Introduction to Quantum Mechanics · Problem 7.43

Time-Independent Perturbation Theory — David J. Griffiths

Physics — Quantum Mechanics · kramers_relation

Problem

Prove Kramers' relation:

\frac{s+1}{n^2}\langle r^s \rangle - (2s+1)a\langle r^{s-1} \rangle + \frac{s}{4}\left[(2\ell+1)^2 - s^2\right]a^2\langle r^{s-2} \rangle = 0, \qquad (7.113)

which relates the expectation values of

r

to three different powers (

s

s-1

, and

s-2

), for an electron in the state

\psi_{n\ell m}

of hydrogen. *Hint:* Rewrite the radial equation (Equation 4.53) in the form

u'' = \left[\frac{\ell(\ell+1)}{r^2} - \frac{2}{ar} + \frac{1}{n^2 a^2}\right]u,

and use it to express

\int (ur^s u'')\,dr

in terms of

\langle r^s \rangle

\langle r^{s-1} \rangle

, and

\langle r^{s-2} \rangle

. Then use integration by parts to reduce the second derivative. Show that

\int (ur^s u')\,dr = -(s/2)\langle r^{s-1} \rangle

, and

\int (u'r^s u')\,dr = -\left[2/(s+1)\right]\int (u''r^{s+1}u')\,dr

. Take it from there.