A Modern Approach to Quantum Mechanics · Problem 11.16

Identical Particles — John S. Townsend

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

Problem

In order to evaluate

\langle 1/r \rangle

consider

\gamma/r

as a perturbation for the hydrogenic atom, where we can think of

\gamma

as some "small" constant. The first-order shift in the energy is given by

E_n^{(1)} = \left\langle \frac{\gamma}{r} \right\rangle_{n,l,m}

which is clearly linear in

\gamma

. (a) First show that the exact eigenvalues are given by

E_n = -\frac{\mu(Ze^2 - \gamma)^2}{2\hbar^2 n^2}

*Suggestion:* Examine (10.32). (b) Since

E_n = E_n^{(0)} + E_n^{(1)} + E_n^{(2)} + \cdots

, we can obtain

E_n^{(1)}

either by explicitly finding the contribution to

E_n

that is linear in

\gamma

, or, more generally, noting that

E_n^{(1)} = \gamma\left(\frac{dE_n}{d\gamma}\right)_{\gamma=0}

since

E_n^{(0)}

is of course independent of

\gamma

and the higher order terms in the expansion are at least of order

\gamma^2

. In this way show that

\left\langle \frac{1}{r} \right\rangle_{n,l,m} = \frac{\mu Z e^2}{\hbar^2 n^2} = \frac{Z\mu c\alpha}{\hbar n^2} = \frac{Z}{a_0 n^2}