A Modern Approach to Quantum Mechanics · Problem 3.19

Time Evolution — John S. Townsend

Physics — Quantum Mechanics · spin-1-rotation-operator

Problem

Find the state with

S_n = \hbar

of a spin-1 particle, where

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

, by rotating a state with

S_z = \hbar

by angle

\theta

counterclockwise about the

y

axis using the rotation operator

\hat{R}(\theta\mathbf{j}) = e^{-i\hat{S}_y\theta/\hbar}

. Suggestion: Use the matrix representation (3.104) for

\hat{S}_y

in the

S_z

basis and expand the rotation operator in a Taylor series. Work out the matrices through the one representing

\hat{S}_y^3

in order to see the pattern and show that

\hat{R}(\theta\mathbf{j}) \xrightarrow{S_z\text{ basis}} \begin{pmatrix} \dfrac{1+\cos\theta}{2} & -\dfrac{\sin\theta}{\sqrt{2}} & \dfrac{1-\cos\theta}{2} \\[6pt] \dfrac{\sin\theta}{\sqrt{2}} & \cos\theta & -\dfrac{\sin\theta}{\sqrt{2}} \\[6pt] \dfrac{1-\cos\theta}{2} & \dfrac{\sin\theta}{\sqrt{2}} & \dfrac{1+\cos\theta}{2} \end{pmatrix}