A Modern Approach to Quantum Mechanics · Problem 11.17

Identical Particles — John S. Townsend

Physics — Quantum Mechanics · hydrogen-expectation-p-squared-virial

Problem

(a) Treat

\gamma p^2/2\mu

as a perturbation for the hydrogen-like atom and, using the techniques of Problem 11.16, show that

\left\langle \frac{\mathbf{p}^2}{2\mu} \right\rangle_{n,l,m} = \frac{\mu c^2 Z^2 \alpha^2}{2n^2}

(b) Use the results of (a) and Problem 11.16 to show that for the hydrogen atom

\langle K \rangle = -\frac{1}{2}\langle V \rangle

in agreement with the virial theorem in quantum mechanics. See Problem 10.13.