A Modern Approach to Quantum Mechanics · Problem 11.19

Identical Particles — John S. Townsend

Physics — Quantum Mechanics · hydrogen-expectation-1-over-r-cubed

Problem

We cannot use the techniques of Problem 11.16 to evaluate

\langle 1/r^3 \rangle

, since there is no term in the Coulomb Hamiltonian that involves

1/r^3

. However, use the fact that

\langle n, l, m|[\hat{H}_0, \hat{p}_r]|n, l, m\rangle = 0

where

\hat{p}_r

is the radial momentum operator introduced in (9.92) and

\hat{H}_0

is the unperturbed hydrogenic Hamiltonian (11.65), to show that

\left\langle \frac{1}{r^3} \right\rangle_{n,l,m} = \frac{Z}{a_0 l(l+1)}\left\langle \frac{1}{r^2} \right\rangle_{n,l,m} = \frac{Z^3}{a_0^3 n^3 l(l+1)(l+\frac{1}{2})}