Demonstrate that the operator $\Theta=i \sigma_{2} \mathscr{K}$ defined in Box 29.1 meets all the criteria given there for a fermion time-reversal operator: it preserves the norm of a wavefunction, is antilinear, and satisfies $\Theta^{2}=-1$.

Data from Box 29.1

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Time-Reversal Symmetry and Kramers Degeneracy Ordinary symmetries are implemented by unitary operators. For example, if II inverts the spatial coordinates its action on a wavefunction is II (r) = (-). Then, if the Hamiltonian H commutes with II, simultaneous eigenvectors of H and II can be constructed, labeled by a parity 1 that is the eigenvalue of II. Time-Reversal Symmetry In contrast to ordinary symmetries, time reversal is implemented by an antiunitary operator that must satisfy (ca) cl)) = cola) + co|)