A bead is threaded on a rigid circular wire of radius lying in the -plane and slides…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Symmetry, angular momentum and spin

The Structure of Quantum Mechanics, Ch Angular momentum from the rotation group

Problem

A bead is threaded on a rigid circular wire of radius aa lying in the xyxy-plane and slides freely around it. Because the wire confines it to the plane, the only rotations available to this system are those about the axis perpendicular to the plane (the zz-axis): the rotation group available here is the two-dimensional rotation group, generated by the single operator J^z\hat J_z. ((a)) Write the 2×22\times2 matrix R(ϵ)R(\epsilon) describing an active rotation of the plane by angle ϵ\epsilon, and show directly from its form that R(ϵ1)R(ϵ2)=R(ϵ2)R(ϵ1)R(\epsilon_1)R(\epsilon_2)=R(\epsilon_2)R(\epsilon_1) for any two angles ϵ1,ϵ2\epsilon_1,\epsilon_2. Explain, by contrast with the derivation of [J^x,J^y]=iℏJ^z[\hat J_x,\hat J_y]=i\hbar\hat J_z in three dimensions (which needed two different rotation axes to build a nontrivial group commutator), why this bead's rotation group offers no analogous relation to derive beyond the exact commutation [J^z,J^z]=0[\hat J_z,\hat J_z]=0, which holds identically for any single operator. ((b)) The bead's state is a wavefunction ψ(ϕ)\psi(\phi), where ϕ\phi and ϕ+2π\phi+2\pi label the same point on the wire. Using R^(θ)=e−iθJ^z/ℏ\hat{\mathcal R}(\theta)=e^{-i\theta\hat J_z/\hbar} and the requirement R^(2π)ψ=ψ\hat{\mathcal R}(2\pi)\psi=\psi, show that the eigenvalues of J^z/ℏ\hat J_z/\hbar for this bead must be integers.

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