A Modern Approach to Quantum Mechanics · Problem 2.6

Angular Momentum — John S. Townsend

Physics — Quantum Mechanics · rotation-operator-y-axis

Problem

Evaluate R^(θj)∣+z⟩\hat{R}(\theta\mathbf{j})|{+z}\rangle, where R^(θj)=e−iJ^yθ/ℏ\hat{R}(\theta\mathbf{j}) = e^{-i\hat{J}_y\theta/\hbar} is the operator that rotates kets counterclockwise by angle θ\theta about the yy axis. Show that R^(π2j)∣+z⟩=∣+x⟩\hat{R}\left(\frac{\pi}{2}\mathbf{j}\right)|{+z}\rangle = |{+x}\rangle. Suggestion: Express the ket ∣+z⟩|{+z}\rangle as a superposition of the kets ∣+y⟩|{+y}\rangle and ∣−y⟩|{-y}\rangle and take advantage of the fact that J^y∣±y⟩=(±ℏ/2)∣±y⟩\hat{J}_y|{\pm y}\rangle = (\pm\hbar/2)|{\pm y}\rangle; then switch back to the ∣+z⟩|{+z}\rangle-∣−z⟩|{-z}\rangle basis.

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