Principles of Quantum Mechanics · Problem 7.4.7

The commutator $[\Omega, \Lambda]$ measures whether two quantum operators are compatible—and when it's nonzero, you'll discover that the order in which you write products of $X$ and $P$ matters crucially. For case (1), use the canonical commutation relation $[X,P] = i\hbar$ to expand $[\Omega, \Lambda]$ systematically and verify the claimed result. Case (2) reveals the subtlety: $X^2 P^2$ can be reordered in multiple ways, and different choices yield different commutators, showing why quantum mechanics requires careful operator algebra—there's no single "right" answer until you specify the ordering convention.

The commutator $[\Omega, \Lambda]$ measures whether two quantum operators are compatible—and when it's nonzero, you'll discover that the order in which you write products of $X$ and $P$ matters crucially. For case (1), use the canonical commutation relation $[X,P] = i\hbar$ to expand $[\Omega, \Lambda]$ systematically and verify the claimed result. Case (2) reveals the subtlety: $X^2 P^2$ can be reordered in multiple ways, and different choices yield different commutators, showing why quantum mechanics requires careful operator algebra—there's no single "right" answer until you specify the ordering convention.