Commutators with the Dirac Equation

  • #1

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]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
 

Answers and Replies

  • #2
213
8
all you have to do is compute

[tex] [\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 [/tex]

i'm sure you can manage the rest like putting in factors and writing it in vector notation
 
  • #3
Thanks, sgd37, writing it in [tex] \epsilon_{ijk} [/tex] notation definitely cleared things up for me.
 

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