Quantum Mechanics: Concepts and Applications · Problem 7.15

Identical Particles — Nouredine Zettili

Physics — Quantum Mechanics · rotation-operators

Problem

(a) Show that

e^{i\pi\hat{J}_y/2\hbar}\,\hat{J}_x\, e^{-i\pi\hat{J}_y/2\hbar} = \hat{J}_z

. (b) Show also that

e^{i\pi\hat{J}_y/2\hbar}\, e^{i\alpha\hat{J}_x/\hbar}\, e^{-i\pi\hat{J}_y/2\hbar} = e^{i\alpha\hat{J}_z/\hbar}

. (c) For any vector operator

\hat{A}

, show that

e^{i\alpha\hat{J}_z/\hbar}\,\hat{A}_x\, e^{-i\alpha\hat{J}_z/\hbar} = \hat{A}_x\cos\alpha + \hat{A}_y\sin\alpha