# Radial wave function matrix elements

For hydrogen atom radial wave function is the analytic form of the matrix elements, $$\langle n'\ell'|r^k|n\ell\rangle,$$ known? I am especially interested in $$k=-2$$ and $$k=-3$$.

Notation: $$|n\ell\rangle=2^{\ell+1} e^{-\frac{r}{a n}} \sqrt{\frac{(-\ell+n-1)!}{a^3 n^4 (\ell+n)!}} \left(\frac{r}{a n}\right)^\ell L_{-\ell+n-1}^{2 \ell+1}\left(\frac{2 r}{a n}\right)$$ and, $$\langle n'\ell'|r^k|n\ell\rangle=\int_0^\infty 2^{\ell^\prime+1} e^{-\frac{r}{a n^\prime}} \sqrt{\frac{(-\ell^\prime+n^\prime-1)!}{a^3 n^{\prime4} (\ell^\prime+n^\prime)!}} \left(\frac{r}{a n^\prime}\right)^{\ell^\prime} L_{-\ell^\prime+n^\prime-1}^{2 \ell^\prime+1}\left(\frac{2 r}{a n^\prime}\right)2^{\ell+1} e^{-\frac{r}{a n}} \sqrt{\frac{(-\ell+n-1)!}{a^3 n^4 (\ell+n)!}} \left(\frac{r}{a n}\right)^\ell L_{-\ell+n-1}^{2 \ell+1}\left(\frac{2 r}{a n}\right)r^kr^2dr$$

• This paper has a formula for the case when $n=n’$. May 10, 2020 at 16:35
• See this answer for an analytic formula for your matrix element. Note: For $k=-3$ some matrix elements are infinite. Jan 30 at 5:45