Hyper-Virial Theorem for Central potentials

Question

I saw an exercise on the hyper virial theorem for central potential and, as I was looking for some clues, I found it hasn't been solved here, so after finding the solution I thought I may include it as well in case anyone finds it useful (for some reason, I only encountered it until graduate school).

Consider a central potential of the form $V(r)=kr^{-s}$. Show that the hypervirial theorem satisfies:

$$\langle p_r r^s\rangle=-\frac{i\hbar s}{2}\langle r^{s-1}\rangle$$

Where $p_r=-i\hbar\partial/\partial_r$ and $s\in Z$

Answer 1

The simplest way to prove it is by considering the commutator:

$[p_r,r^s]=r[p_r,r^{s-1}]+[p_r,r^{s-1}]r=r^2[p_r,r^{s-2}]+2r[p_r,r^{s-2}]+[p_r,r^{s-2}]r^2+...+=r^{s-1}[p_r,r]+...+[p_r,r]r^{s-1}=r^{s-1}(-i\hbar)+.. (\textrm{other} \ (s-2) \ \textrm{terms with} \ r^{s-1})..+(-i\hbar)r^{s-1}=-i\hbar sr^{s-1}$

Taking the expectation value:

$\langle[p_r,r^s]\rangle=-i\hbar\langle r^{s-1}\rangle$

On the other hand:

$\langle[p_r,r^s]\rangle=\langle p_r r^s-r^s p_r \rangle=\langle p_r r^s\rangle-\langle r^s p_r \rangle=\langle p_r r^s\rangle+\langle p_r r^s\rangle$

For stationary states. Therefore:

$\langle[p_r,r^s]\rangle=2\langle p_r r^s\rangle$

$$\langle p_r r^s\rangle=-\frac{i\hbar s}{2}\langle r^{s-1}\rangle$$

Which completes the proof. Feel free to add any other method you may consider useful.