Commutators with the Dirac Equation

[Greener1387]

Homework Statement

(Introduction to Elementary Particles, David Griffiths. Ch 7 Problem 7.8 (c))
Find the commutator of H with the spin angular momentum, S=$$\frac{\hbar}{2}$$$$\vec{\Sigma}$$. In other words find [H,S]

Homework Equations

For the Dirac equation, the Hamiltonian H=c$$\gamma^0$$($$\vec{\gamma} \cdot$$p+mc) where p=$$\frac{\hbar}{i}$$$$\vec{\nabla}$$


Where $$\vec{\gamma} = \left( \begin{array}{cc} 0 & \vec{\sigma}\\ -\vec{\sigma} & 0 \end{array} \right)$$ where $$\vec{\sigma}$$ are the pauli spin matrices

And $$\vec{\Sigma} = \left( \begin{array}{cc} \vec{\sigma} & 0\\ 0 & \vec{\sigma} \end{array} \right)$$

The Attempt at a Solution

In a previous part of the problem we determined that the commutator for orbit angular momentum L, [H,L]=-i$$\hbar$$c$$\gamma^0$$ ($$\gamma$$xp). The point of the problem is to show that Total Angular Momentum J=L+S commutes with the Hamiltonian, so we know that [H,S] should be -[H,L].

My initial attempt at a solution was to explicitly write out all the matrices and decompose p into it's cartesian coordinates and then hopefully recognize a cross product after factoring out an i. The problem I ran into was that there were a few terms on the backwards diagonal that did not involve i and did not cancel out. If nothing else works I'll triple check that, but I think what is needed is to make the quantized p substitution. But when I do that, I'm at a loss for how to handle the expression for the commutator with that definition of p. It also seems a bit strange to me to have a cross product where the first term in the cross product has three components of 4x4 matrices and the second term is a three component vector.

Can anybody verify which method I should follow or where I made any bad assumptions?

Any bold letters are three component vectors, not four

 

[sgd37]

all you have to do is compute

$$[\gamma^0 \gamma^i , \Sigma^j ] = \begin{pmatrix} 0 & [\sigma^i ,\sigma^j ] \\ [\sigma^i ,\sigma^j ] & 0 \end{pmatrix} = 2i \epsilon_{ijk} \begin{pmatrix} 0 & \sigma^k \\ \sigma^k & 0 \end{pmatrix} = 2i \epsilon_{ijk} \gamma^0 \gamma^k$$

i'm sure you can manage the rest like putting in factors and writing it in vector notation

 

[Greener1387]

Thanks, sgd37, writing it in $$\epsilon_{ijk}$$ notation definitely cleared things up for me.