Principles of Quantum Mechanics · Problem 12.3.4

Rotational Invariance and Angular Momentum — R. Shankar

Physics — Quantum Mechanics · eigenvalue-problem-lz

Problem

A particle is described by a wave function

\psi(\rho, \phi) = A\, e^{-\rho^2/2\Delta^2}\left(\frac{\rho}{\Delta}\cos\phi + \sin\phi\right)

Show that

P(l_z = \hbar) = P(l_z = -\hbar) = \frac{1}{2}