# Homework Help: Using the Baker-Campbell-Hausdorff Identity

1. Feb 29, 2012

### Mindstein

1. The problem statement, all variables and given/known data
Given that U = e$^{-i*θ*\hat{n}*\vec{J}/hbar}$

Show that U$^{†}$$\vec{J}$U = $\hat{n}(\hat{n}*\vec{J}) - \hat{n}\times(\hat{n}\times\vec{J})cos(θ) + \hat{n}\times\vec{J}sin(θ)$

2. Relevant equations
The Baker-Campbell-Hausdorff Identity

3. The attempt at a solution
U$^{†}$$\vec{J}$U = e$^{-i*θ*\hat{n}*\vec{J}/hbar}$$\vec{J}$e$^{i*θ*\hat{n}*\vec{J}/h_bar}$

U$^{†}$$\vec{J}$U = $\vec{J} + [i*θ*\hat{n}*\vec{J}/hbar,\vec{J}]+[i*θ*\hat{n}*\vec{J}/hbar,[i*θ*\hat{n}*\vec{J}/hbar,\vec{J}]]/2! + ...$

I guess I just need some help on whether or not I am headed in the right direction.

Last edited: Feb 29, 2012