Chap 4. Quantum Mechanics In Three Dimensions 1.Schrodinger Equation in Spherical Coordinates 2.The Hydrogen Atom 3.Angular Momentum 4.Spin.

Chap 4. Quantum Mechanics In Three Dimensions

1. Schrodinger Equation in Spherical Coordinates

2. The Hydrogen Atom

3. Angular Momentum

4. Spin

4.1. Schrodinger Equation in Spherical Coordinates

1. Separation of Variables

2. The Angular Equation

3. The Radial Equation

Read Prob 4.1

Orthogonal Curvilinear Coordinates

22i

i

ds ds i i ids h dq

Ref: G.Arfken, “Mathematical Methods for Physicists”, 3rd ed., Chap. 2.B.Schutz, “Geometrical Methods of Mathematical Physics”, p.148.

ˆ ii i i

ff

h q

e 1 i

i i i

VH

H q h

V ii

H h

ˆ i ii

VV e

22

1

i i i i

ff H

H q h q

1 1 2 2 3 3

1 2 3

1 1 2 2 3 3

1

h h h

H q q q

hV h V h V

e e e

V

Spherical coordinates : 1, , sinih r r

4.1.1. Separation of Variables

2

2

2

H V E

mr

22 2

2 2 2 2 2

1 1 1sin

sin sin

rr r r r r

V V(r) Spherical coordinates :

2 2

22 2 2 2 2

1 1 1sin

2 sin sin

r V r E

m r r r r r

Ansatz : , , , r R r Y

2 22

2 2 2 2 2sin

2 sin sin

Y d d R R Y R Yr VRY ERY

m r d r d r r r

2 22

2 2 2 2

1 1 1 1sin

2 sin sin

d d R Y Yr V E

m R r d r d r r Y Y

Set

2

2 2

1 1sin

sin sin

Y Y

Y Y dimensionless constant

Or

2 22

2 2 2 2

1 1 1 1sin

2 sin sin

d d R Y Yr V E

m R r d r d r r Y Y

22

2 2

1

2

d d Rr V E

m R r d r d r r

2

2 2

1 1sin

sin sin

Y YY

2

22

2

d d R m r

r V E R Rd r d r

Mnemonics

22 2

2 2 2 2 2

1 1 1sin

sin sin

rr r r r r

22 2

2 2

1r

r r r r

L

22

2 2

1 1sin

sin sin

L

2 1l m l mY l l Y L

2 2 22

2 2

1

2 2H r

m r r r m r

L

Do Prob 4.2

4.1.2. The Angular Equation

2

2 2

1 1sin

sin sin

Y YY

Ansatz : , Y

2

2 2

1 1 1sin

sin sin

d d d

d d d

Set2

22

1

dm

d

2

2

1sin 0

sin sin

d d m

d d

Azimuthal Solutions

22

2

1

dm

d ime

single-valued, i.e., 2

0, 1, 2, ... m

2 1 ime

Legendre Polynomials

Frobenius method shows that convergence requires

2

2

1sin 0

sin sin

d d m

d d

1 l l 0,1, 2,...l

cosx 2 2

22 2

1 2 01

d d mx x

d x d x x

Setting m 0 gives 2

22

1 2 0

d dx x

d x d x

The corresponding solutions are called the Legendre polynomials, which can also be defined by the Rodrigues formula :

211

2 !

ll

l l

dP x x

l d x

See Arfken (3rd ed) Ex 8.5.5

1st few Legendre Polynomials

P0 1P1 x

P2 123 x2 1

P3 125 x3 3 x

P4 1835 x4 30 x2 3

P5 1863 x5 70 x3 15 x

( ), 0, 2,1 5, 4,3,lP x l

Normalization:Pl (1) = 1

1

2

1

2

2 1ld x P xl

Associated Legendre Functions

2 2

22 2

1 2 01

d d mx x

d x d x xSolutions to the m 0 case :

are called associated Legendre functions defined by

2 21

mmm

l l

dP x x P x

d xwhere l m

Thus 0,1, 2,...l

while m takes on 2l + 1 values : , 1, ..., 1, 0, 1, ..., 1, m l l l l

Note : Another independent solution exists but is not physically acceptable ( see Prob. 4.4 ).

2 21mm

ml l

dP x x P x

d x

!

!mm m

l l

n mP x P x

n m

Griffiths:

Arfken,Mathenmatica:

1st few Associated Legendre Polynomials

P00 1 P1

0 cosP1

1 sin P20 1

43 sin2 3 cos2 1

P21 3 sin cos P2

2 3 sin2P3

0 18cos 15 sin2 5 cos2 3 P3

1 32sin 5 cos2 1

P32 15 sin2 cos P3

3 15 sin3 0

0 P 01 P 1

1 P

02 P 1

2 P 22 P

Normalization

, cos exp ml m l m lY A P i m

Normalization : 1 2

2

1 0

cos , 1

l md d Y

1

1 0

cos sin

d d

Griffiths :

!2 1

4 !

l m

l mlA

l mwhere

for 0

1 for 0

mm

m

Note :

Orthonormality : 1 2

*

1 0

cos , ,

l m l m l l mmd d Y Y

Spherical Harmonics

Arfken,Mathenmatica:

!2 1

4 !m

l m

l mlA

l m

1st few Spherical Harmonics

* m

l m l mY Y

Read Prob 4.4, 4.6

Do Prob 4.3

4.1.3. The Radial Equation

2

22

21

d d R m r

r V E R l l Rd r d r

Set u

Rr

2

1

d R u du

d r r r d r2

d d R d dur u r

d r d r d r d r

2

2

d ur

d r

2 2 2

2 2

1

2 2

l ld uV u E u

m d r m r

2

2 2

21

d u m r u

r V E u l ld r r

Effective potential 2

2

1

2

eff

l lV V

m r Centrifugal term 2

2

1

2

l l

m r

Normalization :22

0

1

r d r R2

0

d r u

Example 4.1. Infinite Spherical Well

0 if

if

r aV r

r aFind the wave functions and the allowed energies.Let

Ans :

, if

0 if

ll m

u rY r a

rr a

r

2 2 2

2 2

1

2 2

l

l l

l ld uu E u

m d r m r

22

2 2

10

ll

l ld uk u

d r r

2

mE

k

l l lu r r A j kr B n kr jl = spherical Bessel functionnl = spherical Neumann function

Spherical Bessel & Neumann Functions

1 sin

ll

l

d xj x x

x d x x 1 cos

ll

l

d xn x x

x d x x

jl (0) is finite nl (0)

ll l

u rR r A j kr

r

0lR a

Let n l be the nth zero of jl .

nlka

22

2

nlnlE

m a , if

0 if

nlnl l l m

nl m

A j r Y r aa

r a

r

(2l+1)-fold degeneracy in m.

1st Few Spherical Bessel & Neumann Functions

( ), , 31 20, ,lj x l

( ), ,1, 2 30,ln x l

Do Prob 4.9

Bessel & Neumann Functions

22 2

2

10

d d R mk R

d d

The Bessel & Neumann functions are solutions to the radial part of the Helmholtz equation in cylindrical coordinates :

2 22

2 2 2

1 1

z

2 2 0k

1, , 1ih 22 2 2ds d d dz

cos

sinim i z z

R e e orz

2 2 2 2 0R R m R m mR A J B N

Bessel Neumann functions

2 2 2k

Modified Bessel functions mm mI x i J ix ( for 2 < 0 )

Spherical Bessel & Neumann Functions

2 22 2

11 l ld d Rr k R

r d r d r r

The spherical Bessel & Neumann functions are solutions to the radial part of the Helmholtz equation in spherical coordinates :

22 2

2 2

1 Lr

r r r r

2 2 0k

1, , sinih r r 2 22 2 sinds dr rd r d

mlR Y

2 2 22 1 0r R rR k r l l R

l lR A j kr B n kr

Spherical Bessel Neumann functions

22 2 2 1

02

r Z rZ k r l Z

Z kr

R krkr

1/2

2l

l

J xj x

x

1/2

2l

l

N xn x

x

Asymptotic Forms

0

sin xj x

x 0

cos xn x

x

~ nnj x x for x 0 1~ n

nn x x

sin

2~n

nx

j xx

for x

cos2~n

nx

n xx

4.2. The Hydrogen Atom

1. The Radial Wave Function

2. The Spectrum of Hydrogen

2 2 2 2

2 20

1

2 4 2

l ld u eu E u

m d r r m r

2

04

eV r

r

Bohr’s Model

2

04

eV r

r

2

20

ˆ4

e

rF r

Circular orbit :2 2

204

mv e

r r

2

ˆv

ra r

2

04

ev

m r

22

0

1

2 4

eE mv

r

2 2

0 08 4

e e

r r

2

0

1

2 4

e

r

Quantization of angular momentum : m v r n

2

04

n e

m r m r

2 2

0 24n

nr

me

2

20

1 1

2 4n

eE

a n

12

E

n

2

0 24a

me

0.529 A

Bohr radius22

202 4

m e

13.6 eV

2n a

2

10

1

2 4

eE

a

4.2.1. The Radial Wave Function

2 2 2 2

2 20

1

2 4 2

l ld u eu E u

m d r r m r

Bound States ( E < 0 ) : Set2

mE

2 22

2 2 20

1

2

l ld u m eu u

d r r r

Set r

2

02 20

4

2

e

m

20

2 2

11 0

l ld uu

d

2

202

m e

Asymptotic Behavior

20

2 2

11 0

l ld uu

d

:2

20

d uu

d u A e B e

u finite everywhere u A e

2

2 2

10

l ld u

ud

0 : 1 l lu C D

u finite everywhere 1 lu C

Set 1 lu e v

Factor-Out Asympototic Behavior

1 lu e v 20

2 2

11 0

l ld uu

d

1

ldu d v

e l vd d

2 2

2 21 1 1

ld u l d v d v d v d ve l v l v

d d d d d

2

2

12 2 2 1

l l l d v d ve l v l

d d

2

02 2

1 12 2 2 1 1 0

l l l ld v d vl v l v

d d

2

022 1 2 1 0

d v d vl l v

d d

Frobenius Method

2

022 1 2 1 0

d v d vl l v

d d

0

jj

j

v c 1

1

jj

j

d vj c

d 1

0

1

jj

j

j c

2

112

0

1

jj

j

d vj j c

d

2

120

1

jj

j

d vj j c

d

0

jj

j

d vj c

d

1 1 00

1 2 1 1 2 2 1 0

jj j j j

j

j j c l j c j c l c

1 01 2 1 2 1 0 j jj j l c j l c

01

2 1

1 2 1

j j

j lc c

j j l

Series Termination

01

2 1

1 2 1

j j

j lc c

j j l

j : 1 2

2 j j

jc c

j

2 jc

j

1

2

1 j jc c

j 2

2

2

1 2 jc

j j 0

jj

j

d vj c

d

1

1

2

1 !

j

j

v cj

1

1

2

1 !

j

cj

21

e c

0

jj

j

v c

1 lu e v 21

lu e c ( unacceptable for large )

Series must terminate : max 02 1 0 j l

Eigenenergies

max 02 1 0 j l

Let max 1 n j l = principal quantum number

0 2 n2

202

m e

2

204

m e

n

2 2 4

2

02 2 4

m e

Em n

22

2 20

1

2 4

n

m eE

n1

2

E

nn 1, 2, 3, ...

Bohr radius :2

1002

40.529 10

a mm e

1

an

rr

a n

21

2

nE

m n a

Eigenfunctions

1

l

nle v

r

, , , nl m nl l mr R r Y

nlnl

uR r

r

rr

a n

1

2 1

1 2 1

j j

j l nc c

j j l

1

0

n l

jnl j

j

v c with

21

2

m n a

Eigenfunction belonging to eigenenergy

is

where

22

2 20

1

2 4

n

m eE

n

1l n

Ground State

22

1 20

13.62 4

m e

E eV

100 10 00, , , r R r Y

1010

e v

R rr

2210

0

1

d r r R

10 0 v c/0 r ac

ea

00

1,

4

Y

Normalization :2

2 2 /0

0

r acd r r e

a

220

08

xac

d x x e0

!

n xd x x e n20

4

ac

0

2c

a /10 3/2

2 r aR r ea

/100 3

1, ,

r ar e

a

n 1, l 0.

11

2

1

0

1

,

,

2 1

1 2 1

lnl

n nl

n lj

nl jj

j j

E e vE R

n rr

v ca n

j l nc c

j j l

r

a

1st Excited States

2

13.63.4

4

eVE eV

n 2, l 0, 1.

11

2

1

0

1

,

,

2 1

1 2 1

lnl

n nl

n lj

nl jj

j j

E e vE R

n rr

v ca n

j l nc c

j j l 21 21 1, , , m mr R r Y

221

21

e v

R rr

21 0 v c/ 2024

r acr e

a

m 1, 0, 1

2

r

a

20 20 00, , , m r R r Y

2020

e v

R rr

/ 20 12 2

r ac re

a a

20 0 1 v c c

1 0 0

2 1 2

2

c c c

Normalization : see Prob. 4.11

Degeneracy of nth excited state : 1

0

2 1

n

l

d n l 12 1

2 n n n 2n

Associated Laguerre Polynomials

2 11 2

lnl n lv L

p

ppq p qp

dL x L x

d xAssociated Laguerre polynomial

q

x x qq q

dL x e e x

d xqth Laguerre polynomial ; Used by Griffiths.

p

ppn n pp

dL x L x

d x

1

!

nx x n

n n

dL x e e x

n d xUsed by Arfken & Mathematica. 1/n! of Griffiths’ value.

Used by Arfken & Mathematica.1/(n+p)! Griffiths’ value.

q p q p

Differential eqs. :

1 0p p pn n nx L p x L n L 2 1 2 1 0v l v n l v

0 1nL

!0

! !pn

n pL

n p

1st Few Laguerre & Associated LaguerrePolynomials

Ln : Arfken & Mathematica convention Griffiths’ / n!

Lna : Arfken & Mathematica convention

Griffiths’ / (n+a)!

Orthogonal PolynomialsRef: M.Abramowitz, I.A.Stegun, “Handbook of Mathematical Functions”, Chap 22.

Orthogonality:

1 1n n n n n nf a b x f c f

b

n m nm n

a

d x w x f x f x A w weight function

Recurrence relations:

2 1 0n n n ng x f g x f a f Differential eq.:

1 nn

n nn

df w x g x

e w x d x Rodrigues’ formula:

fn (a,b) en w g Standard An

Pn (1,1) ()n n! 2n 1 x2 1 Pn(1) = 1 2 / (2n+1)

Ln ( 0, ) 1 e x x 1

Lnp ( 0, ) ()p e x xp x (p+n)! / n!

Hn ( , ) ()n exp(x2/2) 1 en = (1)n n! 2

Hydrogen Wave Functions

3

/ 2 113

1 !2 2 2,

2 !

l

r na lnl m n l l m

n l r re L Y

na na nan n lr

Orthonormality :1 2

2 *

0 1 0

cos

nl m n l m nn l l mmd r r d d

22

2 20

1

2 4

n

m eE

n

3

/ 2 11

1 !2 2 2,

2 !

l

r na lnl m n l l m

n l r re L Y

na n n l na nar

Griffithsconvention

Arfkenconvention :[3rd ed., eq(13.60)]

0

!

!x p p p

n m nm

n pd x e x L x L x

n

Arfken

1

0

!2 1

!x p p p

n n

n pd x e x L x L x n p

n

First Few Rnl (r)

Rnl Plots

20( ), 10, , , , 32 1 30 21 ,3nlR r nl

Note: Griffiths’ R31 plot is wrong.

(n 0) has n1 nodes(n, n1) has no node

Density Plots of 4 l 0

(400)

(430)(420)

(410)

White = Off-scale

White = Off-scale

(n 0 m) has n1 nodes(n, n1, m) has no node

Surfaces of constant | 3 l m |

(322)(300)(320)

(310)(321)Warning : These are plots

of | |, NOT | |2 .

Do Prob 4.13,4.15.

4.2.2. The Spectrum of Hydrogen

H under perturbation transition between “stationary” levels:

energy absorbed : to higher excited state

energy released : to lower state

H emitting light ( Ei > Ef ) : i fE E E 2 2

1 113.6

i f

eVn n

E hPlanck’s formula :

c

h

2 2

1 1 1

f i

Rn n

where 1E

Rhc

22

304 4

m e

c7 11.097 10 m Rydberg constant

Rydberg formula

H Spectrum

nf 1 nf 2 nf 3

Series Lyman Balmer Paschen

Radiation UV Visible IR

2 2

1 1 1

f i

Rn n

4.3. Angular Momentum

1. Eigenvalues

2. Eigenfunctions

L r p x y z

x y z

x y z

p p p

e e e

CM :

QM : i

p

z y x x z y y x zy p z p z p x p x p y pe e e

Commutator Manipulation

, A B C A B C B C A AB AC BA CA

, , , A B C A B A C distributive

, A BC ABC BCA ABC BAC BAC BCA

, , , A BC A B C B A C

, , , AB C A C B A B CSimilarly

[ Li , Lj ]

, , x y z y x zL L y p z p z p x p

z y x x z y y x zy p z p z p x p x p y pL e e e

, , , , z x z z y x y zy p z p y p x p z p z p z p x p

, z x z x x zy p z p y p z p z p y p xi y p

, z z z z z zy p x p y p x p x p y p

x z zy p p z z p

0

, y x y x x yz p z p z p z p z p z p 0

, , , A B C A B A C

, y z y z z yz p x p z p x p x p z p y z zp x z p p z yi x p

, x y x yL L i y p x p zi L

Cyclic permutation :

,

,

,

x y z

y z x

z x y

L L i L

L L i L

L L i L

,j k j kx p i

Uncertainty Principle

,

,

,

x y z

y z x

z x y

L L i L

L L i L

L L i L1 ˆ ˆ,2

A B A B

1

2

x yL L zL

Only one component of L is determinate.

[ L2, L ]

2 2 2 2 x y zL L L L

2 2 2 2, , x x y z xL L L L L L 2 2, , y x z xL L L L

, , ,

, , ,

A BC A B C B A C

AB C A C B A B C

, , , , y y x y x y z z x z x zL L L L L L L L L L L L

y z z y z y y zi L L L L L L L L 0

Similarly 2 , 0 iL L for i x, y, z

2 , 0 L Li.e.

L2 & Lz share the same eigenfunction :2 L f f zL f f

4.3.1. Eigenvalues

Ladder operators :

, , z z x yL L L L i L y xi L iL

,

,

,

x y z

y z x

z x y

L L i L

L L i L

L L i L

y xiL L

, zL L L

2 2, , x yL L L L i L 0 2 , 0 zL L

Let 2 L f f zL f f

2 2 L L f L L f L f L f is an eigenfunction of L2.

z zL L f L L L f L f L f is an eigenfunction of Lz.

L raiseslowers eigenvalue of Lz by .

maxz t tL f f

Lz finite max max().

Let 0 tL f 2 t tL f fAlso

x y x yL L L i L L i L 2 2 x y x y y xL L i L L L L 2 2 x y zL L L

2 2 z zL L L L L , 2 zL L L

Now

2 2 t z z tL f L L L L f 2

max max tf tf

max max

Lz finite min min().

minz b bL f fLet 0 bL f

2 2b z z bL f L L L L f 2

min min bf bf

min min

max max min min

Also2

b bL f f N

b tf L f

max max min min

2max min min

14

2 min

12

2 min

min

Since max min max min

Let

m

max min N N = integer

max2 Nmax 2

N

ormax must be integer or half integer

Let max l

, 1, ..., 1, m l l l l

1 30, , 1, , ...

2 2l

z l m l mL f m f 2 21 l m l mL f l l f

where

Diagram Representation of L

L can’t be represented by a vector fixed in space since only ONE of it components can be determinate.

l 2

4.3.2. Eigenfunctions

Gradient in spherical coordinates :

1 1ˆ ˆˆsin

r

r r r

i

L r1ˆ ˆ

sin

i

ˆr rr ˆ ˆˆ r ˆ ˆˆ r

ˆ ˆˆ ˆcos cos cos sin sin i j k

ˆ ˆ ˆsin cos i j

sin cot cos

xL

i

cos cot sin

yL

i

zL

i

2

ˆ ˆˆ sin1

0 0sin

r r r

rr

r

r

sin cot cos

xL

icos cot sin

yL

i

x yL L i L sin cos cot cos sin

i i

i

cot

i ii e ei

cot

ie i

22 2

2 2

1 1sin

sin sin

L Do Prob. 4.21

2 21 l m l mL f l l f z l m l mL f m f l m l mf Y

2 22 2

2 2cot cot

L L i

2 2 22 2 2

1 1

r Lr r r r Read Prob 4.18, 4.19, 4.20

Do Prob 4.24

4.4. Spin

1. Spin 1/2

2. Electrons in a Magnetic Field

3. Addition of Angular Momenta

Spin

Spin is an intrinsic angular momentum satisfying

, x y zS S i S , y z xS S i S , z x yS S i S

2 21 S sm s s sm zS sm m smwith

1 1 1 S sm s s m m s m

1 30, , 1, , ...

2 2s , 1, ..., 1, m s s s s

4.4.1. Spin 1/2

2-D state space spanned by1 1

2 2(spin up )

1 1,

2 2 (spin

down )

In matrix form (spinors) :1

0

0

1

General state :

a

b a b

Operators are 22 matrices.

2 23

4 S 1 1 3

1 12 2 4

s s 2 2 1 03

0 14

S

1

2 zS

1 01

0 12

zS

Pauli Matrices

0 S 3 1 11

4 2 2

S

1 1 , 1 S sm s s m m s m

3 1 11

4 2 2

S 0 S

0 1

0 0

S

0 0

1 0

S

x yS S i S 1

2 xS S S 1

2 yS S Si

0 1

1 02

xS

2

xσ0

02

y

i

iS

2

yσ2

z yS σ

0 1

1 0

xσ0

0

y

i

iσ

1 0

0 1

zσ

Pauli matrices

Spin Measurements

Let particle be in normalized state :

a

b2 2

1 a bwith

Measuring Sz then has a probability | a |2 of getting /2,

and probability | b |2 of getting /2,

Characteristic equation is/ 2

0/ 2

0 1

1 02

xS

2 210

4

1

2

0 1 1

1 02 2

Eigenvectors :

11

12

x 11

12

x

Eigenvalues :

11

12

x 11

12

x

Measuring Sx then has a probability | |2 of getting /2,

and probability | |2 of getting /2,

Writing in terms of (x) :

1 11 1

1 12 2

a

b

2 a

2 b

1

2 a b

1

2 a b

Read last paragraph on p.176.

Do Prob 4.26, 27 Read Prob 4.30

4.4.2. Electrons in a Magnetic Field

Ampere’s law : Current loop magnetic moment .

Likewise charge particle with angular momentum.

μ L gyromagnetic ratio

QM : Spin is an angular momentum : μ S

H μ B

experiences a torque when placed in a magnetic field B : Γ μ B

tends to align with B,

i.e., // B is the ground state with 0 S B

QM : L has no fixed direction can’t be aligned to B

Larmor precession

Example 4.3. Larmor Precession

Consider a spin ½ particle at rest in uniform 0ˆBB k

The Hamiltonian is 0 zBH S 01 0

0 12

B

H & S share the same eigenstates : +

H E+ B0 / 2 E + B0 / 2

Sz / 2 / 2

Time evolution of

+ exp exp

i it a E t b E t

0

0

1exp

2

1exp

2

a i B t

b i B t

0

a

bis

x xS t tS

0* *

0 0

0

1exp

0 1 21 1exp exp

1 02 2 2 1exp

2

a i B t

a i B t b i B t

b i B t

0* *

0 0

0

1exp

21 1exp exp

2 2 2 1exp

2

b i B t

a i B t b i B t

a i B t

* *0 0exp exp

2

a b i B t ab i B t

if a, b are real

Sinilarly

0sin yS ab B t 2 2

2

zS a b

0cos xS ab B t

0cos xS ab B t 0sin yS ab B t 2 2

2

zS a b

Set cos2

a sin

2

b

0sin cos2

xS B t 0sin sin2

yS B t cos2

zS

i.e., S is tilted a constant angle from the z-axis,and precesses with the Larmor frequency 0 B

( same as the classical law )

Example 4.4. The Stern-Gerlach Experiment

Force on in inhomogeneous B : UF μ B

Consider a particle moving in the y-direction in a field

0ˆ ˆ x B zB r i k

0 BNote : The x term is to make sure

ˆ ˆ x zS SF i k

S B

0 B z

Due to precession about B0 , Sx oscillates rapidly & averages to zero.

z zF S Net force is incident beam splits into two.

In contrast, CM expects a continuous spread-out.

0x zS x S B z S B

Alternative Description

In frame moving with particle :

0

0 0

0

0

z

t

H t B z S t T

t T

x component dropped for convenience

Let t a b for t 0

exp exp

i it a E t b E t for 0 t T

where 0 2

E B z

0 0

1 1exp exp

2 2

t T a i B z T b i B z T for t > T

0 0

1 1exp exp

2 2

t a i B z T b i B z T for t > T

0 0

1 1 1 1exp exp exp exp

2 2 2 2

a i B T i T z b i B T i T z

z

dp

i d z21

2 T a

z

dp

i d z21

2 T b

spin up particles move upward

spin up particles move downward

S-G apparatus can be used to prepare particles in particular spin state.

4.4.3. Addition of Angular Momenta

The angular momentum of a system in state can be found by writing

jm

jm jm is proportional to the probability of measuring

2jm

If the system has two types of angular momenta j1 and j2 , its state can be written as

where

2 21J j j zJ m

1 1 2 2

1 2 1 2 1 2 1 2; ;j m j m

j j m m j j m m

where 1 2 1 2 1 1 2 2;j j m m j m j m

The total angular momentum of the system is therefore described by the quantities

1 1 2 2

1 2 1 2 1 2 1 2; ;j m j m

jm jm j j m m j j m m

jmjm

jm c

1 2 1 2

1 1 2 2

1 2 1 2; j j m mj m j m

j j m m c

1 2 1 2

1 1 2 2

1 2 1 2; j j m mj m j m

jm j j m m c

Since either set of basis is complete, we have

1 1 2 2

1 2 1 2 1 2 1 2; ;j m j m

jm j j m m j j m m jm

1 2 1 2 1 2 1 2; ;jm

j j m m jm jm j j m m

The transformation coefficients

are called Clebsch-Gordan coeffiecients (CGCs).

*

1 2 1 2 1 2 1 2; ;j j m m jm jm j j m m

The problem is equivalent to writing the direct product space

as a direct sum of irreducible spaces

1 1 2 2j m j m

jm

4.4.3. Addition of Angular Momenta

Rules for adding two angular momenta : 1 2 J J J

j1 j2 j1j2 J1J2...Jn

State | j1 m1 | j2 m2 | j1 m1 | j2 m2 | J1 M1 ...| Jn Mn

1. Possible values of S are

j1 + j2 , j1 + j2 1, ..., | j1 j2 | + 1, | j1 j2 |

2. Only states with M = m1 + m2 are related.

3. Linear transformation between | j1 m1 | j2 m2 and | Jk Mk can be obtained

by applying the lowering operator to the relation between the “top” states.

4. Coefficients of these linear transformation are called the Clebsch-Gordan

( C-G ) coefficients.

1 2

1 2

1 22 1 2 1 2 1j j

j j j

j j j

Example: Two Spin ½ Particles

s1 s2 s1s2 S1 S2

States

| | | | 1 1

| 1 0

| 1 1

| 0 0 | | |

| | |

| | |

Possible total S 1, 0

s1 = ½ , s2 = ½ s1 + s2 = 1, s1 s2 = 0

11 “Top” state for S = 1 :

11 2 10 S

1 2 S S

1 1 , 1 S sm s s m m s m

110

2

10 2 1, 1 S

1 2 1

2 S S 2

1, 1

“Top” state for S = 0 : 00 a b must be orthogonal to | 10 .

Normalization then gives 100

2

J1 J2 ...

M = m1 + m2

m1 , m2

M

Clebsch-Gordan ( C-G ) coefficients

1 2

1 2 1 2 1 2 1 2; ;m m M

SM s s m m s s m m SM

Shaded column gives

1 3 130 21 1 1 20 10 2 1 11

55 5

Shaded row gives

3 1 3 5 1 1 3 1 1 1 110

2 2 5 2 2 2 2 2 215 3

1 2

1 21 1 2 2s s S

m m MS

s m s m C SM

Sum of the squares of each row or column is 1.

Do Prob 4.36

1 2

1 2

1 2

1 1 2 2s s S

m m Mm m M

C s m s m

1 2 1 2 1 2 1 2; ;S

s s m m SM SM s s m m 1 2M m m