A Modern Approach to Quantum Mechanics · Problem 5.25

Wave Mechanics in One Dimension — John S. Townsend

Physics — Quantum Mechanics · density-operator-measurement-probability

Problem

Show for the density operator for a mixed state

\hat{\rho} = \sum_{k} p_k |\psi^{(k)}\rangle\langle\psi^{(k)}|

that the probability of obtaining the state

|\phi\rangle

as a result of a measurement is given by

\text{tr}(\hat{P}_{|\phi\rangle}\hat{\rho})

, where

\hat{P}_{|\phi\rangle} = |\phi\rangle\langle\phi|