A Modern Approach to Quantum Mechanics · Problem 2.24

Angular Momentum — John S. Townsend

Physics — Quantum Mechanics · hermitian-operator-spectral-decomposition

Problem

The Hermitian operator

\hat{A}

corresponding to the observable

A

has two eigenstates

|a_1\rangle

and

|a_2\rangle

with eigenvalues

a_1

and

a_2

, respectively. Assume

a_1 \neq a_2

. Show that

\hat{A}

can be written in the form

\hat{A} = a_1|a_1\rangle\langle a_1| + a_2|a_2\rangle\langle a_2|

and that

\langle\psi|\hat{A}|\psi\rangle = \langle A\rangle