A Modern Approach to Quantum Mechanics · Problem 9.4

Bound States of Central Potentials — John S. Townsend

Physics — Quantum Mechanics · center-of-mass-commutation-relations

Problem

Use the commutation relations of

\hat{\mathbf{r}}_1

\hat{\mathbf{r}}_2

\hat{\mathbf{p}}_1

, and

\hat{\mathbf{p}}_2

to establish that the center-of-mass and relative position and momentum operators satisfy

[\hat{x}_i, \hat{P}_j] = 0 \qquad [\hat{X}_i, \hat{P}_j] = i\hbar\delta_{ij}

[\hat{x}_i, \hat{p}_j] = i\hbar\delta_{ij} \qquad [\hat{X}_i, \hat{p}_j] = 0