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

[Homo Novus]

Homework Statement

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.

Homework 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]

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?

 

[mark726]

To solve this problem, we can start by writing out the expression for the dot product between the Pauli matrices and the momentum operator:

σ \cdot \vec{p} = σ_{x}p_{x} + σ_{y}p_{y} + σ_{z}p_{z}

Next, we can use the expressions given in the homework equations to substitute for the momentum components:

p_{x} = \frac{\hbar}{i} \frac{\partial}{\partial x}, p_{y} = \frac{\hbar}{i} \frac{\partial}{\partial y}, p_{z} = \frac{\hbar}{i} \frac{\partial}{\partial z}

Using this, we can rewrite the dot product as:

σ \cdot \vec{p} = \frac{\hbar}{i} (σ_{x} \frac{\partial}{\partial x} + σ_{y} \frac{\partial}{\partial y} + σ_{z} \frac{\partial}{\partial z})

Next, we can use the definition of the angular momentum operator given in the homework equations to substitute for the components of \vec{L}:

L_{x} = yp_{z} - zp_{y}, L_{y} = zp_{x} - xp_{z}, L_{z} = xp_{y} - yp_{x}

Using this, we can rewrite the dot product as:

σ \cdot \vec{p} = \frac{\hbar}{i} (σ_{x} \frac{\partial}{\partial x} + σ_{y} \frac{\partial}{\partial y} + σ_{z} \frac{\partial}{\partial z})
= \frac{\hbar}{i} (σ_{x} (yp_{z} - zp_{y}) + σ_{y} (zp_{x} - xp_{z}) + σ_{z} (xp_{y} - yp_{x}))

Using the definitions of the Pauli matrices, we can simplify this expression to:

σ \cdot \vec{p} = \frac{\hbar}{i} (σ_{x}yp_{z} - σ_{x}zp_{y} + σ_{y}zp_{x} - σ_{y}xp_{z} + σ_{z}xp_{y} - σ_{z}