Rotation and Pauli matrices

[Wiemster]

I am given the formula (valid for any a)

$$A ( \vec{ \sigma } \cdot \vec{a} ) A^{-1} = \vec{ \sigma } \cdot R_A \vec{a}$$

with [itex]A=exp(i \phi \cdot \vec{\sigma} /2) = exp(i \phi \vec{\sigma} \cdot \hat{n} /2)[/tex] R_A the rotation matrix and sigma the Pauli matrices.

And am supposed to derive the relation

$$\vec{b} \cdot R_A \vec{a} = \vec{b} \cdot \vec{a} cos( \phi) + \hat{n} \cdot (\vec{a} \times \vec{b})sin(\phi) + 2 (\vec{b} \cdot \hat{n})(\vec{a} \cdot \hat{n}) sin^2(\phi /2)$$

As a hint some formulas for the traces of products of Pauli matrices are supplied. I have absolutey no idea how to begin and what the problem has to do with the traces of Pauli matrices. Can anybody see where to start?!

 

[Nino MMP]

Hint: Let $\vec{b}$ be an arbitrary vector. Then use the formula for the trace of two Pauli matrices to show that $R_A \vec{a} = \vec{a} cos(\phi) + \hat{n} \times \vec{a} sin(\phi) + 2 (\vec{a} \cdot \hat{n}) \hat{n} sin^2(\phi /2)$. Multiplying both sides by $\vec{b}$ and using the dot product identity should then give the desired result.