What is the physical significance of the commutator in the Virial Theorem?

[maverick280857]

Hi everyone

I have a question regarding a step in the proof of the Virial Theorem.

Specifically suppose $|E\rangle$ is a stationary state with energy $E$, i.e.

$$\hat{H}|E\rangle = E|E\rangle$$

Now,

$$[\hat{r}\bullet\hat{p},\hat{H}] = i\hbar\left(\frac{p^2}{m} - \vec{r}\bullet\nabla V\right)$$

Taking the expectation value of the left hand side over stationary states, we see that

$$\langle E|[\hat{r}\bullet\hat{p},\hat{H}]|E\rangle = 0$$

(The Virial Theorem for central potentials then assumes $V(r) = \alpha r^{n}$ and one gets <T> = (n/2)<V>.)

My question is: what is the physical significance of this commutator and what does it mean physically that the expectation of this commutator wrt a basis of stationary states is zero?

Thanks in advance.

Cheers,
Vivek.

 

[maverick280857]

Anyone?