A Modern Approach to Quantum Mechanics · Problem 9.13

Bound States of Central Potentials — John S. Townsend

Physics — Quantum Mechanics · angular-momentum-uncertainty-relation

Problem

A particle is in the orbital angular momentum state

|l, m\rangle

. Evaluate

\Delta L_x

and

\Delta L_y

for this state. Which states satisfy the equality in the uncertainty relation

\Delta L_x \Delta L_y \geq \frac{\hbar}{2}|\langle L_z \rangle|

Suggestion: One approach is to use

\hat{L}_x = (\hat{L}_+ + \hat{L}_-)/2

, and so on. Another is to take advantage of the symmetry of the expectation values of

L_x^2

and

L_y^2

in an eigenstate of

\hat{L}_z