- #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: