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

so that, in addition to the complex conjugation, the up and down components are interchanged.

(a) Show that ^Θ² = -1 for a spin-1/2 particle.
(b) Consider an eigenstate |Ψn> of a time-reversal invariant Hamiltonian (Equation 6.83) with energy E_n. We know that |Ψ'n> = ^Θ|Ψ'n> is also an eigenstate of Ĥ with the same energy E_n. There two possibilities: either |Ψ'n> and |Ψn> are the same state (meaning |Ψ'n> = 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.
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-1/2 for that matter), every energy level (of a time-reversal-invariant Hamiltonian) is at least twofold degenerate. This is because—as you just showed—for half-integer spin a state and its time-reversed state are necessarily distinct.

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(6.84) (2-)-(3) =