Griffiths Introduction to Elementary Particles · Problem 7.8

Electrodynamics of Quarks and Hadrons — David J. Griffiths

Physics — QFT · dirac-equation-angular-momentum

Problem

[The purpose of this problem is to demonstrate that particles described by the Dirac equation carry ``intrinsic'' angular momentum (S\mathbf{S}) in addition to their orbital angular momentum (L\mathbf{L}), neither of which is separately conserved, although their sum is. It should be attempted only if you are reasonably familiar with quantum mechanics.] (a) Construct the Hamiltonian, HH, for the Dirac equation. [Hint: Solve equation (7.19) for p0/cp^0/c. Solution: H=cγ0(γ⋅p+mc)H = c\gamma^0(\boldsymbol{\gamma}\cdot\mathbf{p} + mc), where p=(ℏ/i)∇\mathbf{p} = (\hbar/i)\nabla is the momentum operator.] (b) Find the commutator of HH with the orbital angular momentum L=r×p\mathbf{L} = \mathbf{r}\times\mathbf{p}. [Solution: [H,L]=−iℏcγ0(γ×p)[H,\mathbf{L}] = -i\hbar c\gamma^0(\boldsymbol{\gamma}\times\mathbf{p})] Since [H,L][H,\mathbf{L}] is not zero, L\mathbf{L} by itself is not conserved. Evidently there is some other form of angular momentum lurking here. Introduce the ``spin angular momentum,'' S\mathbf{S}, defined by equation (7.48). (c) Find the commutator of HH with the spin angular momentum, S≡(ℏ/2)Σ\mathbf{S} \equiv (\hbar/2)\boldsymbol{\Sigma}. [Solution: [H,S]=iℏcγ0(γ×p)[H,\mathbf{S}] = i\hbar c\gamma^0(\boldsymbol{\gamma}\times\mathbf{p})] It follows that the total angular momentum, J=L+S\mathbf{J} = \mathbf{L}+\mathbf{S}, is conserved. (d) Show that every bispinor is an eigenstate of S2\mathbf{S}^2, with eigenvalue ℏ2s(s+1)\hbar^2 s(s+1), and find ss. What, then, is the spin of a particle described by the Dirac equation?

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