[NOTES/CM-08011] Rotations about a fixed axis

The set of all rotations about a fixed axis form a subgroup.

1. Closure property \begin{equation} R_{\hat{n}}(\theta)R_{\hat{n}}(\phi)=R_{\hat{n}}(\theta+\phi) \end{equation}
2. Associative property is obeyed for matrices. Also because of$$\Big(R_{\hat{n}}(\theta_1)R_{\hat{n}}(\theta_2)\Big)R_{\hat{n}}(\theta_3)=R_{ \hat{n}}(\theta_1+\theta_2+\theta_3)$$\begin{equation} =R_{\hat{n}}\Big(R_{\hat{n}}(\theta_2)R_{\hat{n}}(\theta_3)\Big) \end{equation}
3. Existence of identity The rotation by $\theta=0$ is identity transformation and is represented by the identity matrix $I$
4. Inverse of a rotation $(\hat{n},\theta)$ is the rotation $(\hat{n},-\theta)$\\

$\divideontimes$ The group of rotations about a fixed axis is a commutative (Abelian) group because \begin{equation} R_{\hat{n}}(\theta)R_{\hat{n}}(\phi)=R_{\hat{n}}(\phi)R_{\hat{n}}(\theta). \end{equation} both sides being equal to $R_{n}(\theta+\phi)$.

One parameter group nature is important because an application of Stone's theorem on one parameter unitary groups gives us the form of \(O(3)\) matrices as \begin{equation} R_{\hat n}(\theta) = \exp( -\theta I_n),\end{equation}where \(I_n\) is a real anti-symmetric matrix dependending only on \(\hat n\).