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Need help with proof for expectation value relation.

  1. Jan 29, 2009 #1
    1. The problem statement, all variables and given/known data
    I have to prove the following:
    [tex]
    \hbar \frac{d}{dt}\langle L\rangle = \langle N \rangle
    [/tex]

    Edit: L = Angular Momentum & N = Torque

    2. Relevant equations
    I used Ehrenfest's theorem, and I've got the equation in the following form:

    [tex]
    \frac{1}{i} \left(\left[L,H\right]\right) + \hbar \left\langle \frac{\partial L}{\partial t}\right\rangle
    [/tex]

    3. The attempt at a solution

    I pretty much need to prove the commutator term vanishes, but I'm not sure if it does. I've done the following with the commutation:

    [tex]
    i\hbar\frac{d}{dt}(\mathbf{r}\times\mathbf{p}) - i\hbar(\mathbf{r}\times\mathbf{p})\frac{d}{dt}
    [/tex]

    [tex]
    -i^2\hbar^2\frac{d}{dt}(\mathbf{r}\times\mathbf{\nabla}_r) - -(i^2)\hbar^2(\mathbf{r}\times\mathbf{\nabla}_r)\frac{d}{dt}
    [/tex]

    [tex]
    \hbar^2\frac{d}{dt}(\mathbf{r}\times\mathbf{\nabla}_r) - \hbar^2(\mathbf{r}\times\mathbf{\nabla}_r)\frac{d}{dt}
    [/tex]

    [tex]
    \hbar^2\frac{d}{dt}\left(\epsilon_{ijk}\mathbf{r}^j\mathbf{\nabla}^k\right) - \hbar^2\left(\epsilon_{ijk}\mathbf{r}^j\mathbf{\nabla}^k\right) \frac{d}{dt}
    [/tex]

    Can I switch the order of the derivatives/operators in such a way that I get the following:
    [tex]
    \hbar^2\frac{d}{dt}\left(\epsilon_{ijk}\mathbf{r}^j\mathbf{\nabla}^k\right) - \hbar^2\frac{d}{dt}\left(\epsilon_{ijk}\mathbf{r}^j\mathbf{\nabla}^k\right) = 0
    [/tex]
     
    Last edited: Jan 29, 2009
  2. jcsd
  3. Jan 29, 2009 #2
    Never mind, figured it out. I forgot that the angular momentum operator is time-independent, so it disappears. Then I took the commutation of [L,H] a little differently then the way I did it above, and I got the answer.
     
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