16. Angular Momentum

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 twithin volume d 3raround 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.  OperatorsA & 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 Awith probability and the particle will be in aafterwards.  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.3.10.32  with

6. Central Force  Cartesian commonents Ex.3.10.31 :  eigenstates of H can be labeled by eigenvalues of L2 & Lz, i.e., by l,m. Ex.3.10.29-30 

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

8.    is an eigenfunction of L2with eigenvalue l2 . i.e.    lm normalized  Ylmthus generated agrees with the Condon-Shortley phase convention. areal 

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 16.1.2. 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 : or eigen states : Adding (coupling) means finding Solution always exists & unique since is complete.

17. Vector Model    Total number of states :   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 16.2.2.Two Spinors   

25. Simpler Notations 

26. Example 16.2.3.Coupling of p & d Electrons  Simpler notations : where Mathematica

27. Mathematica