For a spin one-half particle

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

For a spin one-half particle, time reversal is Θ^=−iσ^yK\hat\Theta = -i\hat\sigma_y K, where KK denotes complex conjugation of the components in the {∣ ⁣↑⟩,∣ ⁣↓⟩}\{|\!\uparrow\rangle,|\!\downarrow\rangle\} basis and σ^y=(0−ii0)\hat\sigma_y=\begin{pmatrix}0&-i\\ i&0\end{pmatrix} in that basis. For components (c1,c2)(c_1,c_2) this gives Θ^(c1,c2)=(−c2∗, c1∗)\hat\Theta(c_1,c_2) = (-c_2^*,\,c_1^*).Someone argues: “−iσ^y-i\hat\sigma_y is a unitary matrix and complex conjugation KK just relabels each component by its conjugate, which preserves lengths; composing two length-preserving maps gives a unitary operator, so Θ^\hat\Theta must satisfy ⟨Θ^ϕ∣Θ^ψ⟩=⟨ϕ∣ψ⟩\langle\hat\Theta\phi|\hat\Theta\psi\rangle = \langle\phi|\psi\rangle for all ∣ϕ⟩,∣ψ⟩|\phi\rangle,|\psi\rangle, exactly as for the unitary implementation of parity.”Take ∣ψ⟩|\psi\rangle with components (12,i2)\left(\tfrac{1}{\sqrt2},\tfrac{i}{\sqrt2}\right) (spin along +y+y) and ∣ϕ⟩|\phi\rangle with components (12,12)\left(\tfrac{1}{\sqrt2},\tfrac{1}{\sqrt2}\right) (spin along +x+x). ((a)) Compute ⟨ϕ∣ψ⟩\langle\phi|\psi\rangle and ⟨ψ∣ϕ⟩\langle\psi|\phi\rangle (recall ⟨ϕ∣ψ⟩=ϕ1∗ψ1+ϕ2∗ψ2\langle\phi|\psi\rangle=\phi_1^*\psi_1+\phi_2^*\psi_2). ((b)) Compute Θ^∣ϕ⟩\hat\Theta|\phi\rangle and Θ^∣ψ⟩\hat\Theta|\psi\rangle. ((c)) Compute ⟨Θ^ϕ∣Θ^ψ⟩\langle\hat\Theta\phi|\hat\Theta\psi\rangle and compare it with ⟨ϕ∣ψ⟩\langle\phi|\psi\rangle and with ⟨ψ∣ϕ⟩\langle\psi|\phi\rangle. ((d)) State whether the numbers support the unitary relation ⟨Θ^ϕ∣Θ^ψ⟩=⟨ϕ∣ψ⟩\langle\hat\Theta\phi|\hat\Theta\psi\rangle=\langle\phi|\psi\rangle or the antiunitary relation ⟨Θ^ϕ∣Θ^ψ⟩=⟨ϕ∣ψ⟩∗\langle\hat\Theta\phi|\hat\Theta\psi\rangle=\langle\phi|\psi\rangle^*.

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