Quantum Mechanics (1) · Problem 5.1

Rotations and angular momentum

Physics — Quantum Mechanics · rotation-addition-tangents

Problem

Prove the following theorem: If

R_{\mathbf{e}}(\theta) = R_{\mathbf{e}''}(\theta'') R_{\mathbf{e}'}(\theta')

, and defining

\begin{aligned} \boldsymbol{\tau} &= \mathbf{e} \tan\theta/2 \\ \boldsymbol{\tau}' &= \mathbf{e}' \tan\theta'/2 \\ \boldsymbol{\tau}'' &= \mathbf{e}'' \tan\theta''/2, \end{aligned}

then

\boldsymbol{\tau} = \frac{\boldsymbol{\tau}' + \boldsymbol{\tau}'' + \boldsymbol{\tau}'' \times \boldsymbol{\tau}'}{1 - \boldsymbol{\tau}' \cdot \boldsymbol{\tau}''}. \tag{5.22}