Robert Mann - An Introduction to Particle Physics and the Standard Model · Problem 11.6.3

Quantum Electrodynamics — Robert Mann

Physics — Particle Physics · dirac-equation-hamiltonian-angular-momentum

Problem

(a) Write the Dirac equation in Hamiltonian form by isolating the time-derivative of the spinor

\psi

on the left-hand side of the equation. The Hamiltonian

H

will be the operator on the right-hand side of the equation. What is

H

? (b) Find the commutator of

H

with the orbital angular momentum operator

\vec{L}

. (c) Find the commutator of

H

with the spin angular momentum operator

\vec{S}

. (d) Find the commutator of

H

with the total angular momentum operator

\vec{J} = \vec{L} + \vec{S}

. (e) Show that all spinors are eigenstates of

\vec{S}\cdot\vec{S} = S^2

. Since

S^2 = \hbar^2 s(s+1)

, what is the value of

s

for a Dirac spinor?