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Homework Help: Find the inner product of the Pauli matrices and the momentum operator?

  1. Apr 13, 2012 #1
    1. The problem statement, all variables and given/known data

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

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

    where [itex]\vec{L}[/itex] is the angular momentum operator and [itex]\vec{r}[/itex] is the displacement vector.

    2. Relevant equations

    p[itex]_{x}[/itex] = [itex]\frac{\hbar}{i}[/itex] [itex]\frac{\partial}{\partial x}[/itex]
    [itex]\vec{L}[/itex] = [itex]\vec{r}[/itex] × [itex]\vec{p}[/itex]

    3. The attempt at a solution

    I figured that I could write:

    [itex]\vec{p}[/itex] = [itex]\frac{\hbar}{i}[/itex] [itex]\frac{\partial}{\partial r}[/itex] [itex]\hat{r}[/itex]

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

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