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## 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**=[tex] \frac{\hbar}{2}[/tex][tex]\vec{\Sigma}[/tex]. In other words find [H,

**S**]

## Homework Equations

For the Dirac equation, the Hamiltonian

*H*=c[tex]\gamma^0[/tex]([tex]\vec{\gamma} \cdot [/tex]

**p**+mc) where

**p**=[tex]\frac{\hbar}{i}[/tex][tex]\vec{\nabla}[/tex]

Where [tex]

\vec{\gamma} = \left(

\begin{array}{cc}

0 & \vec{\sigma}\\

-\vec{\sigma} & 0

\end{array}

\right)

[/tex] where [tex]\vec{\sigma}[/tex] are the pauli spin matrices

And [tex]

\vec{\Sigma} = \left(

\begin{array}{cc}

\vec{\sigma} & 0\\

0 & \vec{\sigma}

\end{array}

\right)

[/tex]

## 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[tex]\hbar[/tex]c[tex]\gamma^0[/tex] ([tex]\gamma[/tex]x

**p**). 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