# Find the inner product of the Pauli matrices and the momentum operator?

1. Apr 13, 2012

### Homo Novus

1. The problem statement, all variables and given/known data

Show that the inner product of the Pauli matrices, σ, and the momentum operator, $\vec{p}$, is given by:

σ $\cdot$ $\vec{p}$ = $\frac{1}{r^{2}}$ (σ $\cdot$ $\vec{r}$ )($\frac{\hbar}{i}$ r $\frac{\partial}{\partial r}$ + iσ $\cdot$ $\vec{L}$),

where $\vec{L}$ is the angular momentum operator and $\vec{r}$ is the displacement vector.

2. Relevant equations

p$_{x}$ = $\frac{\hbar}{i}$ $\frac{\partial}{\partial x}$
$\vec{L}$ = $\vec{r}$ × $\vec{p}$

3. The attempt at a solution

I figured that I could write:

$\vec{p}$ = $\frac{\hbar}{i}$ $\frac{\partial}{\partial r}$ $\hat{r}$

So then:
σ $\cdot$ $\vec{p}$ = (σ $\cdot$ $\hat{r}$) $\frac{\hbar}{i}$ $\frac{\partial}{\partial r}$
= $\frac{1}{r}$ (σ $\cdot$ $\vec{r}$) $\frac{\partial}{\partial r}$

... But that clearly gets me nowhere. Help?