Franz Schwabl Quantum Mechanics 4th Edit · Problem 6.3

Motion in an Electromagnetic Field

Physics — Quantum Mechanics · recursion-formula-matrix-elements

Problem

Prove the following recursion formula for the matrix elements of

r^k

\langle r^k \rangle = \langle nl | r^k | nl \rangle

in the hydrogen atom:

\frac{(k+1)}{n^2} \langle r^k \rangle - (2k+1)a\langle r^{k-1} \rangle + \frac{k}{4}\left[(2l+1)^2 - k^2\right] a^2 \langle r^{k-2} \rangle = 0 \quad.

Hint: It is easier to formulate the equation first for the dimensionless variable

\varrho = \kappa r

with

\kappa = \sqrt{-2mE}/\hbar

and, for the computation of the

\langle \varrho^k \rangle

, to begin with the differential equation

\left[\frac{d^2}{d\varrho^2} - \frac{l(l+1)}{\varrho^2} + \frac{2n}{\varrho} - 1\right] u(\varrho) = 0 \quad.

Multiplying this equation by

\varrho^{k+1} u'_{nl}(\varrho)

and

\varrho^k u

, respectively, and partially integrating, you can prove the claim.