1. 16. Angular Momentum Angular Momentum Operator
Angular Momentum Coupling
Spherical Tensors
Vector Spherical Harmonics

2. Principles of Quantum Mechanics
State of a particle is described by a wave function (r,t).
Probability of finding the particle at time t within volume d 3r around r is
Dynamics of particle is given by the time-dependent Schrodinger eq.
Hamiltonian
SI units:
Stationary states satisfy the time-independent Schrodinger eq.
with 

3. Let  be an eigenstate of A with eigenvalue a, i.e.
Measurement of A on a particle in state  will give a and the particle will remain in  afterwards.
 Operators A & B have a set of simultaneous eigenfunctions.
 A stationary state is specified by the eigenvalues of the maximal set of operators commuting with H.
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.
 uncertainty principle

4. 1. Angular Momentum Operator
Quantization rule :
Kinetic energy of a particle of mass  :
Angular momentum :   
Rotational energy :
angular part of T

5.   Ex 
with

6. Central Force  Ex.3.10.31 : Cartesian commonents
 eigenstates of H can be labeled by eigenvalues of L2 & Lz , i.e., by l,m. Ex 

7. Ladder Operators Ladder operators  
Let lm be a normalized eigenfunction of L2 & Lz such that  
 is an eigenfunction of Lz with eigenvalue ( m  1)  .
Raising
Lowering
i.e.
 L are operators

8.  is an eigenfunction of L2 with eigenvalue l 2 .
 is an eigenfunction of L2 with eigenvalue l 2 .
i.e.   
lm normalized 
a real 
Ylm thus generated agrees with the Condon-Shortley phase convention.

9. 
For m  0 : 
 0
For m  0 : 
 0
m = 1    
Multiplicity = 2l+1

10. Example 16.1.1. Spherical Harmonics Ladder  
for l = 0,1,2,…

11. Spinors
Intrinsic angular momenta (spin) S of fermions have s = half integers.
E.g., for electrons
Eigenspace is 2-D with basis
Or in matrix form :
spinors
S are proportional to the Pauli matrices.

12. Example Spinor Ladder
Fundamental relations that define an angular momentum, i.e.,
can be verified by direct matrix calculation.
Mathematica
Spinors: 

13. Summary, Angular Momentum Formulas
General angular momentum :
Eigenstates JM :
J = 0, 1/2, 1, 3/2, 2, …
M = J, …, J

14. 2. Angular Momentum Coupling
Let 
Implicit summation applies only to the k,l,n indices  

15. Example 16.2.1. Commutation Rules for J Components
e.g.  

16. Maximal commuting set of operators :
eigen states :
Adding (coupling) means finding
Solution always exists & unique since is complete.

17. Vector Model    Total number of states :   i.e. Triangle rule
Mathematica
i.e.
Triangle rule

18. Clebsch-Gordan Coefficients
For a given j1 & j2 , we can write the basis as
&
Both set of basis are complete : 
Clebsch-Gordan Coefficients (CGC)
Condon-Shortley phase convention

19. Ladder Operation Construction
Repeated applications of J then give the rest of the multiplet
Orthonormality :

20. Clebsch-Gordan Coefficients
Full notations :
real
Only terms with no negative factorials are included in sum.

21. Table of Clebsch-Gordan Coefficients
Ref: W.K.Tung, “Group Theory in Physics”, World Scientific (1985)

22. Wigner 3 j - Symbols
Advantage : more symmetric

23. Table 16.1 Wigner 3j-Symbols
Mathematica

24. Example Two Spinors   

25. Simpler Notations 

26. Example 16.2.3. Coupling of p & d Electrons l 1 2 3  s p d f
Simpler notations : where
Mathematica

27. Mathematica