A single unpaired electron (spin one-half) is bound at a lattice site of a crystal…
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Symmetry, angular momentum and spin
The Structure of Quantum Mechanics, Ch Symmetries, generators and conservation laws
Problem
A single unpaired electron (spin one-half) is bound at a lattice site of a crystal whose local electric field has no rotational, reflection or inversion symmetry at all: a generic low-symmetry site described by a real, time-independent potential energy V(r) that does not commute with any component of orbital angular momentum. There is no external magnetic field and no spin-orbit term, so H^=2mp^2+V(r^) acts only on the orbital motion, with the spin degree of freedom present but absent from H^.
((a)) Explain why time reversal Θ^ commutes with this H^ even though V(r) has no spatial symmetry at all.
((b)) For spin one-half, Θ^2=−1. Starting from ⟨Θ^ψ∣Θ^2ψ⟩=⟨Θ^ψ∣ψ⟩ and the antiunitary identity ⟨Θ^ϕ∣Θ^χ⟩=⟨χ∣ϕ⟩, show that ⟨Θ^ψ∣ψ⟩=0, and deduce that ∣ψ⟩ and ∣ψ~⟩≡Θ^∣ψ⟩ are linearly independent.
((c)) Given [H^,Θ^]=0, show that ∣ψ~⟩ has the same energy as ∣ψ⟩, and conclude what this means for the degeneracy of every level of this electron, however asymmetric the crystal field, and what alone can lift it.