# Showing that the Virial Theorem holds

by Ed Quanta
Tags: holds, showing, theorem, virial
 P: 297 So I have already calculated correctly that the expectation of the interaction potential for hydrogenic atoms is =-(uZ^2e^4)/((hbar^2)*n^2) Note that u= mass,and V(r)=-Ze^2/r I now have to calculate where T is the kinetic energy operator, and T=p^2/(2u) + L^2/(2ur^2) Note that p is the radial momentum operator and L is the angular momentum operator I know hbar^2l(l+1)=L^2 and I know (pretty sure) that <1/r^2>=e^2/(2n^3hbar^2). However, I am unsure how to find the expectation value of the p^2/(2u) term, and don't see how is going to equal which the book hints at being true since and are said to satisfy the Virial Theorem. Help anyone?
Emeritus
 Sci Advisor HW Helper P: 11,915 There's a piece of genius on my behalf: $$\hat{H}=\hat{T}+\hat{V}$$ (1) $$\langle nlm|\hat{H}|nlm\rangle = E_{n} =-\frac{Z^{2}\mu e^{4}}{2\hbar^{2}(4\pi\epsilon_{0})} \frac{1}{n^{2}}$$ (2) $$\langle nlm|\hat{V}|nlm\rangle =-\frac{Ze^{2}}{(4\pi\epsilon_{0})}\langle \frac{1}{r}\rangle _{|nlm\rangle}$$ (3) Compute (3) using the average of 1/r. Then: $$\langle nlm|\hat{T}|nlm\rangle =\langle nlm|\hat{H}-\hat{V}|nlm\rangle=E_{n}+\frac{Ze^{2}}{(4\pi\epsilon_{0})}\langle \frac{1}{r}\rangle _{|nlm\rangle}$$ (4) Tell where u get stuck. Daniel.
It must use the cgs unit system.. in cgs a factor of $$( 4 \pi \epsilon_0 )^{\frac{1}2}$$ is "absorbed" into the unit of charge, so that Coulomb's law can be written as $$\vec{F} = \frac{e^2}{r^2}\hat{r}$$