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Virial Theorem

  1. Nov 23, 2008 #1
    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.
     
  2. jcsd
  3. Nov 24, 2008 #2
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