- #1
Lovemaker
- 11
- 0
Homework Statement
I have to prove the following:
[tex]
\hbar \frac{d}{dt}\langle L\rangle = \langle N \rangle
[/tex]
Edit: L = Angular Momentum & N = Torque
Homework 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]
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: