Quantum Mechanics: Concepts and Applications · Problem 3.16

One-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · energy-measurement-matrix-hamiltonian

Problem

Consider a system whose Hamiltonian

H

and an operator

A

are given by the matrices

H = \mathcal{E}_0 \begin{pmatrix} 0 & -i & 0 \\ i & 0 & 2i \\ 0 & -2i & 0 \end{pmatrix}, \quad A = a_0 \begin{pmatrix} 0 & -i & 0 \\ i & 1 & 1 \\ 0 & 1 & 0 \end{pmatrix}.

(a) If we measure energy, what values will we obtain? (b) Suppose that when we measure energy, we obtain a value of

\sqrt{5}\mathcal{E}_0

. Immediately afterwards, we measure

A

. What values will we obtain for

A

and what are the probabilities corresponding to each value? (c) Calculate the expectation value

\langle A \rangle