The spin- representation has four basis states

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Symmetry, angular momentum and spin

Invariance, Ch Representations of Lie algebras: spin, products and SU(3)

Problem

The spin-

\tfrac32

representation

D_{3/2}

has four basis states

|\tfrac32,m\rangle

, taken in the order

m = \tfrac32, \tfrac12, -\tfrac12, -\tfrac32

. ((a)) From

J_z|j,m\rangle = m|j,m\rangle

and

J_\pm|j,m\rangle = \sqrt{j(j+1)-m(m\pm1)}\,|j,m\pm1\rangle

, write the

4\times4

matrices

J_z

J_+

and

J_-

. ((b)) Multiply your matrices to check that

[J_+,J_-] = 2J_z

, and that

J^2 = J_+J_- + J_z^2 - J_z

is a multiple of the identity. Which multiple? ((c)) A rotation by

2\pi

about

z

is represented by

e^{-2\pi iJ_z}

. Find its trace from your

J_z

, and say which result of Chapter 12 the sign reproduces.