# Need help with proof for expectation value relation.

1. Jan 29, 2009

### Lovemaker

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

Edit: L = Angular Momentum & N = Torque

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

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

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:

$$i\hbar\frac{d}{dt}(\mathbf{r}\times\mathbf{p}) - i\hbar(\mathbf{r}\times\mathbf{p})\frac{d}{dt}$$

$$-i^2\hbar^2\frac{d}{dt}(\mathbf{r}\times\mathbf{\nabla}_r) - -(i^2)\hbar^2(\mathbf{r}\times\mathbf{\nabla}_r)\frac{d}{dt}$$

$$\hbar^2\frac{d}{dt}(\mathbf{r}\times\mathbf{\nabla}_r) - \hbar^2(\mathbf{r}\times\mathbf{\nabla}_r)\frac{d}{dt}$$

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

Can I switch the order of the derivatives/operators in such a way that I get the following:
$$\hbar^2\frac{d}{dt}\left(\epsilon_{ijk}\mathbf{r}^j\mathbf{\nabla}^k\right) - \hbar^2\frac{d}{dt}\left(\epsilon_{ijk}\mathbf{r}^j\mathbf{\nabla}^k\right) = 0$$

Last edited: Jan 29, 2009
2. Jan 29, 2009

### Lovemaker

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.