Quantum Mechanics: Concepts and Applications · Problem 3.8

One-Dimensional Problems — Nouredine Zettili

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

Problem

Consider a system whose state and two observables are given by

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

(a) What is the probability that a measurement of

A

at time

t

yields

-1

? (b) Let us carry out a set of two measurements where

B

is measured first and then, immediately afterwards,

A

is measured. Find the probability of obtaining a value of

0

for

B

and a value of

1

for

A

. (c) Now we measure

A

first then, immediately afterwards,

B

. Find the probability of obtaining a value of

1

for

A

and a value of

0

for

B

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

\{\hat{A}\}

\{\hat{B}\}

, and

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

form a complete set of commuting operators (CSCO)?