Quantum Formula Sheet

fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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 Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: [email protected]

Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 1

fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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 Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: [email protected]


fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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

Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: [email protected]


fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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


where  

h p


fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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

   



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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

ˆ applied on  if an operator A

results in a1 A 1  a2 A 2

ˆ is said to be linear operator. then operator A



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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

ˆ . The variable. For every observable there corresponds a linear Hermitian operator A ˆ let say   form complete basis. Completeness relation is given Eigen vector of A n 

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

ˆ associated with Eigen value an. u n is the Normalised Eigen vector of A 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 .



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Projection operator Pˆ : An operator Pˆ 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

ˆ is also an Eigen vector of B . degenerate Eigen value, then each Eigen vector of A We can also construct the common orthonormal basis that will be joint Eigen state of A and B .

The anti commutator is defined as

 ,     



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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

ˆ and B ˆ are two operator related to observable A and B then If A ˆ B ˆ,B ˆ 1 A ˆ A 2

 

ˆ  where A

ˆ2  A ˆ A

2

ˆ  and A

ˆ2  B ˆ B

2

.



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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  .



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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.



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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 






k2 

2mE  Vo  2

1


fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES  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 Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: [email protected]


fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES



 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



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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 Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: [email protected]


fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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




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



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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








m x n x dx   mn


fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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 



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES  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



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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...




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.



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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


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 


fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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


m W n

2

En 0   Em 0 


fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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 Hˆ 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







fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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.



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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 Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: [email protected]


fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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:




 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



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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 .



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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 Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: [email protected]


fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 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:



fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES  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



Quantum Formula Sheet

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