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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES
Website: www.physicsbyfiziks.com Email: [email protected] 1
Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498
Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16
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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Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498
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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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Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498
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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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Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498
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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
mvh
ph where h is Plank’s constant
For relativistic case, the mass becomes
2
2
0
1cv
mm
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 qp are two canonical variable then
2
qpq
where ∆q is the error in measurement of q and ∆pq is error in measurement of pq and h is
Planck’s constant ( / 2 )h .
Important uncertainty relations
2
xPx (x is position and xp is momentum in x direction )
2
tE ( E is energy and t is time).
2
L (L is angular momentum, θ is angle measured)
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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 tx,1 and tx,2 are two waves of slightly different wavelength and frequency.
tdxdkkAtkxA sin,sin 21
21 tkxtddkA
sin
22cos2
The velocity of individual wave is known as Phase
velocity which is given ask
vp
. The velocity of
amplitude is given by group velocity vg i.e.dkdvg
The relationship between group and phase velocity is given by
ddv
vvdkdv
kvdkdv p
pgp
pg ;
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
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.
The Maxima condition is given by phdnor
dn
where2
cos2
sin2
S D
gv
phv
t
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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
cos1' cm
ho
where cm
ho
known as c which is Compton wavelength ( c = 2.426 x 10-12 m) and mo is
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
max 0k h h
Stopping potential sV is potential required to stop electron which contain maximum
kinetic energy maxk .
0seV h h , which is known as Einstein equation
photonincident photon scattered
ElectronTarget Electron Scattered
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2. Mathematical Tools for Quantum Mechanics
2.1 Dimension and Basis of a Vector Space
A set of N vectors N ,........, 21 is said to be linearly independent if and only if the
solution of equation
N
iiia
1
0 is 1 2 Na = a = ..... a =0
N dimensional vector space can be represent as
N
iiia
1
0 where i = 1, 2, 3 … are
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.
Square Integrable: If the integration or scalar product 2, 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 rdtrtr 3* ,, in three dimensions.
Properties of kets, bras and brakets:
*
** aa
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 rr
For even parity rr , for odd parity rr
Momentum operator xP ix
Energy operator H it
Laplacian operator 2
2
2
2
2
22
zyx
Position operator rxrX
Linear operator
For 2211 aa if an operator A applied on results in 1 1 2 2a A a A
then operator A 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 A . The
Eigen vector of A let say n form complete basis. Completeness relation is given
by1
n nn
I
Eigen value: The only possible result of measurement of a physical quantity na 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
corresponding observable A is
2
1
ngin
in
aP a
where gn is degeneracy of state and
nu is the Normalised Eigen vector of A 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 na ), Then the state of the system
immediately after the measurement is the normalized projection
n
n
P
P where Pn is
projection operator defined by n n .
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Projection operator P : An operator P is said to be a projector, or projection operator, if
it is Hermitian and equal to its own square i.e. PP ˆˆ 2
The projection operator is represented by n nn
Postulate 5: The time evolution of the state vector t is governed by Schrodinger
equation: ttHtdtdi , where H(t) is the observable associated with total
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
† † †
, , ; , , ,
, , , ; , ,
, , , , , 0 (Popularly known as Jacobi identity).
C C
C C B C
C C C
, 0f
If X is position and xP is conjugate momentum then
1,n nxX P nX i and 1, n n
x xX P nP i
If b is scalar and A is any operator then , 0b
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 A 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 n na
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
In
nnmnmn
1
,
where I is unity operator.
2.6 Time evaluation of the expectation of A (Ehrenfest theorem)
1 AA,Hd Adt 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 .
1d R Pdt m
, ,d P V R tdt
where R is position, P is momentum and ,V R t is potential operator.
2.7 Uncertainty relation related to operator
If A and B are two operator related to observable A and B then
B,A21B A
where 22 AA A and
22 BB A .
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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.
12
12
ikx
ikx
x k e dk
k x e dx
2.9 Expectation value and uncertainty principle
The expectation value A of A in direction of is given by
AA
or
A n na P where nP 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 2
2PH Vm
Time dependent Schrödinger wave equation is given by t
iH
Time independent Schrödinger wave equation is given by EH
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 xn 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 non-
degenerate. 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
xxim
J x
**
2
which satisfies the continually equation 0 J
t
Where * in general J v 2J 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
i
JRJ
and t
i
JTJ
3.4 The free particle in one dimension
Hψ = Eψ xEdxd
m
2
22
2
ikxikx AeeAx
Energy eigen value E mk
2
22 where 2
2
mEk the eigen values are doubly degenerate
3.5 The Step Potential
The potential step is defined as
000
xVx
xVo
Case I: E > Vo
00
22
11
2
1
xBeAexBeAe
xikxik
xikxik
Hence, a particle is coming from left so D = 0.
R = reflection coefficient = incident
reflected
JJ
=
2
1 2
1 2
k kRk k
xo
oV
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T = transmitted coefficient = incident
dtransmitte
JJ
=
1 22
1 2
4k kTk k
where 2221
22
oVEmkmEk
Case II: E < Vo
212011
mEkxBeAe xikxikI
22
202
EVmkxce oxkII
1r
t
JRJ
and 0t
i
JTJ
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.
Potential barrier is given by
axaxV
xxV
00
00
0
Case I: E > Vo
0
00
1
22
11
33
22
11
xEexx
axDeCexxxBeAexx
xik
xikxik
xikxik
Where 212
mEk 22
2
oVEmk
Transmission coefficient 1
2 1sin14
11
T
a
Energy
o x
oV
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Reflection coefficient
1
2 1sin141
R Where 2
2,
o
o
mVaVE
Case II: E < Vo
3.7.1 Tunnel Effect
1 1
2 2
1
0
0
0
ik x ik xI
k x k xII
ik xIII
Ae Be x
Ce Be x aFe x
21 sin 14 1
R h
and 211 sin 1
4 1T h
Where o
o
VEmVa ,2
2
For E << Vo
222
116
EVma
oo
o
eVE
VET
Approximate transmission probability akeT 22 where 22
2
EVmk o
3.8 The Infinite Square Well Potential
The infinite square well potential is defined as as shown in the figure
axax
xxV 00
0
Since V(x) is infinite in the region 0x 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
xV
o a x
ao
oV
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Edxd
m 2
22
2
sin cosA kx B kx
B = 0 since wave function must be vanished at boundary ie 0x so sinA kx
Energy eigen value for bound state can be find by ka n where 1,2,3...n
The Normalized wave function is for thn state is given by
2 sinnn xx
a a
Which is energy Eigen value correspondence to thn
2
222
2manEn
where n = 1, 2, 3 .....
othonormality relation is given by
0
sin sin i.e.2
0
12
a
mnm x n x adx
L Lm n
a m
If x is position operator x P Px is momentum operator and n x is wave function of
particle in nth state in one dimensional potential box then 2 sinnn xx
a a then
*
2 22 * 2
2 2
*
2 2 2 22 *
2 2
2
2 2
0
2
n n
n n
x n n
x n n
ax x x x
a ax x x xn
P x i xx
nP x xm x a
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The uncertainty product is given by 2221
121
nnPxx
The wave function and the probability density function of particle of mass m in one
dimensional potential box is given by
3.8.1 Symmetric Potential
The infinite symmetric well potential is given by
220
22axa
axoraxxV
Schrondinger wave function is given by
)(02
cos2
sin
)(02
cos2
sin
0,2
2cossin
22;
2 2
22
iikaBkaA
ikaBkaAso
xaxat
mEkwherekxBkxA
axaEdxd
m
Hence parity ( ) commute with Hamiltonian ( )H then parity must conserve
So wave function have to be either symmetric or anti symmetric
2a
2aa
xV
mEn
21
22
1
12 42 EEn
13 93 EEn
ax
a
sin21
ax
a
2sin2
2
ax
a
3sin2
3
21
22
23
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For even parity
cosB kx and Bound state energy is given by 02
cos ka
,....5,3,102
na
nkka
Wave function for even parity is given as a
xna
cos2
For odd parity
Ψ (x) = A sin kx and Bound state energy is given by
sin 02ka
, 02ka nk
a
2,4,6,........n
For odd parity Wave function is given as a
xna
sin2
The energy eigen value is given by ,......3,2,12 2
222
nma
nEn
First three wave function is given by
a
xa
x
ax
ax
ax
ax
3cos2
2sin2
cos2
3
2
1
where a2 is normalization constant.
2
22
2ma
2a
2a
2
22
29
ma
2
22
24
ma
2a
2a
2a
2a
x3
x2
x1
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3.9 Finite Square Well Potential
For the Bound state E < Vo
Again parity will commute to Hamiltonian
So wave function is either symmetric or
Anti symmetric
For even parity
2
22cos
2
axAex
axakxCx
axAex
xIII
II
xI
wave function must be continuous and differentiable at boundaries so using boundary
condition at 2a
one will get 2
tan kak
tan where 2a
2ka
For odd parity
2
22sin
2
2
1
axAex
axakxDx
axAex
xIII
x
using boundary condition one can get
cot where 2
2
EVm o 22
mEk
and 2
222
2amVn o which is equation of circle.
The Bound state energy will be found by solving equation
tann for even
2a
2a x
oV
xV
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cotn for odd
222
2
amVn o
one can solve it by graphical method.
The intersection point of = tan (solid curve) and 2
2 222
omV a
(circle) give
Eigen value for even state and intersection point at = - cot (dotted curve) and
22 2
22omV a
(circle) give Eigen value for odd state.
The table below shows the number of bound state for various range of 20V a where Voa2
is strength of potential.
Voa2 Even eigen function Odd eigen function No. of Bound
states
m2
22 1 0 1
maV
m 24
2
222
0
22
1 1 2
maV
m 29
2
222
0
22
2 1 3
maV
m 216
29 22
20
22
2 2 4
n
tann
o2
23 2
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The three bound Eigen function for the square well
3.10 One dimensional Harmonic Oscillator
One dimensional Harmonic oscillator is given by
xxmxV 22
21
The schrodinger equation is given by
Exmdxd
m
222
22
21
2
It is given that xm
and Ek 2
The wave function of Harmonic oscillator is given by.
22/14/1 2
!21
eH
nm
nnn
and energy eigen value is given by
nE = (n+1/2) ; n = 0, 1, 2, 3, ....
The wave function of Harmonic oscillator is shown
0H ( ) = 1 , 1H ( ) = 2 , 22H ( ) = 4 -2
It n and m wave function of Harmonic oscillator then
mnnm dxxx
x2a
2a
x2
x2a
2a
x1
x
x3
1n
20E
xV
0n
2n
23
1E
25
2
E
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Number operator
The Hamiltonian of Harmonic operator is given by 222
21
2Xm
mPH x
Consider dimensioned operator as HPX ˆ,ˆ,ˆ where
Xm
X ˆ
ˆx xP m P HH ˆ so 22 ˆˆ
21ˆ XPH x .
Consider lowering operator xiPXa ˆ2
1 and raising † 1 ˆ2 xa X iP
2/1ˆ NH
where †N a a and 1/ 2H N N is known as number operator
n is eigen function of N with eigen value n.
N n n n and 12
H n N n
nnnH
21 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
11
001
nnna
abutnnna
Expectation value of 22 ,,, XX PPXX in stationary states
mnPnm
X
PX
miaaPaa
mX
X
X
X
21,12
2
0,02
,2
22
2;0,
21
XX PXnfornPX
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4. Angular Momentum Problem
Angular momentum in Quantum mechanics is given kLjLiLL zyxˆˆˆ
Where xyzzxYyzX YPXPLXPZPLZPYPL ;;
andx
iPX
, y
iPY
, z
ipZ
Commutation relation
,x y zL L i L , ,y z xL L i L , ,z x yL L i L
2 2 2 2X Y ZL L L L and 2 , 0XL L , 2 , 0YL L , 2 , 0ZL L
4.1.1 Eigen Values and Eigen Function
iL
iL
SiniL
Z
Y
X
cotsincos
cotcos
Eigen function of ZL is
ime
21 .
and Eigen value of ZL m where m = 0, ± 1, ± 2...
L2 operator is given by
2
2
222
sin1sin
sin1
L
Eigen value of 2L is 2( 1)l l where l = 0, 1, 2, ... l
Eigen function of 2L is mlP where m
lP is associated Legendre function
L2 commute with Lz so both can have common set of Eigen function.
, ( )m m iml lY P e is common set of Eigen function which is known as spherical
harmonics .
The normalized spherical harmonics are given by
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2 1 !
( , ) 1 cos4 !
mm m iml l
l l mY P e
l m
l m l
41,0
0 Y
ieY sin83,1
1 , 01
3, cos4
Y
, 11
3, sin8
iY e
,1, 22 ml
ml YllYL l = 0, 1, 2, ……. And
,, ml
mlZ YmYL m = -l ,(-l +1) ..0,….. (l – 1)( l ) there is 2 1l
Degeneracy of 2L is 2 1l .
Orthogonality Relation 2
' '0 0
, , sinm ml l ll mmd Y Y d
4.1.2 Ladder Operator
Let X YL L iL and X YL L iL
Let us assume ml, is ket associated with 2L and ZL operator.
22 , 1 ,L l m l l l m 0,1, 2,.......l
, , , ...0,...zL l m m l m m l l
Action of L+ and L- on ml, basis
, 1 1 , 1L l m l l m m l m
, 1 1 , 1L l m l l m m l m
Expectation value of XL and YL in direction of ml,
0xL , 0yL 2
2 2 212X YL L l l m
2
212X YL L l l m
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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
ˆˆ ˆX Y ZS S i S j S k , 2 2 2 2
X Y ZS S S S
2 , 0XS S , 2 , 0YS S 2 , 0ZS S and
,X Y ZS S iS ,Y Z XS S iS ,Z X YS S iS
2 2, 1 ,s sS s m s s s m , ,z s sS s m m s m where ss m s
X YS S iS and X YS S iS
, 1 1 , 1s s s sS s m s s m m s m
4.2.3 Pauli Spin Matrices
For Spin 12
Pauli matrix 12
s , 1 1,2 2sm
Pauli Matrix is defined as
e.anticommutMatrix Spin Pauli;0
1001
00
0110
222
kjI
ii
jkkj
zyx
zyx
1 if is an even permutation of , ,1 if is an odd permutation of , ,
0 if any two indices among , , are equal jkl
jkl x y zjkl x y z
j k l
zzyyxx SSS 2
,2
,2
, 1 1 , 1s s s sS s m s s m m s m
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1001
200
20110
2
zyx Si
iSS
01
000010
SS
1001
43 2
2 S
For spin ½ the quantum number m takes only two values 21
sm and 21
. So that two
states are
21,
21and
21,
21, smS
21,
21
43
21,
21 2
2 S ,
21,
21
43
21,
21 2
2 S
21,
21
21
21,
21
zS , 21,
21
21
21,
21
zS
021,
21
S , 21,
21
21,
21
S
21,
21
21,
21
S , 021,
21
S
4.3 Total Angular Momentum
Total angular momentum J = L + S , kJjJiJJ zyxˆˆˆ
jmj, is the Eigen ket at J2 and Jz and x yJ J iJ x yJ J iJ
jj mjjjmjJ ,1, 22 , jjjz mjmmjJ ,,
1,11, jjjj mjmmjjmjJ
1,11, jjjj mjmmjjmjJ
J L S l s j l s
jmjSLJ jzzz and l s jm m m
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5. Two Dimensional Problems in Quantum Mechanics
5.1 Free Particle
Hψ = Eψ
Eyxm
2
2
2
22
2
x and y are independent variable. Thus
1,21
2
yx ik yik xn
i k r
x y e e
e
Energy Eigen value 22
222
22k
mkk
m yx
As total orientation of k
which preserve its magnitude is infinite. So energy of free
particle is infinitely degenerate.
5.2 Square Well Potential
0V x x a and 0 y a
otherewise
2 2 2
2 22H E
m x y
The solution of Schrödinger wave equation is given by Wave function
, 24 sin sin
x y
yxn n
n yn xa aa
Correspondence to energy eigen value
2
2
2
222
, 2 an
an
mE yx
nynx 1,2,3...xn and 1,2,3...yn
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Energy of state (nx, ny) Degeneracy
Ground state 2
22
22
ma
(1, 1)
Non degenerate
First state 2
22
25
ma
(1, 2), (2,1)
2
Second state 2
22
28
ma
(2, 2)
Non-degenerate
5.3 Harmonic oscillator
Two dimensional isotropic Harmonic oscillators is given by
22
2 2 212 2 2
yx ppH m x y
m m
1x yn n x yE n n where 0,1,2,3...xn 0,1,2,3...yn
1nE n
degeneracy of the nth 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
EH
Ezyxm
2
2
2
2
2
22
2
Hence x, y, z are independent variable. Using separation of variable one can find the
zikyikxikk
zyn eeezyx 2/321,,
3/ 2 .2 ik re
Energy Eigen value 22
2222
22k
mkkk
m zyx
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
, , 0 ,xV x y z x L 0 ,yy L 0 ,zz L
= other wise .
The Schrodinger wave equation for three dimensional box is given by
Ezyxm
H
2
2
2
2
2
22
2
Solution of the Schrödinger is given by Eigen function x y zn n n and energy eigen value
isx y zn n nE is given by
z
z
y
y
x
x
zyxnnn L
znL
ynL
xnLLLzyx
sinsinsin8
2
2
2
2
2
222
2 z
z
y
y
x
xnnn L
nLn
Ln
mE
zyx
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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
azn
ayn
axn
azyx
nnn zyx
sinsinsin8
3
2222
22
2 zyxnnn nnnma
Ezyx
1,2,3...xn 1,2,3...yn 1, 2,3...zn
Energy of state (nx, ny, nz) Degeneracy
E of ground state 2
22
23
ma
(1, 1, 1)
Non degenerate
E of first excited state 2
22
26
ma
(2, 1, 1) (1, 2, 1) (1, 1, 2
3
E of 2nd excited state 2
22
29
ma
(2, 2, 1) (2, 1, 2) (1, 2, 2)
3
6.3 Harmonic Oscillator
6.3.1 An Anistropic Oscillator
222222
21
21
21,, ZmYmXmZYXV ZYX
zzyyxxnnn nnnEzyx
21
21
21 where
0,1,2,3...xn 0,1, 2,3...yn 0,1,2,3...xn
6.3.2 The Isotropic Oscillator
x y z
32x y zn n n x y zE n n n
where x y zn n n n 0,1,2,3...n
Degeneracy is given by = 2121
nn
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6.4 Potential in Spherical Coordinate (Central Potential)
Hamiltonion in spherical polar co-ordinate
2 2
22 2 2 2 21 1 1sin
2 sin sinr V r E
m r rr r r
2 2
22 21
2 2Lr V r
m r rr mr
L2 is operator for orbital angular momentum square.
So ,mlY are the common Eigen state of and L2 because [H, L2] = 0 in central force
problem and 2 2, 1 ( , )m ml lL Y l l Y
So ,,r can be separated as ,mlf r Y
2 22
2 212 0
2 2
d f r l lf f r V r E f rm r rdr mr
To solve these equations rrurf
So one can get
02
12 2
2
2
22
uE
mrllrV
drud
m
Where 2
2
21
mrll is centrifugal potential and
2
2
21
mrllrV
is effective potential
The energy Eigen function in case of central potential is written as
, , , ,m ml l
u rr f r Y Y
r
The normalization condition is
2, , 1r d
2 2
22
0 0 0, sin 1m
lu r
r dr Y d dr
2
0
1u r dr
or 2 2
0
1R r r dr
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6.4.1 Hydrogen Atom Problem
Hydrogen atom is two body central force problem with central potential is given by
2
04eV r
r
Time-independent Schrondinger equation on centre of mass reference frame is given by
2 2
2 2 , ,2 2R r V r R r E R rm
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
h2
2M1
R R2 R
h2
21
r R2 r V r
E
Separating R and r part
RERM R 2
2
2
rrVrr
22
2
Total energy R rE = E + E
ER is Energy of centre of mass and Er is Energy of reduce mass µ
3/ 2
12
i k RR e
2 2
2RkEM
i.eCentre of mass moves with constant momentum so it is free particle.
Solution of radial part
h2
2d2u r
dr2 l l 1 h2
2r2 e2
4 0 r
u r E r
For Hydrogen atom em
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The energy eigen value is given by nE =
4
2 2 202 4em e
n
eV
n2
6.13 1,2,3...n
And radius of thn orbit is given by 2 2
02
4n
nrm e
1,2,3...n where me is mass of
electron and e is electronic charge.
n is known as participle quantum number which varies as 1, 2,...n l n For the
given value of n the orbital quantum number n can have value between 0 and 1n (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 = 2n and if spin is included the degeneracy
is given by 1
2
02 2 1 2
n
nl
g l n
For hydrogen like atom
2
26.13n
zEn where z is atomic number of Hydrogen like atom.
Normalized wave function for Hydrogen atom i.e. Rnl (r) where n is principle quantum
number and l is orbital quantum.
n l E(eV) Rnl
1
0
2-13.6z
0
3/ 2/
02 zr az e
a
2
0
2- 3.4 z
0
3/ 2/ 2
0 02
2zr az zr e
a a
2
0
2- 3.4 z
0
3/ 2/ 20
0 0
124
zr az zr ea a
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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 .
For the nth state there is n – l – 1 node
If Rnl is represented by ln, then
20
1 3 12
nl r nl n l l a
2 2 2 20
1 5 1 3 12
nl r nl n n l l a
12
0
1nl r nln a
2
3 20
22 1
nl r nln l a
rR10
0/ arr
rR20
0/arr
21R
0/ar
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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
o PH H H
Where H Hamiltonian can be divided into two parts in that oH can be solved exactly
known as unperturbed Hamiltonian and pH is perturbation in the system eigen value
of oH is non degenerate
It is known nonno EH and 1 pH W where 1
Now o n n nH W E where n is eigen function corresponds to eigen value nE
for the Hamiltonian H
Using Taylor expansion
........2210 nnnn EEEE and .........221 nnnn
First order Energy correction 1nE is given by 1
n n nE W
And energy correction up to order in is given by 1n n n nE E W
First order Eigen function correction 10 0
m nn m
m n n m
WE E
And wave function up to order correction in 0 0m n
n n mm n n m
WE E
Second order energy correction
2
20 0
m nn
m n n m
WE
E E
Energy correction up to order of 2
2
20 0 0
m nn n n
m n n m
WE E W
E E
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7.1.2 Degenerate Theory
( )n o p n n nH H H E
0nE is f fold degenerate fEH nnno ,.....3,2,10
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 pH
ff
f
f
ppfpf
ppp
ppp
p
HHH
HHH
HHH
H
...........
.........
.........
ˆ
21
22221
11211
where p n p nH H
then diagonalised pH and find Eigen value and Eigen vector of diagonalized pH which
are nE and n respectively.
10 nnn EEE and
f
nn q1
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
'
0
'fiiw tif f i
iP t V t e dt
Where f ifi
E Ew
and
f o f i o ifi
H Hw
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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
noo ,.....,, 321 where 321 ,, are parameter.
2. Find nn
nnn
HE
,.....,,,.....,,
,.....,,,.....,,,.....,,
32103210
321032103210
3. Find 0,.....,, 321
ni
oE
Find the value of ,.....,, 321 so that it minimize E0.
4. Substitute the value of ,.....,, 321 in ,.....,, 3210E 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 1ddx where ( )
2 ( ( )x
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 1x x
11 1exp ' '
x
Ix
c P x dxP x
WKB wave function in region II: i.e 1 2x x x
' "2 2exp ' exp ' 'II
x
c ci iP x dx P x dxP x P x
WKB wave function in region III: i.e 2x x
2
3 1exp ' 'x
IIIx
c P x dxP x
9.1.1 Quantization of the Energy Level of Bound state
Case I: When both the boundary is smooth
1( ) ( )2
p x dx n where 0,1,2...n
2
1
12 22
x
nx
m E V x dx n where 0, 1, 2,......n and 1x and 2x are turning
point
Case II: When one the boundary is smooth and other is rigid
2
1
3( ) ( )4
x
x
p x dx n where 0,1,2...n
2
1
324
x
nx
m E V x dx n where 0, 1, 2,......n and 1x and 2x are turning
points
xV
I II III
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Case III: When both boundary of potential is rigid
2
1
( ) ( 1)x
x
p x dx n where 0,1,2...n
2
1
2 1x
nx
m E V x dx n where 0, 1, 2,......n and 1x and 2x are turning
points .
9.1.2 Transmission probability from WKB
T is defined as transmission probability through potential barrier V is given by
exp 2T where 2
1
1 2x
nx
m E V x dx and 1x and 2x are turning points.
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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 ijP 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 1 2 3 1 2 3( .... ... ... ) ( .... ... ... )ij i j N j i NP
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.
1 2 1 2 2 11 , ,2s
For three identical particle:
123213312
132231321321 ,,,,,,
,,,,,,6
1
s
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:
122121 ,,2
1 a
For three identical particle:
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1 2 3 1 3 2 2 3 1
1 2 32 1 3 3 1 2 3 2 1
, , , , , ,1, , , , , ,6a
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
Incident wave is given by .oik rinc r Ae
. If particle is scattered with angle θ which is
angle between incident and scattered wave vector ok
and k
Scattered wave is given by
.
,ik r
scer Af
r
, where ,f is called scattering amplitude wave function.
is superposition of incident and scattered wave
refeA
rikrik
oo ,
differential scattering cross section is given by 2,o
d k fd k
where is solid angle
For elastic collision 2,d fd
If potential is given by V and reduce mass of system is µ then
k
ok
2 22 ' 3
2, ' ' '4
ikrd f e V r r d rd
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11.1 Born Approximation
Born approximation is valid for weak potential V(r)
20
2, ' ' sin ' 'f r V r qr drq
Where 2 sin2oq k k k
for and for ok k
22
4 20
4 ' ' sin ' 'd r V r qr drd q
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 z-
direction hence exp cosinc r ikr . So it can be expressed in term of a superposition of
angular momentum Eigen state, each with definite angular momentum number l
cos
02 cosik r ikr l
l ll
e e i l l J kr p
, where Jl is Bessel’s polynomial function and Pl
is Legendre polynomial.
0
, 2 1 cosikr
ll l
l
er i l J kr P fr
1 2 1 sin coslil lf l e P
k
Total cross section is given by
22
0 0
4 2 1 sinl ll l
lk
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
2
2p Em and
2
2p Em
2
2p im 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: 2 2 2 4p c m c E Then, just
inserting the quantum mechanical operators for momentum and energy yields the
equation 2 2 2 2 4c m c it
This, however, is a cumbersome expression to work with because the differential operator
cannot be evaluated while under the square root sign.
which simplifies to2
2 2 2 2 42c m c
t
Rearranging terms yields 2 2 2
22 2 2
1 m c Ec t
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 2 2
2 0m c
where
2( ) 0
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In covariant notation. This is often abbreviated as 2( ) 0 where mc
and
22
2 21c 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
22 2
1 Ec t
with the same solution as in the non-relativistic case:
dispersion relation from free wave equation ( , ) exp ( . )r t i k r t which can be
obtained by putting the value of in 2
22 2
1 Ec t
equation we will get
dispersion relation which is given by 2 2 2
22 2
m ckk
.
12.2 Dirac Equation
searches for an alternative relativistic equation starting from the generic form describing
evolution of wave function:
Ht
i ˆ
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:
t
iE
ˆ and z
ipy
ipx
ip zyx
ˆ,ˆ,ˆ
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mpppE zyx ˆˆˆˆ321
By acting with left-and right-hand operators twice, we get
21 2 3 1 2 3
ˆ ˆ ˆ ˆ ˆ ˆ ˆx y z x y zE p p p m p p p m
which must be compatible with the Klein-Gordon equation
22222 ˆˆˆˆ mpppE zyx
This implies that
,0 ijji for ji
0 ii
12 i
12
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
00
i
ii
and
10
01
where i are 2 2 Pauli matrices:
1001
00
0110
321 i
i
The equation is usually written using γµ-matrices, where i i for
The equation is usually written using γµ-matrices, where i i for 1,2,3i and
0 (just multiply the above equation with matrix β and move all terms on one side of
the equation):
0
mx
i
where
0
0
i
ii
and
10
010
Find solution for particles at rest, i.e. p=0:
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0
4
3
2
1
0
mt
i
B
A
B
A mt
i10
01
It has two positive energy solutions that correspond to two spin states of spin-½
electrons:
01imt
A e and
10imt
A e
and two symmetrical negative-energy solutions
01imt
B e and
10imt
B e