A Modern Approach to Quantum Mechanics · Problem 3.18

Time Evolution — John S. Townsend

Physics — Quantum Mechanics · spin-1-eigenstates-arbitrary-axis

Problem

Determine the eigenstates of

\hat{S}_n = \hat{\mathbf{S}} \cdot \mathbf{n}

for a spin-1 particle, where the spin operator

\hat{\mathbf{S}} = \hat{S}_x\mathbf{i} + \hat{S}_y\mathbf{j} + \hat{S}_z\mathbf{k}

and

\mathbf{n} = \sin\theta\cos\phi\,\mathbf{i} + \sin\theta\sin\phi\,\mathbf{j} + \cos\theta\,\mathbf{k}

. Use the matrix representation of the rotation operator in Problem 3.19 to check your result when

\phi = 0