Consider first a single spinless particle in one dimension with canonical commutator .

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

Consider first a single spinless particle in one dimension with canonical commutator [x^,p^]=iℏ[\hat x,\hat p]=i\hbar. ((a)) A candidate time-reversal operator Θ^\hat\Theta must implement Θ^x^Θ^−1=x^\hat\Theta\hat x\hat\Theta^{-1}=\hat x and Θ^p^Θ^−1=−p^\hat\Theta\hat p\hat\Theta^{-1}=-\hat p (positions unchanged, momenta reversed, when the film is run backwards). Show that no unitary (linear) Θ^\hat\Theta can do this consistently with [x^,p^]=iℏ[\hat x,\hat p]=i\hbar, and that an antiunitary (antilinear) Θ^\hat\Theta resolves the difficulty provided Θ^iΘ^−1=−i\hat\Theta i\hat\Theta^{-1}=-i. ((b)) For a spin one-half particle at rest (spin only, no orbital motion), Θ^\hat\Theta must reverse the spin, Θ^S^⃗Θ^−1=−S^⃗\hat\Theta\vec{\hat S}\hat\Theta^{-1}=-\vec{\hat S}, with S^⃗=ℏ2σ⃗\vec{\hat S}=\frac{\hbar}{2}\vec\sigma the Pauli spin operator. Take Θ^=−iσyK\hat\Theta = -i\sigma_y K, where KK complex-conjugates the components of a spinor in the S^z\hat S_z eigenbasis. Using KσxK−1=σxK\sigma_xK^{-1}=\sigma_x, KσyK−1=−σyK\sigma_yK^{-1}=-\sigma_y, KσzK−1=σzK\sigma_zK^{-1}=\sigma_z, and σyσiσy=−σi\sigma_y\sigma_i\sigma_y=-\sigma_i for i=x,zi=x,z, verify Θ^S^⃗Θ^−1=−S^⃗\hat\Theta\vec{\hat S}\hat\Theta^{-1}=-\vec{\hat S} and compute Θ^2\hat\Theta^2.

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