A Modern Approach to Quantum Mechanics · Problem 5.19

Wave Mechanics in One Dimension — John S. Townsend

Physics — Quantum Mechanics · density-operator-unpolarized-any-direction

Problem

Show that

\hat{\rho} = \frac{1}{2}|{+\mathbf{n}}\rangle\langle{+\mathbf{n}}| + \frac{1}{2}|{-\mathbf{n}}\rangle\langle{-\mathbf{n}}| = \frac{1}{2}|{+z}\rangle\langle{+z}| + \frac{1}{2}|{-z}\rangle\langle{-z}|

where

\begin{aligned} |{+\mathbf{n}}\rangle &= \cos\frac{\theta}{2}|{+z}\rangle + e^{i\phi}\sin\frac{\theta}{2}|{-z}\rangle \\ |{-\mathbf{n}}\rangle &= \sin\frac{\theta}{2}|{+z}\rangle - e^{i\phi}\cos\frac{\theta}{2}|{-z}\rangle \end{aligned}