mo c
where
scattered photon
incident photon


Target Electron
Scattered Electron
h
known as c which is Compton wavelength ( c = 2.426 x 10-12 m) and mo is
mo c
rest mass of electron.
Photoelectric effect
When a metal is irradiated with light, electron may get emitted. Kinetic energy k of
electron leaving when irradiated with a light of frequency    o , where o is threshold
frequency. Kinetic energy is given by
kmax  h  h 0
Stopping potential Vs is potential required to stop electron which contain maximum
kinetic energy  k max  .
eVs  h  h 0 , which is known as Einstein equation
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2. Mathematical Tools for Quantum Mechanics
2.1 Dimension and Basis of a Vector Space
A set of N vectors 1 ,  2 ,........ N is said to be linearly independent if and only if the
N
solution of equation
a 
i i
 0 is a1 = a 2 = ..... a N =0
i 1
N
N dimensional vector space can be represent as    a ii  0 where i = 1, 2, 3 … are
i 1
linearly independent function or vector.
Scalar Product: Scalar product of two functions is represented as  ,  , which is
defined as   * dx . If the integral diverges scalar product is not defined.
2
Square Integrable: If the integration or scalar product  ,     dx is finite then the
integration is known as square integrable.
Dirac Notation: Any state vector  can be represented as  which is termed as ket
and conjugate of  i.e.  * is represented by  which is termed as bra.
The scalar product of  and ψ in Dirac Notation is represented by   (bra-ket). The
value of   is given by integral   * r , t  r , t  d 3 r in three dimensions.
Properties of kets, bras and brakets:
a  
*

*
 a* 
  
Orthogonality relation: If  and  are two ket and the value of bracket    0
then  ,  is orthogonal.
Orthonormality relation: If  and  are two ket and the value of bracket    0
and    1    1 then  and  are orthonormal.
Schwarz inequality:  
2
   
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2.2 Operators
An operator A is mathematical rule that when applied to a ket  transforms it into
another ket  i.e.
A    
Different type of operator
Identity operator I   
Parity operator   r     r 
For even parity   r    r  , for odd parity   r     r 
Momentum operator Px   i 
Energy operator H   i 

x

Laplacian operator  2  
 t
2 
x 2

2 
y 2

2 
z 2
Position operator X  r   x  r 
Linear operator
For   a1 1  a 2  2
if an operator  applied on 
results in a1 A 1  a2 A 2
then operator  is said to be linear operator.
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2.3 Postulates of Quantum Mechanics
Postulate 1: State of system
The state of any physical system is specified at each time t by a state vector  t  which
contains all the information about the system. The state vector is also referred as wave
function. The wave function must be: Single valued, Continuous, Differentiable, Square
integrable (i.e. wave function have to converse at infinity).
Postulate 2: To every physically measurable quantity called as observable dynamical
variable. For every observable there corresponds a linear Hermitian operator  . The
Eigen vector of  let say n  form complete basis. Completeness relation is given

by  n n  I
n 1
Eigen value: The only possible result of measurement of a physical quantity an is one of
the Eigen values of the corresponding observable.
Postulate 3: (Probabilistic outcome): When the physical quantity A is measured on a
system in the normalized state  the probability P(an) of obtaining the Eigen value an of
2
gn
corresponding observable A is P  an  

i
n
a 
i 1
  
where gn is degeneracy of state and
u n is the Normalised Eigen vector of  associated with Eigen value an.
Postulate 4: Immediately after measurement.
If the measurement of physical quantity A on the system in the state  gives the result
an (an is Eigen value associated with Eigen vector an ), Then the state of the system
immediately after the measurement is the normalized projection
Pn 
 Pn 
where Pn is
projection operator defined by n n .
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Projection operator P̂ : An operator P̂ is said to be a projector, or projection operator, if
it is Hermitian and equal to its own square i.e. Pˆ 2  Pˆ
The projection operator is represented by

n
n
n
Postulate 5: The time evolution of the state vector  t  is governed by Schrodinger
equation: i
d
 t   H t   t  , where H(t) is the observable associated with total
dt
energy of system and popularly known as Hamiltonian of system. Some other operator
related to quantum mechanics:
2.4 Commutator
If A and B are two operator then their commutator is defined as  A,B  AB-BA
Properties of commutators
 ,    , ;
 ,   C    ,    , C 
†
 , C    ,  C  B  , C  ;  ,   † , † 
 ,  , C    C ,   C ,  ,    0
(Popularly known as Jacobi identity).
 , f     0
If X is position and Px is conjugate momentum then
 X n , Px   nX n 1  i  and  X , Pxn  nPx n 1  i  
If b is scalar and A is any operator then
 , b  0
If [A, B] = 0 then it is said that A and B commutes to each other ie AB  BA .
If two Hermition operators A and B , commute ie  ,        0 and if A has non
degenerate Eigen value, then each Eigen vector of  is also an Eigen vector of B .
We can also construct the common orthonormal basis that will be joint Eigen state of
A and B .
The anti commutator is defined as
 ,     
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2.5 Eigen value problem in Quantum Mechanics
Eigen value problem in quantum mechanics is defined as
 n  an n
where an is Eigen value and n is Eigen vector.

In quantum mechanics operator associated with observable is Hermitian, so its Eigen
values are real and Eigen vector corresponding to different Eigen values are
orthogonal.

The Eigen state (Eigen vector) of Hamilton operator defines a complete set of
mutually orthonormal basis state. This basis will be unique if Eigen value are non
degenerate and not unique if there are degeneracy.
Completeness relation: the orthonormnal realtion and completeness relation is given by


 n m   mn ,
n
n  I
n 1
where I is unity operator.
2.6 Time evaluation of the expectation of A (Ehrenfest theorem)
d
1
A
A 
A,H  

dt
i
t
where
 A, H  is
commutation between operator A and
Hamiltonian H operator .Time evaluation of expectation of A gives rise to Ehrenfest
theorem .
d
1
R 
P
dt
m
,
d
P   V  R, t 
dt
where R is position, P is momentum and V  R, t  is potential operator.
2.7 Uncertainty relation related to operator
If  and B̂ are two operator related to observable A and B then
Â B̂ 
where Â 
 2  Â
2
1
Â, B̂
2
 
and Â 
B̂ 2  B̂
2
.
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2.8 Change in basis in quantum mechanics
If  k  are wave function is position representation and  k  are wave function in
momentum representation, one can change position to momentum basis via Fourier
transformation.
  x 
 k  
1
2
1
2

  k  e
ikx
dk


   xe
 ikx
dx

2.9 Expectation value and uncertainty principle
The expectation value A of A in direction of  is given by
A 
 A

or
A   an Pn where Pn is probability to getting Eigen value an in state  .
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3. Schrödinger Wave Equation and Potential Problems
3.1 Schrödinger Wave Equation
Hamiltonian of the system is given by H 
P2
V
2m
Time dependent Schrödinger wave equation is given by H  i

t
Time independent Schrödinger wave equation is given by H  E
where H is Hamiltonian of system.
It is given that total energy E and potential energy of system is V.
3.2 Property of bound state
Bound state
If E > V and there is two classical turning point in which particle is classically trapped
then system is called bound state.
Property of Bound state
The energy Eigen value is discrete and in one dimensional system it is non degenerate.
The wave function  n x  of one dimensional bound state system has n nodes if n = 0
corresponds to ground state and (n – 1) node if n = 1 corresponds to ground state.
Unbound states
If E > V and either there is one classical turning point or no turning point the energy
value is continuous. If there is one turning point the energy eigen value is nondegenerate. If there is no turning point the energy eigen value is degenerate. The particle
is said to be unbounded.
If E < V then particle is said to be unbounded and wave function will decay at ± ∞. There
is finite probability to find particle in the region.
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3.3 Current density
If wave function in one Dimension is  x  then current density is given by
  * 
 * 


Jx 


2im 
x
x 

J  0
t
which satisfies the continually equation


2
Where    * in general J   v J   v where v is velocity of particle?
If Ji, Jr, Jt are incident, reflected and transmitted current density then reflection coefficient
R and transmission coefficient T is given by
R
J
Jr
and T  t
Ji
Ji
3.4 The free particle in one dimension
Hψ = Eψ

 2 d 2
 E  x 
2m dx 2
  x   A e ikx  Ae ikx
Energy eigen value E 
 2k 2
where k 
2m
2 mE
the eigen values are doubly degenerate
2
3.5 The Step Potential
The potential step is defined as
0
V x   
Vo
x0
x0
Vo
Case I: E > Vo
 1  Aeik1 x  Beik1 x
x0
 2  Aeik 2 x  Be ik 2 x
x0
o
x
Hence, a particle is coming from left so D = 0.
R = reflection coefficient =
J reflected
J incident
 k k 
R  1 2 
=
 k1  k2 
2
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T = transmitted coefficient =
where k1 
2mE
2
k2 
J transmitted
4k1k2
= T
2
J incident
 k1  k2 
2mE  Vo 
2
Case II: E < Vo
 I  Aeik1x  Be ik1x
x0
k1 
2mE
2
 II  ce k 2 x
x0
k2 
2mVo  E 
2
R
Jr
 1 and
Jt
T
Jt
0
Ji
For case even there is Transmission coefficient is zero there is finite probability to find
the particle in x > 0 .
3.7 Potential Barrier
Potential barrier is shown in figure.
Energy
Vo
o
0

Potential barrier is given by V x   V0
0

x
a
x0
0 xa
xa
Case I: E > Vo
 1  x    1 x   Aeik1x  Beik1x x  0
 2 x    2 x   Ceik2 x  Deik2 x 0  x  a
 3 x    3  x   Eeik1x
Where k1 
x0
2mE
2


1
Transmission coefficient T  1 
sin 2   1 
 4   1


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
k2 
2mE  Vo 
2
1
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
4   1 
Reflection coefficient R  1 

2
 sin   1 

1
Where 

E
,
Vo
a
2mVo
2
Case II: E < Vo
3.7.1 Tunnel Effect
Vo
 I  Aeik1 x  Beik1 x
x0
 II  Cek2 x  Be k2 x
0 xa
 III  Fe
R
ik1 x
o
x0
1
sin h 2  1 
4  1 
Where   a

2mVo
,
2

and T  1 

a
1
sin h 2  1 
4  1 


E
Vo
For E << Vo
2 a 2 m Vo  E 
T
16 E 
E 
1  e 
Vo  Vo 
2
Approximate transmission probability T  e2 k 2 a
2mVo  E 
2
where k2 
3.8 The Infinite Square Well Potential
The infinite square well potential is defined as as shown in the figure
 

V x   0




x0
0 xa
V x 
xa
o
x
a
Since V(x) is infinite in the region x  0 and x  a so the wave function corresponding
to the particle will be zero.
The particle is confined only within region 0 ≤ x ≤ a.
Time independent wave Schrödinger wave equation is given by
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
 2 d 2
 E
2m dx 2
  A sin kx  B cos kx
B = 0 since wave function must be vanished at boundary ie x  0 so   A sin kx
Energy eigen value for bound state can be find by ka  n where n  1, 2,3...
The Normalized wave function is for n th state is given by
 n  x 
2
n x
sin
a
a
Which is energy Eigen value correspondence to nth
En 
n 2 2  2
where n = 1, 2, 3 .....
2ma 2
othonormality relation is given by
a
m x
n x
a
sin
dx   mn
L
L
2
 sin
0
0
a

2
i.e.
mn
m 1
If x is position operator Px Px is momentum operator and  n  x  is wave function of
particle in n th state in one dimensional potential box then  n  x  

x    n*  x x n  x  

a
2

x 2    n*  x x 2 n  x  


Px    n*  x   i 


2
x
P
   n*  x  

2
n x
sin
then
a
a
a2
a2
 2 2
2 2n 

 n  x  0
x
2 
n 2 2  2

x



n
2m x 2
a2
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The uncertainty product is given by xPx  n
1
1
 2 2
12 2n 
The wave function and the probability density function of particle of mass m in one
dimensional potential box is given by
n3
E 3  9 E1
 3 
2
3x
sin
a
a
3
n2
E 2  4 E1
 2 
2
2x
sin
a
a
2
2 x
sin
a
a
1
n 1
E1 
 22
2m
 1 
2
2
2
3.8.1 Symmetric Potential
The infinite symmetric well potential is given by
V x   
0
a
x   or
2
a
a
 x
2
2
a
x
2
Schrondinger wave function is given by

 2 d 2
 E ;
2m dx 2
so

a
2
a
a
2
a
a
 x
2
2
  A sin kx  B cos kx where k 
at
V x 
2mE

a
x   ,  x  0
2
ka
ka
A sin  B cos  0   (i )
2
2
ka
ka
 A sin  B cos  0
  (ii )
2
2
Hence parity (  ) commute with Hamiltonian ( H ) then parity must conserve
So wave function have to be either symmetric or anti symmetric
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For even parity
  B cos kx and Bound state energy is given by cos
ka
n
0 k 
2
a
ka
0
2
n  1,3,5,....
Wave function for even parity is given as
2
n x
cos
a
a
For odd parity
Ψ (x) = A sin kx and Bound state energy is given by
sin
ka
ka
n
0 ,
0 k
2
2
a
n  2, 4,6,........
For odd parity Wave function is given as
2
n x
sin
a
a
n 2 2  2
The energy eigen value is given by En 
2ma 2
n  1,2,3,......
 3 x 
First three wave function is given by
 1 x  
2
x
cos
a
a
 2 x 
2
2 x
sin
a
a
 3 x 
where

a
2
a
2
9 2  2
2 ma 2
a
2
4 2  2
2 ma 2
a
2
 2 2
2ma 2
 2 x 
a

2
2
3 x
cos
a
a
 1 x 
2
is normalization constant.
a

a
2
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3.9 Finite Square Well Potential
V x 
For the Bound state E < Vo
Again parity will commute to Hamiltonian
So wave function is either symmetric or
Vo
Anti symmetric
I x   Aex
x
II x   C cos kx

III  x   Aex
a
2

For even parity
a
2
x
a
2
a
a
x
2
2
a
x
2
wave function must be continuous and differentiable at boundaries so using boundary
condition at 
a
2
one will get   k tan
ka
2
where  
   tan 
a
2

k
2mE
2
ka
2
For odd parity
1  x   Aex
x
2 x   D sin kx

 III x    Ae x
a
2
a
a
x
2
2
a
x
2
using boundary condition one can get
   cot  where  
and n 2   2 
2mVo  E 
2
mVo a 2
which is equation of circle.
2 2
The Bound state energy will be found by solving equation
n   tan 
for even
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n   cot 
n2   2 
for odd
mVo a 
2 2
one can solve it by graphical method.
n   tan 
n
o


2
3
2
2

(solid curve) and  2   2 
mVo a 2
(circle) give
2 2
Eigen value for even state and intersection point at  = -  cot
(dotted curve) and
The intersection point of  =  tan
2  2 
mVo a 2
(circle) give Eigen value for odd state.
2 2
The table below shows the number of bound state for various range of V0 a 2 where Voa2
is strength of potential.
Voa 2
Even eigen function
Odd eigen function
No. of Bound
states
 2 2
2m
1
0
1
 2 2
4 2  2
2
 V0 a 
2m
2m
1
1
2
 2 2
9 2  2
 V0 a 2 
2m
2m
2
1
3
9 2  2
16 2  2
 V0 a 2 
2m
2m
2
2
4
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The three bound Eigen function for the square well
 2 x 
1  x 
 3 x 


a
2
a
2
a
2
a
2
x
x
x
3.10 One dimensional Harmonic Oscillator
V  x
One dimensional Harmonic oscillator is given by
V x 
1
m 2 x 2
2
n2
  x  
n 1
E2 
5
2
E1 
3
2
E0 

2
The schrodinger equation is given by
n0
2
2
 d  1
 m 2 x 2  E
2m dx 2 2
It is given that  
m
2E
x and k 


The wave function of Harmonic oscillator is given by.
1/ 4
1/ 2
 m   1 
n    
  n 
    2 n! 
H n  e

2
2
and energy eigen value is given by
E n = (n+1/2) ;
n = 0, 1, 2, 3, ....
The wave function of Harmonic oscillator is shown
H 0 ( ) = 1 , H1 ( ) = 2 , H 2 ( ) = 4 2 -2
It n and m wave function of Harmonic oscillator then
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


m x n x dx   mn
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Number operator
The Hamiltonian of Harmonic operator is given by H 
Px2 1
 m 2 X 2
2m 2
Consider dimensioned operator as Xˆ , Pˆ , Hˆ where
X 
 ˆ
X
m
1
H   Hˆ so Hˆ  Pˆx2  Xˆ 2 .
2

Px  m  Pˆx
1

Xˆ  iP 
2
Consider lowering operator a 
x

and raising a† 
1 ˆ
X  iPx 
2
Hˆ  N  1 / 2
where N  a † a and H   N  1/ 2  
N is known as number operator
n is eigen function of N with eigen value n.
1

and H n    N   n
2

N n n n
1

H n   n   n
2

where n = 0, 1, 2, 3,………
Commutation of a and a † :
[a, a † ] = 1 , [a † , a] = -1 and [N, a] = -a , [N a † ] = a †
Action of a and a † operator on n
a n  n n 1
a0 0
but
a n  n 1 n  1
Expectation value of X , X 2 , PX , PX2 in stationary states

a  a ,
2m
 X   0,
X
X 2 



2n  1,
2m
1

X PX   n  ,
2

a  a 

PX 
2i
m
 PX   0
1

 PX2    n  m
2

for n  0; X PX 

2
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4. Angular Momentum Problem
Angular momentum in Quantum mechanics is given L  L x iˆ  L y ˆj  Lz kˆ
Where L X  YPz  ZPy ; LY  ZPx  XPz ; Lz  XPy  YPx
and PX  i

,
x
PY  i

,
y
p Z  i

z
Commutation relation
 Lx , Ly   i Lz ,  Ly , Lz   iLx ,  Lz , Lx   iLy
L2  L2X  L2Y  L2Z and  L2 , LX   0 ,  L2 , LY   0 ,  L2 , LZ   0
4.1.1 Eigen Values and Eigen Function


 

LX  i Sin
 cos cot 

 



 
LY  i  cos 
 sin  cot 


 


LZ  i

Eigen function of LZ is
1 im
e .
2
and Eigen value of LZ  m where m = 0, ± 1, ± 2...
 1  

L2 operator is given by L2   2 
 sin 

 sin   
1 2 



2
2
 sin   
Eigen value of L2 is l (l  1) 2 where l = 0, 1, 2, ... l
Eigen function of L2 is Pl m   where Pl m   is associated Legendre function
L2 commute with Lz so both can have common set of Eigen function.
Yl m  ,    Pl m ( )eim
is common set of Eigen function which is known as spherical
harmonics .
The normalized spherical harmonics are given by
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Yl m ( ,  )   1
m
 2l  1  l  m 
4  l  m 
!
!
Pl m  cos   eim
l  m  l
1
Y00  ,   
4
Y11  ,    
3
3
3
sin  ei , Y10  ,   
cos  , Y11  ,   
sin  ei
8
4
8
L2Yl m  ,   l l  1 2Yl m  ,  l = 0, 1, 2, ……. And
LZ Yl m  ,   mYl m  ,   m = -l ,(-l +1) ..0,….. (l – 1)( l ) there is 2l  1
Degeneracy of L2 is 2l  1 .
2
Orthogonality Relation

m
m
 d  Yl  ,  Yl  ,  sin  d   ll ' mm '
0
0
4.1.2 Ladder Operator
Let L  LX  iLY and L  LX  iLY
Let us assume l, m is ket associated with L2 and LZ operator.
2
L2 l , m  l  l  1 l , m
Lz l ,, m  m l , m
l  0,1, 2,.......
m  l ...0,...l
Action of L+ and L- on l, m basis
L l , m  l  l  1  m  m  1 l , m  1
L l , m  l  l  1  m  m  1 l , m  1
Expectation value of L X and LY in direction of l, m
Lx  0 , Ly  0
LX LY 
L2X  LY2 
2
l  l  1  m 2 
2
2
l  l  1  m 2 
2 
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4.2 Spin Angular Momentum
4.2.1 Stern Gerlach experiment
When silver beam is passed to the inhomogeneous Magnetic field, two sharp trace found
on the screen which provides the experimental evidence of spin.
4.2.2 Spin Algebra
S  S X iˆ  SY ˆj  S Z kˆ , S 2  S X2  SY2  S Z2
 S 2 , S X   0 ,
 S 2 , SY   0  S 2 , S Z   0 and
 S X , SY   iSZ  SY , SZ   iS X  SZ , S X   iSY
S 2 s, ms  s  s  1  2 s, ms
S z s, ms  m s, ms where  s  ms   s
S   S X  iSY and S   S X  iSY
S  s, ms   s  s  1  ms  ms  1 s, ms  1
S  s, ms   s  s  1  ms  ms  1 s, ms  1
4.2.3 Pauli Spin Matrices
For Spin
1
1
1 1
Pauli matrix s  , ms   , 
2
2
2 2
Pauli Matrix is defined as
1
0

 x  
0 
1
 x2   y2   z2  I
0
 y  
 i
 j k   k  j  0
 j  k ;
1

 jkl  1
0

i 

0 
1
 z  
0
0 

1
Pauli Spin Matrix anticommute.
if jkl is an even permutation of x, y, z
if jkl is an odd permutation of x, y , z
if any two indices among j , k , l are equal



Sx   x , Sy   y , Sz   z
2
2
2
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 0

2 1
1

0 
Sy 
0
S   
0
1

0 
0
S   
1
Sx 
S2 
3 2
4
1

0
 0

2  i
i 

0 
Sz 
 1

2  0
0 

1
0

0 
0

1 
For spin ½ the quantum number m takes only two values ms 
1
1
and  . So that two
2
2
states are
S , ms 
1 1
1 1
, and ,
2 2
2 2
S2
1 1
3 2 1 1
, 
,
2 2
4 2 2
Sz
1 1
1 1 1
,   ,
2 2
2 2 2
S
1 1
1 1
1 1
,
 0 , S  ,   ,
2 2
2 2
2 2
S
1 1
1 1
,   ,
2 2
2 2
, S2
, Sz
, S
1 1
3 2 1 1
, 
,
2 2
4 2 2
1 1
1 1 1
,    ,
2 2
2 2 2
1 1
,  0
2 2
4.3 Total Angular Momentum
Total angular momentum J = L + S , J  J xiˆ  J y ˆj  J z kˆ
j, m j is the Eigen ket at J2 and Jz and J   J x  iJ y
J   J x  iJ y
J 2 j , m j  j  j  1 2 j , m j , J z j , m j  m j  j , m j
J  j, m j 
j  j  1  m j m j  1 j , m j  1
J  j, m j 
j  j  1  m j m j  1 j, m j  1
J  LS
ls  j  ls
J z  Lz  S z
 j  m j  j and ml  ms  m j
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5. Two Dimensional Problems in Quantum Mechanics
5.1 Free Particle
Hψ = Eψ

 2   2  2


2 m  x 2 y 2

  E

x and y are independent variable. Thus
1 ikx x ik y y
e e
2

1 i k r 

e
2
 n  x, y  
2 2
2 2
k x  k y2 
k
2m
2m

As total orientation of k which preserve its magnitude is infinite. So energy of free

Energy Eigen value 

particle is infinitely degenerate.
5.2 Square Well Potential
V  x   0  x  a and 0  y  a
 otherewise
H 
 2   2  2 


  E
2m  x 2 y 2 
The solution of Schrödinger wave equation is given by Wave function
 n x ,n y 
 n  x   ny y 
sin  x  sin 

 a   a 
a
4
2
Correspondence to energy eigen value
Enx , ny
2
 2  2  nx2 n y 


2 m  a 2 a 2 
nx  1, 2,3... and n y  1, 2,3...
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Energy of state
2 2 2
2ma 2
Ground state
(n x, ny)
Degeneracy
(1, 1)
Non degenerate
First state
5 2 2
2ma 2
(1, 2), (2,1)
2
Second state
8 2 2
2ma 2
(2, 2)
Non-degenerate
5.3 Harmonic oscillator
Two dimensional isotropic Harmonic oscillators is given by
H
2
p 2x p y 1

 m 2 x 2  y 2
2m 2m 2



Enx n y  nx  n y  1 

where nx  0,1, 2,3... n y  0,1, 2,3...
En   n  1 
degeneracy of the n th state is given by (n + 1) where n = nx + ny.
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6. Three Dimensional Problems in Quantum Mechanics
6.1 Free Particle
H  E

 2   2  2  2



2 m  x 2 y 2 z 2

  E

Hence x, y, z are independent variable. Using separation of variable one can find the
 k  x, y , z  
1
2 
3/ 2
e ikn x e
Energy Eigen value 
ik y y ik z z
  2 
e
3/ 2

eik .r
2 2
2 2
k x  k y2  k z2 
k
2m
2m


As total orientation of k which preserve its magnitude is infinite, the energy of free
particle is infinitely degenerate.
6.2 Particle in Rectangular Box
Spinless particle of mass m confined in a rectangular box of sides Lx, Ly, Lz
V  x, y, z   0  x  Lx , 0  y  Ly , 0  z  Lz ,
=  other wise .
The Schrodinger wave equation for three dimensional box is given by
H 
 2   2  2  2



2m  x 2 y 2 z 2

  E

Solution of the Schrödinger is given by Eigen function  nx ny nz and energy eigen value
is Enx n y nz is given by
 nx n y nz 
En x n y n z 
 n x   n yy   n zz 
8
 sin 
sin  x  sin 


Lx L y Lz
 Lx   L y   Lz 
 2 2
2m
 n x2 n 2y n z2 
 2  2  2
L L
Lz 
y
 x
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6.2.1 Particle in Cubical Box
For the simple case of cubic box of side a, the
i.e. Lx = Ly = Lz = a
 nx n y nz 
En x n y n z 
8
 n x   n yy   n z z 
sin x  sin 
 sin 

3
a
 a   a   a 
 2 2 2
n x  n 2y  n z2
2
2ma


nx  1, 2,3...
Energy of state
n y  1, 2, 3... nz  1, 2,3...
(nx, n y, n z)
Degeneracy

3 2  2
2ma 2
(1, 1, 1)
Non degenerate
E of first excited state 
6 2  2
2 ma 2
(2, 1, 1) (1, 2, 1) (1, 1, 2
3
E of 2 nd excited state 
9 2  2
2ma 2
(2, 2, 1) (2, 1, 2) (1, 2, 2)
3
E of ground state
6.3 Harmonic Oscillator
6.3.1 An Anistropic Oscillator
V X ,Y , Z  
1
1
1
m X2 X 2  mY2Y 2  m Z2 Z 2
2
2
2
1
1
1



En x n y n z   n x   x   n y   y   n z   z where
2
2
2



nx  0,1, 2,3... n y  0,1, 2,3... nx  0,1, 2,3...
6.3.2 The Isotropic Oscillator
x   y  z  
3

Enx n y nz   nx  n y  nz    where n  nx  n y  nz
2

Degeneracy is given by =
n  0,1, 2,3...
1
n  1n  2
2
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6.4 Potential in Spherical Coordinate (Central Potential)
Hamiltonion in spherical polar co-ordinate

 2  1   2 

r
2m  r 2 r 
r

1
 


 2
 sin 

 r sin   
1
 2 

  V  r   E
 2 2
 r sin   2 
 2  1   2    L2
 V  r 
r
 
2m  r 2 r 
r   2mr 2
L2 is operator for orbital angular momentum square.
So Ylm  ,   are the common Eigen state of and L2 because [H, L2] = 0 in central force
problem and L2Ylm  ,    l  l  1  2Ylm ( ,  )
So  r , ,  can be separated as   f  r  Ylm  ,  

2
2
2  d f  r  2 f  l  l  1 



f  r   V  r   E  f  r   0
2m  dr 2
r r 
2mr 2


To solve these equations f r  
u r 
r
So one can get


 2 d 2u 
l l  1 2

V

r


 E u  0

2
2
2m dr
2mr


Where
l l  1 2
is centrifugal potential and
2 mr 2
V r  
l l  1 2
is effective potential
2mr 2
The energy Eigen function in case of central potential is written as
  r , ,    f  r  Ylm  ,   
u r 
r
Ylm  ,  
The normalization condition is
   r , ,  


0
u r 
2
 u r 
0
d  1
 2
r 2 dr 
r

2

2
Ylm  ,   sin  d d  1
0 0
2

dr  1 or
 R r 
2
r 2 dr  1
0
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6.4.1 Hydrogen Atom Problem
Hydrogen atom is two body central force problem with central potential is given by
V r   
e2
4 0 r
Time-independent Schrondinger equation on centre of mass reference frame is given by
 
 2 2  2 2
  
R 
r  V  r  R, r  E R, r

2
 2m

 
 
where R is position of c.m and r is distance between proton and electron.
    
 R,r   R  r
The Schronginger equation is given by
 2
  2

1
h 1
 h
 2R  R  
 2R   r  V r E
 2M  R
  2   r

 






Separating R and r part

2 2
  R  E R R
2M


2 2

 r r   V r  r 
2
Total energy E = E R + E r
ER is Energy of centre of mass and Er is Energy of reduce mass µ
 R 
 
1
 2 
3/ 2
ei k R
ER 
2 k 2
2M
i.eCentre of mass moves with constant momentum so it is free particle.
Solution of radial part
h2 d 2 ur ll 1h2
e2




2
2
4  0
 2 r
 2 dr

ur  E r
r 
For Hydrogen atom   me
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The energy eigen value is given by En = 
And radius of n th orbit is given by rn 
me e4
2
2  4 0   n
2
2

13.6
eV n  1, 2,3...
n2
4 0  2 n 2
n  1, 2,3... where me is mass of
m e2
electron and e is electronic charge.
n is known as participle quantum number which varies as n  l  1, n  2,... For the
given value of n the orbital quantum number n can have value between 0 and n  1 (i.e. l
= 0, 1, 2, 3, ….. n – 1) and for given value of l the Azimuthal quantum number m varies
from – l to l known as magnetic quantum mechanics .
Degeneracy of Hydrogen atom without spin = n 2 and if spin is included the degeneracy
n1
is given by g n  2 2l  1  2n 2
l 0
For hydrogen like atom
13.6 z 2
En  
where z is atomic number of Hydrogen like atom.
n2
Normalized wave function for Hydrogen atom i.e. Rnl (r) where n is principle quantum
number and l is orbital quantum.
n
1
2
2
l
0
0
0
E(eV)
-13.6z
- 3.4 z
- 3.4 z
Rnl
 z 
2 
 a0 
2
2
 z 


 2a0 
2
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3/ 2
1  z0 
 
24  a0 
3/ 2
e zr / a0

zr   zr / 2 a0
2  e
a

0
3/ 2
 zr   zr / 2a0
  e
 a0 
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The radial wave function for hydrogen atom is Laguerre polynomials and angular part of
the wave function is associated Legendre polynomials .
R20 r 
R10 r 
r / a0
r
R21
r
r / a0
r / a0
For the nth state there is n – l – 1 node
If R nl is represented by n, l then
nl r nl 
1 2
3n  l  l  1  a0

2
1
nl r 2 nl  n2 5n 2  1  3l  l  1  a02

2 
nl r 1 nl 
1
2
n a0
nl r 2 nl 
2
3
n
 2l  1 a02
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7. Perturbation Theory
7.1 Time Independent Perturbation Theory
7.1.1 Non-degenerate Theory
For approximation methods of stationary states
H  Ho  H P
Where H Hamiltonian can be divided into two parts in that H o can be solved exactly
known as unperturbed Hamiltonian and H p is perturbation in the system eigen value
of H o is non degenerate
It is known H o  n  Eno n
and   1 H p  W where   1
Now  H o  W   n  En  n where  n is eigen function corresponds to eigen value En
for the Hamiltonian H
Using Taylor expansion
 n   n    1n  2  n2  .........
En  E n0  En1  2 En2  ........ and
First order Energy correction En1 is given by En1  n W n
And energy correction up to order in  is given by En  En1   n W n
First order Eigen function correction  n1  
m n
m W n
m
En0  Em0
And wave function up to order correction in   n  n   
m n
 2
Second order energy correction En  
m n
Energy correction up to order of 
2
m W n
m W n
m
En0  Em0
2
En 0  Em 0
En  E0   n W n  
2

m n
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m W n
2
En 0   Em 0 
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7.1.2 Degenerate Theory
H  n  ( H o  H p )  n  En  n
En0  is f fold degenerate H o  n  E n0  n
  1,2,3,..... f 
To determine the Eigen values to first-order and Eigen state to zeroth order for an f-fold
degenerate level one can proceed as follows
First for each f-fold degenerate level, determine f x f matrix of the perturbation Ĥ p
 H p11

 H p21

Hˆ p   .
.

 H pf 1

H p12 ......... H p1 f 

H p22 .........H p2 f 




H pf 2 .........H p f f 

where H p  n H p n
then diagonalised H p and find Eigen value and Eigen vector of diagonalized H p which
are En and n respectively.
f
En  En0  En1 and  n   q n
 1
7.2 Time Dependent Perturbation Theory
The transition probability corresponding to a transition from an initial unperturbed
state i to perturbed  f is obtained as
1
Pif  t   
i
iw t '
 f V  t   i e fi dt '
 0
Where w fi 
E f  Ei

and w fi 

f
Ho  f   i Ho  i


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8. Variational Method
Variational method is based on energy optimization and parameter variation on the basis
of choosing trial wave function.
1. On the basis of physical intuition guess the trial wave function. Say
 o   o 1 , 2 , 3 ,..... n  where 1 , 2 , 3 are parameter.
2. Find E0 1 ,  2 ,  3 ,..... n  
3. Find
 0 1 ,  2 ,  3 ,..... n  H  0 1 ,  2 ,  3 ,..... n 
 0 1 ,  2 ,  3 ,..... n   0 1 ,  2 ,  3 ,..... n 
Eo
 1 , 2 , 3 ,..... n   0
 i
Find the value of 1 ,  2 ,  3 ,.....  so that it minimize E0.
4. Substitute the value of 1 ,  2 ,  3 ,.....  in E0 1 , 2 , 3 ,.....  one get minimum value
of E0 for given trial wave function.
5. One can find the upper level of  1 on the basis that it must be orthogonal to  0 i.e.
1  0  0
Once  1 can be selected the 2, 3, 4 step can be repeated to find energy the first Eigen
state.
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9. The Wentzel-Kramer-Brillouin (WKB) method
WKB method is approximation method popularly derived from semi classical theory
For the case
d
dx
1 where  ( x) 

2 m ( E  V ( x)
If potential is given as V(x) then and there is three region
WKB wave function in the first region i.e x  x1
I 
V x 
 1 x1

exp    P  x ' dx '
P  x
  x

c1
WKB wave function in region II: i.e x1  x  x2
 II 
I
II
III
c2'
 i

c2"
i

exp   P  x ' dx  
exp    P  x ' dx '


P  x
P  x
 x

WKB wave function in region III: i.e x  x2
 III 
 1 x

exp    P  x ' dx '
P  x
  x2

c3
9.1.1 Quantization of the Energy Level of Bound state
Case I: When both the boundary is smooth
1
 p( x)dx  (n  2 ) where n  0,1, 2...
x2
1

2  2m  En  V  x   dx   n    where n  0, 1, 2,...... and x1 and x2 are turning
2

x1
point
Case II: When one the boundary is smooth and other is rigid
x2
3
 p( x)dx  (n  4 ) 
where n  0,1, 2...
x1
x2

x1
3

2m  En  V  x   dx   n     where n  0, 1, 2,...... and x1 and x2 are turning
4

points
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Case III: When both boundary of potential is rigid
x2
 p( x)dx  (n  1) 
where n  0,1, 2...
x1
x2

2m  En  V  x   dx   n  1   where n  0, 1, 2,...... and x1 and x2 are turning
x1
points .
9.1.2 Transmission probability from WKB
T is defined as transmission probability through potential barrier V is given by
x
T  exp  2 where  
1 2
2m  En  V  x   dx and x1 and x2 are turning points.
 x1
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10. Identical Particles
Identical particle in classical mechanics are distinguishable but identical particle in
quantum mechanics are Indistinguishable.
Total wave function of particles are either totally symmetric or totally anti-symmetric.
11.1 Exchange Operator
Exchange operator Pij as an operator as that when acting on an N-particle wave
function (1 23 ....i ... j ... N ) interchanges i and j.
i.e Pij (1 23 ....i ... j ... N )   (1 23 .... j ...i ... N )
 sign is for symmetric wave function s and  sign for anti symmetric wave
function a .
11.2 Particle with Integral Spins
Particle with integral spins or boson has symmetric states.
 s 1 2  
1
 1 ,  2    2 , 1  
2
For three identical particle:
 s 1  2  3  
1
6
 1 ,  2 , 3    1 , 3 ,  2     2 , 3 , 1  
  ,  ,      ,  ,     ,  ,  
2 1
3
3 1
2
3
2
1 

For boson total wave function(space and spin) is symmetric i.e if space part is symmetric
spin part will also symmetric and if space part is ant symmetric space part will also also
anti symmetric.
11.3 Particle with Half-integral Spins
Particle with half-odd-integral spins or fermions have anti-symmetric.
For two identical particle:
 a 1 2  
1
 1, 2   2 ,1 
2
For three identical particle:
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 a 1  2  3  
1
6
 1 ,  2 , 3   1 , 3 ,  2     2 , 3 , 1  


   2 , 1 , 3   3 , 1 ,  2   3 ,  2 , 1 
For fermions total wave function(space and spin) is anti symmetric .ie if space part is
symmetric spin part will anti symmetric and if space part is ant symmetric space part will
also symmetric.
11. Scattering in Quantum Mechanics
k

k
  o
Incident wave is given by inc  r   Aeiko .r . If particle is scattered with angle θ which is


angle between incident and scattered wave vector ko and k Scattered wave is given by

eik .r
sc  r   Af  ,  
, where f  ,   is called scattering amplitude wave function.
r
 iko r
e iko r 
is
superposition
of
incident
and
scattered
wave

  Ae  f  ,  

r 

differential scattering cross section is given by
For elastic collision
d
k
2

f  ,   where
d  ko
 is solid angle
d
2
 f  ,  
d
If potential is given by V and reduce mass of system is µ then
d
2
2
 f  ,   
d
4  2
e
 ikr '
V  r '   r '  d 3r '
2
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11.1 Born Approximation
Born approximation is valid for weak potential V(r)
f  ,    
2 
 r 'V  r '  sin  qr ' dr '
2 q 0
 
Where q  ko  k  2k sin
  d 

4 2
for and for ko  k

2
d  4 q 2
2

 r 'V  r ' sin  qr ' dr '
0
11.2 Partial Wave Analysis
Partial wave analysis for elastic scattering
For spherically symmetric potential one can assume that the incident plane wave is in zdirection hence inc  r   exp  ikr cos   . So it can be expressed in term of a superposition of
angular momentum Eigen state, each with definite angular momentum number l
eik r  eikr cos  

 i l  2l  l  J l  kr  pl  cos   , where Jl is Bessel’s polynomial function and Pl
l 0
is Legendre polynomial.

  r ,  
 il  2l  1 J l  kr  Pl  cos    f  
l 0
f   
eikr
r
1
  2l  1 eil sin l Pl  cos 
k
Total cross section is given by

   l 
l 0
4 
  2l  1 sin 2  l
k 2 l 0
Where σl is called the partial cross section corresponding to the scattering of particles in
various angular momentum states and  l is phase shift .
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12. Relativistic Quantum Mechanics
12.1 Klein Gordon equation
The non-relativistic equation for the energy of a free particle is
p2
E
2m
and

p2
  E
2m

p2

  i
2m
t
where
p  i is the momentum operator (  being the del operator).
The Schrödinger equation suffers from not being relativistic ally covariant, meaning it
does not take into account Einstein's special relativity .It is natural to try to use the
identity from special relativity describing the energy:
p 2 c 2  m2 c 4  E Then, just
inserting the quantum mechanical operators for momentum and energy yields the
equation
  2 c 2 2  m 2c 4  i

t
This, however, is a cumbersome expression to work with because the differential operator
cannot be evaluated while under the square root sign.
which simplifies to  2c 2 2  m 2c 4  
Rearranging terms yields
 2
t 2
1  2
m 2 c 2
2




 E
c 2 t 2
2
Since all reference to imaginary numbers has been eliminated from this equation, it can
be applied to fields that are real valued as well as those that have complex values.
Using the inverse of the Murkowski metric we ge     
m2c 2
 0
2
where
(   2 )  0
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In covariant notation. This is often abbreviated as (   2 )  0 where


mc
and

1 2
 2
2
2
c t
This operator is called the d’Alembert operator . Today this form is interpreted as the
relativistic field for a scalar (i.e. spin -0) particle. Furthermore, any solution to the Dirac
equation (for a spin-one-half particle) is automatically a solution to the Klein–Gordon
equation, though not all solutions of the Klein–Gordon equation are solutions of the Dirac
equation.
The Klein–Gordon equation for a free particle and dispersion relation
Klein –Gordon relation for free particle is given by  2 
1  2
 E
c 2 t 2
with the same solution as in the non-relativistic case:
dispersion relation from free wave equation  ( r , t )  exp i( k.r  t ) which can be
obtained by putting the value of in  2 
dispersion relation which is given by  k 2 
1  2
 E equation we will get
c 2 t 2
 2 m 2c 2
 2 .
k2

12.2 Dirac Equation
searches for an alternative relativistic equation starting from the generic form describing
evolution of wave function:
i

 Hˆ 
t
If one keeps first order derivative of time, then to preserve Lorentz invariance, the space
coordinate derivatives must be of the first order as well. Having all energy-related
operators (E, p, m) of the same first order:




Eˆ  i
and pˆ x  i
, pˆ y  i
, pˆ z  i
t
x
y
z
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Eˆ    1 pˆ x   2 pˆ y   3 pˆ z   m 
By acting with left-and right-hand operators twice, we get
Eˆ 2   1 pˆ x   2 pˆ y   3 pˆ z   m 1 pˆ x   2 pˆ y   3 pˆ z   m  
which must be compatible with the Klein-Gordon equation


Eˆ 2   pˆ x2  pˆ 2y  pˆ z2  m 2 
This implies that
 i j   j i  0, for i  j
 i    i  0
 i2  1
 2 1
Therefore, parameters α and β cannot be numbers. However, it may and does work if they
are matrices, the lowest order being 4×4. Therefore, ψ must be 4-component vectors.
Popular representations are
 0 i 

 i  
 i 0 
and
1 0 

  
 0  1
where  i are 2  2 Pauli matrices:
0 1

 1  
1 0
0  i
1 0 
  3  

 2  
i 0 
 0  1
The equation is usually written using γµ-matrices, where  i   i for
The equation is usually written using γµ-matrices, where  i   i for i  1, 2,3
and
 0   (just multiply the above equation with matrix β and move all terms on one side of
the equation):


0
 i    m   0 where  i  



x 
i


i 
1 0 
 and  0  

0
 0  1
Find solution for particles at rest, i.e. p=0:
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 1 
 

 2 

 i 0
 m    0
t

 3 
 
 4
i
 
  1 0   A 
  m A 


 t  0  1 B 
 B 
It has two positive energy solutions that correspond to two spin states of spin-½
electrons:
1
0
A  e imt   and  A  e imt  
0
1
and two symmetrical negative-energy solutions
1
 0
B  e imt   and B  e imt  
 0
1
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