Proof that $\langle[\hat{H},\hat{O}]\rangle=0$

Question

How can I show that for a particle in an infinite square well in a stationary state, that the expectation value $\langle[\hat{H},\hat{O}]\rangle=0$ where $\hat{H}$ is the Hamiltonian operator and $\hat{O}$ is an arbitrary operator?

Answer 1

The stationary states may have even or odd symmetry. If $\hat{O}$ is a unitary operator representing some symmetry, then it can be said that $\hat{H}$ is invariant under that symmetry provided that the Hamiltonian is invariant under conjugation by $\hat{O}$, i.e. \begin{align} \hat{O}^{-1} \hat{H} \hat{O} = \hat{H}. \end{align} This condition can also be written as $[\hat{H},\hat{O}]=0$.

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