A Modern Approach to Quantum Mechanics · Problem 2.1

Rotation of Basis States and Matrix Mechanics — John S. Townsend

Physics — Quantum Mechanics · rotation-operator-taylor-series

Problem

Establish the identity

\hat{R}(\phi\mathbf{k}) = \lim_{N\to\infty}\left[1 - \frac{i}{\hbar}\hat{J}_z\left(\frac{\phi}{N}\right)\right]^N = e^{-i\hat{J}_z\phi/\hbar}

by expanding both sides in a Taylor series and showing that they agree term by term.