Symmetries and Conservation Laws — David J. Griffiths
Physics — Quantum Mechanics · time_reversal_spin
Problem
As an angular momentum, a particle's spin must flip under time reversal (Problem 6.36). The action of time-reversal on a spinor (Section 4.4.1) is in fact
Θ^(ab)=(−b∗a∗)(6.84)
so that, in addition to the complex conjugation, the up and down components are interchanged.
(a) Show that Θ^2=−1 for a spin-1/2 particle.
(b) Consider an eigenstate ∣ψn⟩ of a time-reversal invariant Hamiltonian (Equation 6.83) with energy En. We know that ∣ψn′⟩=Θ^∣ψn⟩ is also an eigenstate of H^ with the same energy En. There are two possibilities: either ∣ψn′⟩ and ∣ψn⟩ are the same state (meaning ∣ψn′⟩=c∣ψn⟩ for some complex constant c) or they are distinct states. Show that the first case leads to a contradiction in the case of a spin-1/2 particle, meaning the energy level must be (at least) two-fold degenerate in that case.
*Comment:* What you have proved is a special case of Kramer's degeneracy: for an odd number of spin-1/2 particles (or any half-integer spin for that matter), every energy level (of a time-reversal-invariant Hamiltonian) is at least two-fold degenerate. This is because—as you just showed—for half-integer spin a state and its time-reversed state are necessarily distinct.