Expectation Values

Expectation Values Consider an ensemble of microscopic systems prepared in the same initial state \(\mid A \rangle\). Suppose a measurement of the observable \(\xi'\) is made on each system. We know that each measurement yields the value \(\xi'\) with probability \(P(\xi')\). What is the mean value of the measurement? This quantity, which is generally referred to as the expectation value of \(\xi'\), is given by \((58)\;\;\;\;\;\ \langle\xi\rangle = \displaystyle \sum^{}_{\xi'} \xi'P(\xi') = \displaystyle \sum^{}_{\xi'} \xi' \mid\langle A\mid\ xi\rangle\mid^{2}\) \( = \displaystyle \sum^{}_{\xi'} \xi' \langle A\mid\xi\rangle\langle \xi'\mid A\rangle = \displaystyle \sum^{}_{\xi'} \xi' \langle A\mid\xi\mid\xi'\rangle\langle \xi'\mid A\rangle\)