Post on 17-Jan-2016
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