A Modern Approach to Quantum Mechanics · Problem 10.13

Time-Independent Perturbations — John S. Townsend

Physics — Quantum Mechanics · virial-theorem-quantum

Problem

Expectation values are constant in time in an energy eigenstate. Hence d⟨r⋅p⟩dt=iℏ⟨E∣[H^,r^⋅p^]∣E⟩=0\frac{d\langle \mathbf{r} \cdot \mathbf{p} \rangle}{dt} = \frac{i}{\hbar}\langle E|[\hat{H}, \hat{\mathbf{r}} \cdot \hat{\mathbf{p}}]|E\rangle = 0 Use this result to show for the Hamiltonian H^=p^22μ+V(∣r∣)\hat{H} = \frac{\hat{\mathbf{p}}^2}{2\mu} + V(|\mathbf{r}|) that ⟨K⟩=⟨p22μ⟩=12⟨r⋅∇V(r)⟩\langle K \rangle = \left\langle \frac{\mathbf{p}^2}{2\mu} \right\rangle = \frac{1}{2}\langle \mathbf{r} \cdot \nabla V(r) \rangle which can be considered a quantum statement of the virial theorem.

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