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). 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 

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

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

  Ex.3.10.32  with

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.3.10.29-30 

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

 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.

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

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

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.

Example 16.1.2. Spinor Ladder Fundamental relations that define an angular momentum, i.e., can be verified by direct matrix calculation. Mathematica Spinors: 

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

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

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

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

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

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

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

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

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

Wigner 3 j - Symbols Advantage : more symmetric

Table 16.1 Wigner 3j-Symbols Mathematica

Example 16.2.2. Two Spinors   

Simpler Notations 

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

Mathematica