An electron (spin one-half) is trapped at a point defect in a crystal
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Symmetry, angular momentum and spin
The Structure of Quantum Mechanics, Ch Symmetries, generators and conservation laws
Problem
An electron (spin one-half) is trapped at a point defect in a crystal. The local electrostatic potential energy V(r) at the defect is real and time-independent and has no rotational symmetry, no reflection symmetry and no inversion symmetry: its only exact symmetry operation is the identity. No magnetic field is present. As a model of a coupling between the electron's motion and its spin, take
H^=2mp^2+V(r^)+λp^xS^z,
where λ is a real constant.
((a)) Using the procedure for reading a degeneracy off a symmetry group, what multiplet dimension does the spatial symmetry of V(r) alone predict for a generic orbital energy level? State the group.
((b)) Set λ=0 for a moment. Show that every level of H^ is then at least two-fold degenerate without using time reversal, and name the symmetry responsible.
((c)) Now take λ=0. Which components of S^ still commute with H^? Explain why the symmetry they generate forces no degeneracy.
((d)) Show that Θ^ still commutes with H^ when λ=0. State Θ^2 for a spin one-half particle, and use the Kramers argument to find the minimum degeneracy of every level.
((e)) A colleague claims that, because the defect has no spatial symmetry at all, a suitably shaped V(r) and a suitable λ could split every level completely. Say what is wrong with this claim, and state what kind of term would lift the degeneracy found in part (d).