# Rotation of spin projection different to general rotation?

1. May 17, 2017

### bananabandana

1. The problem statement, all variables and given/known data
Prove, for the matrix $S = exp \bigg(-\frac{i}{\hbar}\mathbf{\hat{n}}\cdot \mathbf{\hat{S}}\bigg)$ (spin-rotation matrix), and for an arbitary vector $\mathbf{a}$ that:

$$S^{-1} \mathbf{a} \cdot \mathbf{\hat{x}} S = a(-\theta) \cdot \mathbf{x} = a \cdot \mathbf{x}(\theta)$$

2. Relevant equations

3. The attempt at a solution
Let $\langle \psi| \mathbf{V} \rangle = \mathbf{V}_{0}$. Where $\mathbf{V}$ is some vector operator. Then (after Binney, Physics of Qunatum Mechanics,pg74):

Where $\boldsymbol{\alpha}$ is an axis of rotation, and let $\mathbf{R}(\boldsymbol{\alpha})$ be a rotation operator about that axis.

$$\mathbf{R}(\boldsymbol{\alpha})\mathbf{V} = U^{\dagger}(\boldsymbol{\alpha})\mathbf{V}U(\boldsymbol{\alpha})$$

i.e as standard. Apart from that, I have no idea where to begin! Apparently this is based on some 'standard relation in tensor calculus (read, matrix manipulations here, I think) - but I have no idea what is happening!

Thanks

Last edited by a moderator: May 19, 2017
2. May 22, 2017

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.