- #1
maverick280857
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Hi everyone
I have a question regarding a step in the proof of the Virial Theorem.
Specifically suppose [itex]|E\rangle[/itex] is a stationary state with energy [itex]E[/itex], i.e.
[tex]\hat{H}|E\rangle = E|E\rangle[/tex]
Now,
[tex][\hat{r}\bullet\hat{p},\hat{H}] = i\hbar\left(\frac{p^2}{m} - \vec{r}\bullet\nabla V\right)[/tex]
Taking the expectation value of the left hand side over stationary states, we see that
[tex]\langle E|[\hat{r}\bullet\hat{p},\hat{H}]|E\rangle = 0[/tex]
(The Virial Theorem for central potentials then assumes [itex]V(r) = \alpha r^{n}[/itex] 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 [itex]|E\rangle[/itex] is a stationary state with energy [itex]E[/itex], i.e.
[tex]\hat{H}|E\rangle = E|E\rangle[/tex]
Now,
[tex][\hat{r}\bullet\hat{p},\hat{H}] = i\hbar\left(\frac{p^2}{m} - \vec{r}\bullet\nabla V\right)[/tex]
Taking the expectation value of the left hand side over stationary states, we see that
[tex]\langle E|[\hat{r}\bullet\hat{p},\hat{H}]|E\rangle = 0[/tex]
(The Virial Theorem for central potentials then assumes [itex]V(r) = \alpha r^{n}[/itex] 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.