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16. Angular Momentum. Angular Momentum Operator Angular Momentum Coupling Spherical Tensors Vector Spherical Harmonics. Principles of Quantum Mechanics. State of a particle is described by a wave function ( r , t ). - PowerPoint PPT Presentation
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16. Angular Momentum
1. Angular Momentum Operator
2. Angular Momentum Coupling
3. Spherical Tensors
4. Vector Spherical Harmonics
Principles of Quantum Mechanics
, ,i t H tt
r r
Dynamics of particle is given by the time-dependent Schrodinger eq.
2
2 ,2
H V tm
r
341.05 10 J s SI units:
23 3, ,P t d r t d r r r
Probability of finding the particle at time t within volume d 3r around r is
State of a particle is described by a wave function (r,t).
H E r r
Hamiltonian
Stationary states satisfy the time-independent Schrodinger eq.
/, i E tE Et e r r
V V rwith
Operators A & B have a set of simultaneous eigenfunctions. , 0A B
A stationary state is specified by the eigenvalues
of the maximal set of operators commuting with H.
, xx p i uncertainty principle
Let be an eigenstate of A with eigenvalue a, i.e. A a
Measurement of A on a particle in state will give a and the particle will remain in afterwards.
Measurement of A on a particle in state will
give one of the eigenvalues a of A with probability
and the particle will be in a afterwards.
2
a
Kinetic energy of a particle of mass :
1. Angular Momentum Operator
22
2 2
1 1sin
sin sin
L2 2 2
2
1r r
L2
22
2r r r r
2 22
2 2T
p
Quantization rule :i
p
Rotational energy : 2 2
22 22 2RT
r r
L L
Angular momentum : i
L r p r
ˆ ˆ ˆ p r p r r p2
2 22rp
r
Lp
22 2 2 2 2
2r r
L
angular part of T
2 2 2 L L
i L r
i L r
ˆ ˆ ˆ
0 0
1 1
sin
r
ri
r r r
e e e
1ˆ ˆ
sini
L e e
2 2 2 2 2 2x y zL L L L L L L Ex.3.10.32
22
2 2
1 1sin
sin sin
L
2 , 1 ,m ml lY l l Y L 2 2, 1 ,m m
l lY l l Y L
with 0,1,2,l , 1, ,m l l l
Central Force
2
22rH T V rr
L
22
2 2
1 1sin
sin sin
L2 2 2L L
2 2
2 222 2r V r
r
L
sin cot cos
cos cot sin
x
y
z
Li
Li
Li
2, 0r L 2 , 0H L
eigenstates of H can be labeled by eigenvalues of L2 & Lz , i.e., by l,m.
Ex.3.10.31 : ,j k j k n nL L i L 2 , 0jL L Cartesian commonents
Ex.3.10.29-30
m mz l lL Y m Y
L are operators
is an eigenfunction of Lz with eigenvalue ( m 1) .
Ladder Operators
x yL L i L
RaisingLowering
, , ,z z x z yL L L L i L L y xi L L ,zL L L
Let lm be a normalized eigenfunction of L2 & Lz such that
2 2m ml l l L m m
z l lL m
, m mz l z z lL L L L L L
mlL m
z lL L L m
mlL
1m mz l lL L m L
Ladder operators
1m ml l l
l
L c
i.e.
2 , 0jL L 2 , 0L L
2 2m ml l lL L L
is an eigenfunction of L2 with eigenvalue l 2 . m
lL
1m ml l l
l
L c
m ml m l
m
L b
i.e. 1,m ml lL a l m
2,m m
l lL L a l m
†L L x yL L i L
lm normalized
x y x yL L L i L L i L 2 2x y x y y xL L i L L L L
2 2x y zL L L L L
2 21
2 zL L L L L L
, 2 zL L L
m ml lL L 2 2
l m m
2 2z zL L L L L
a real 11m ml l lL m m
Ylm thus generated agrees with the
Condon-Shortley phase convention.
2 2m ml l lL L m m
max
11 1 4
2 lm
11m ml l lL m m
0 ,m ml lL L l m 2 0 ,l m m l m
2 0l m m For m 0 : 0
For m 0 : 2 0l m m min
11 1 4
2 lm 0
1 4 1l n
1l l l , ,m l l
max min integerm m m = 1
212
4l n n 12 2
n n
1l l
0n
2
nl
max
1
2m n l min
1
2m n l
1 30, ,1, ,
2 2l Multiplicity = 2l+1
Example 16.1.1.Spherical Harmonics Ladder
sin cot cos
cos cot sin
x
y
z
Li
Li
Li
x yL L i L cotiL e i
01
11
3, cos
4
3, sin
8i
Y
Y e
0
1
3, cot cos
4iL Y e i
3sin
4ie
112 ,Y
11m ml l lL m m
0 11 12L
m m
l lY
for l = 0,1,2,…
Spinors
Intrinsic angular momenta (spin) S of fermions have s = half integers.
E.g., for electrons
1
2s 2 3
14
s s S1 1
or 2 2sm
Eigenspace is 2-D with basis1 1
,2 2
,
Or in matrix form :1 0
,0 1
spinors
S are proportional to the Pauli matrices.
Example 16.1.2. Spinor Ladder
0 1
1 02x
S
0
02y
i
i
S
1 0
0 12z
S
Fundamental relations that define an angular momentum, i.e.,
can be verified by direct matrix calculation.
,j k j k n ni S S S 2 21Eigenvalue of s s S
1
2s
1/21/2 Spinors:
1
0
1/2
1/2 0
1
0 1
0 02
S
0 0
1 02
S
0
0
S
S
S
S
3 1
1 1 0 14 4
s s m m or
Mathematica
Summary, Angular Momentum Formulas
,k l k l n nJ J i J
, ,x y zJ J JJGeneral angular momentum :
2 , 0kJ J
2 21M MJ JJ J J
Eigenstates JM :
M Mz J JJ M
J = 0, 1/2, 1, 3/2, 2, …
M = J, …, J
M MJ J J J M M
x yJ J i J
11 1M MJ JJ J J m m
2 2z zJ J J J J , 2 zJ J J
2. Angular Momentum Coupling
Let 1 1 1 1, ,x y zj j jj
1 2 , ,x y zJ J J J j j
2 2 2 2, ,x y zj j jj
1 2 1 2 1 2, ,x x y y z zj j j j j j
2 2 21 2 1 22 J j j j j
,k l k l n nj j i j
, 1, 2
,k l k l n nJ J i J
, , , ,k l n x y z2 , 0kj j
2 21 2, , 0k k kJ j j J J
2 2 2x y zJ J J 1 2 2 1 j j j j
Implicit summation applies only to the k,l,n indices
Example 16.2.1.Commutation Rules for J Components
1 22 , kj j j
2 ,l l kj j j
2 l k n l ni j j 2 , 2k kj i J j j
2 1
1 2
2 2 21 2 1 22 J j j j j
,k l k l n nj j i j
,k l k l n nJ J i J
2 , 0kj j2 , 0kJ J
21 1 2, 2z zj i J j je.g. 1 2 1 22 x y y xi j j j j
2 2 21 2 1 2, 2 ,k kj j J j j j j
2 2 2 2 21 2 1 2, 2 , J j j j j j j 22 ,k kj j j
2 2, 0 J j21 22 , j j j
2 21 2, ,k k kJ j j j j 2 , 0kJ j
Solution always exists & unique since is complete.
2 , 2k kj i J j j
2 , 0kj j
2 , 0kJ J 2 2, 0 J j
2 , 0kJ j
2 21 2, , , zJj j J
Maximal commuting set of operators :
2 21 2 1 2, , ,z zj jj jor
eigen states : 1 2, ;j j J M
Adding (coupling) means finding1 2 j j J
1 2
1 2
1 2 1 1 2 2,
, ; ,m mm m
j j J M C j m j m
1 1 2 2,j m j m
1 1 2 2 1 1 2 2,j m j m j m j m
Vector Model 2 2
1 2, , , zJj j J 2 21 2 1 2, , ,z zj jj j
1 2, ;j j J M
1 2z z zJ j j
1 2M m m max 1 2M j j
max 1 2J j j 1 2
3 2 1
2 1 1
2 2 0
1 0 1
1 1 0
1 2 1
0 1 1
0 0 0
0 1 1
1 2 1
1 1 0
1 0 1
2 2 0
2 1 1
3 2 1
M m m
1 2;2 1j j
Mathematica
Total number of states :
max
min
1 22 1 2 1 2 1J
J J
j j J
min 1 2J j j
i.e. 1 2 1 2 1 2, 1 , ,J j j j j j j
max min max min1 1J J J J
1 2 min 1 2 min1 1j j J j j J
22min 1 2 1 21 2 1 2 1J j j j j
min 0J
2
1 2j j
Triangle rule
1 1 2 2,j m j m
Clebsch-Gordan Coefficients 2 2
1 2, , , zJj j J 2 21 2 1 2, , ,z zj jj j
1 2, ;j j J M 1 1 2 2,j m j m
max 1 2J j j min 1 2J j j
For a given j1 & j2 , we can write the basis as J M 1 2,m m&
1 2
1 1 2 2
1 2 1 2, ,j j
m j m j
J M m m m m J M
Both set of basis are complete :
1 2
1 2
1 2 1 2, ,j j J
M JJ j j
m m J M J M m m
*
1 2 1 2, ,m m J M J M m m Clebsch-Gordan Coefficients (CGC)
1 2M m m
Condon-Shortley phase convention 1 2 1 2, ,m m J M J M m m
1 1, 0j J j J J
Ladder Operation Construction
max 1 2 1 2,J j j j j
1
1 1
1 1 1 1, ,j
m j
J M m M m m M m J M
1 1 1j jm j j m m j m
max 1 2 1 2 1 2,J J j j j j j j
max 1 2J j j
1 2 max 1 2 1 1 2 2 1 22 1 2 1 , 2 , 1j j J j j j j j j j j
1 2max 1 2 1 2 1 2
1 2 1 2
1 1 , , 1j j
J j j j j j jj j j j
Repeated applications of J then give the rest of the multiplet
max max max; , ,J M M J J
Orthonormality : 1 2max 1 2 1 2 1 2
1 2 1 2
1 1 1 , , 1j j
J j j j j j jj j j j
1 1, 0j J j J J
Clebsch-Gordan Coefficients
Full notations :
1 1 2 2,j m j m J M 1 2 1 2, , | , ,C j j J m m M real1 2 3F F F
1 2 1 2 2 1
1
1 2
! ! ! 2 1
1 !
j j J J j j J j j JF
j j J
2 1 1 1 1 2 2 2 2! ! ! ! ! !F J M J M j m j m j m j m
3
1 1 2 2 2 1 1 2 1 2! ! ! ! !
s
s
Fj m s j m s J j m s J j m s j j J s
Only terms with no negative factorials are included in sum.
1
1 1
1 1 1 1, ,j
m j
J M m M m m M m J M
Table of Clebsch-Gordan Coefficients
Ref: W.K.Tung, “Group Theory in Physics”, World Scientific (1985)
, , | , ,C j j J m m M
Wigner 3 j - Symbols
1 2 3
1 2 3
1 2 3 1 2 31 2 3 3
, , | , ,2 1
j j mj j jC j j j m m m
m m m j
Advantage : more symmetric
1 2 3
1 2 3
1 2 3
1,2,3 , , is even
1,2,3 , , is odd
k l n
k l n
j j j k l n
k l n
j j jk l n
m m mj j j
m m m j j jk l n
m m m
1 2 31 2 3 1 2 3
1 2 3 1 2 3
j j jj j j j j j
m m m m m m
2 1 1 2
1 2 1 21 2
, , | , , 2 1j j M j j J
C j j J m m M Jm m M
Table 16.1 Wigner 3j-Symbols
1 1, ,1 :
2 2
1,2,3 :
Mathematica
1 1, ,0 :
2 2
Example 16.2.2.Two Spinors
1 2
1
2j j 1, 0J
1 111 ,
2 2
1 1 1 12 1 0 , ,
2 2 2 2
1 1 1 1 1 11 0 , ,
2 2 2 22 2
1 1 1 1 1 12 1 1 , ,
2 2 2 22 2
1 111 ,
2 2
1 1 1 1 1 10 0 , ,
2 2 2 22 2
1 1 1 1, ,1 , ,1 1
2 2 2 2C
1 1 1 1 1 1 1 1 1, ,1 , ,0 , ,1 , ,0
2 2 2 2 2 2 2 2 2C C
1 1 1 1, ,1 , , 1 1
2 2 2 2C
1 1 1 1 1 1 1 1 1, ,0 , ,0 , ,0 , ,0
2 2 2 2 2 2 2 22C C
1 11
2 231 1
12 2
1 1 1 11 1
2 2 2 23 31 1 1 1
0 02 2 2 2
1 11
2 231 1
12 2
1 1 1 10 0
2 2 2 21 1 1 1
0 02 2 2 2
2 1 1 2
1 2 1 21 2
, , | , , 2 1j j M j j J
C j j J m m M Jm m M
1 1 1j jm j j m m j m
1 1, 0j J j J J
1
1 1
1 1 1 1, ,j
m j
J M m M m m M m J M
Simpler Notations
1 111 ,
2 2
1
2
1 1 1 1 1 11 0 , ,
2 2 2 22 2
1 1
2 2
1 111 ,
2 2
1 1 1 1 1 10 0 , ,
2 2 2 22 2 1 1
2 2
Simpler notations : where
Example 16.2.3.Coupling of p & d Electrons
1 21 , 2j j 3,2,1J
mlm l 0 1 2 3
s p d f
3 3 1 , 21 2p d
3 2 1,2,3 0, 2, 2 0 , 2 1,2,3 1,1, 2 1 ,1C C
0 2 1 1
1 2
3 3p d p d
1 2 3 1 2 37 0 , 2 7 1 ,1
0 2 2 1 1 2
31 1,2,3 1, 2,1 1 , 2 1,2,3 0,1,1 0 ,1 1,2,3 1, 0,1 1 , 0C C C
1 2 0 1 1 0
1 8 2
15 15 5p d p d p d
1 2 3 1 2 3 1 2 37 1 , 2 7 0 ,1 7 1 , 0
1 2 1 0 1 1 1 0 1
2 1 1 2
1 2 1 21 2
, , | , , 2 1j j M j j J
C j j J m m M Jm m M
1
1 1
1 1 1 1, ,j
m j
J M m M m m M m J M
Mathematica
2 1 1 2
1 2 1 21 2
, , | , , 2 1j j M j j J
C j j J m m M Jm m M
2 2 1,2,2 0, 2, 2 0 , 2 1,2,2 1,1, 2 1 ,1C C
0 2 1 1
2 1
3 3p d p d
1 2 2 1 2 25 0 , 2 5 1 ,1
0 2 2 1 1 2
2 1 1,2,2 1,2,1 1 , 2 1,2,2 0,1,1 0 ,1 1,2,2 1, 0,1 1 , 0C C C
1 2 0 1 1 0
1 1 1
3 6 2p d p d p d
1 2 2 1 2 2 1 2 25 1 , 2 5 0 ,1 5 1 , 0
1 2 1 0 1 1 1 0 1
1
1 1
1 1 1 1, ,j
m j
J M m M m m M m J M
11 1,2,1 1,2,1 1 , 2 1,2,1 0,1,1 0 ,1 1,2,1 1, 0,1 1 , 0C C C
1 2 0 1 1 0
3 3 1
5 10 10p d p d p d
1 2 1 1 2 1 1 2 13 1 , 2 3 0 ,1 3 1 , 0
1 2 1 0 1 1 1 0 1
Mathematica
