Quantum Mechanics: Concepts and Applications · Problem 5.3

Angular Momentum — Nouredine Zettili

Physics — Quantum Mechanics · matrix-representation-angular-momentum-j1

Problem

Consider the case where

j = 1

. (a) Find the matrices representing the operators

\hat{J}^2

\hat{J}_z

\hat{J}_{\pm}

\hat{J}_x

, and

\hat{J}_y

. (b) Find the joint eigenstates of

\hat{J}^2

and

\hat{J}_z

and verify that they form an orthonormal and complete basis. (c) Use the matrices of

\hat{J}_x

\hat{J}_y

and

\hat{J}_z

to calculate

[\hat{J}_x, \hat{J}_y]

[\hat{J}_y, \hat{J}_z]

, and

[\hat{J}_z, \hat{J}_x]

. (d) Verify that

\hat{J}_z^3 = \hbar^2 \hat{J}_z

and

\hat{J}_{\pm}^3 = 0