maverick280857
Nov23-08, 01:02 PM
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.
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.