Quantum Mechanics: Concepts and Applications · Problem 3.9

One-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · quantum-measurement-sequential-observables-degenerate

Problem

Consider a system whose state and two observables

A

and

B

are given by

|\psi(t)\rangle = \frac{1}{6} \begin{pmatrix} 1 \\ 0 \\ 4 \end{pmatrix}, \quad A = \frac{1}{\sqrt{2}} \begin{pmatrix} 2 & 0 & 0 \\ 0 & 1 & i \\ 0 & -i & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix}.

(a) We perform a measurement where

A

is measured first and then, immediately afterwards,

B

is measured. Find the probability of obtaining a value of

0

for

A

and a value of

1

for

B

. (b) Now we measure

B

first then, immediately afterwards,

A

. Find the probability of obtaining a value of

1

for

B

and a value of

0

for

A

. (c) Compare the results of (a) and (b). Explain. (d) Which among the sets of operators

\{\hat{A}\}

\{\hat{B}\}

, and

\{\hat{A}, \hat{B}\}

form a complete set of commuting operators (CSCO)?