An electron (spin ) of mass moves in the isotropic three-dimensional oscillator potential

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Central potentials and canonical three-dimensional systems

The Structure of Quantum Mechanics, Ch Central potentials as a rotation problem

Problem

An electron (spin s=12s=\tfrac12) of mass mm moves in the isotropic three-dimensional oscillator potential V(r)=12mω2r2V(r) = \tfrac12 m\omega^2 r^2, with the spin decoupled from the orbital motion (no spin-orbit coupling). As derived for this potential, the oscillator level N=2N=2 contains only l=2l=2 and l=0l=0 orbital multiplets. ((a)) Using the canonical commutator [x^i,p^j]=iℏδij[\hat x_i,\hat p_j]=i\hbar\delta_{ij} to confirm [H^,L⃗^]=0[\hat H,\hat{\vec L}]=0 still holds once spin is included as a spectator, the 2j+12j+1 counting rule for a rotation multiplet, and the addition rule for combining an orbital multiplet ll with a spin-12\tfrac12 multiplet into total j=l±12j=l\pm\tfrac12, find the total number of degenerate states at N=2N=2 once the electron's spin is taken into account. ((b)) Ignoring spin, state the total spatial degeneracy of the N=2N=2 level, and explain why [H^,L⃗^]=0[\hat H,\hat{\vec L}]=0 by itself accounts only for the degeneracy within l=2l=2 (five states) and within l=0l=0 (one state) separately, not for their being at the same energy. Name the group whose extra conserved generators are responsible for the equality between the l=2l=2 and l=0l=0 sectors.

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