Principles of Quantum Mechanics · Problem 12.5.12

Rotational Invariance and Angular Momentum — R. Shankar

Physics — Quantum Mechanics · parity-spherical-harmonics

Problem

Since

L^2

and

L_z

commute with

\Pi

, they should share a basis with

Y_l^m

. Verify that

Y_l^m \Pi (-1)^l Y_l^m

. (First show that

\theta \to \pi - \theta

\phi \to \phi + \pi

under parity. Prove the result for

Y_l^l

. Verify that

L_-

does not alter the parity, thereby proving the result for all

Y_l^m