Quantum Mechanics: Concepts and Applications · Problem 3.18

One-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · sequential-measurements-two-observables

Problem

Consider a system whose state

|\psi(t)\rangle

and two observables

A

and

B

are given by

|\psi(t)\rangle = \begin{pmatrix} 5 \\ 1 \\ 3 \end{pmatrix}, \quad A = \frac{1}{\sqrt{2}} \begin{pmatrix} 2 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}.

(a) We perform a measurement where

A

is measured first and then

B

immediately afterwards. Find the probability of obtaining a value of

\sqrt{2}

for

A

and a value of

-1

for

B

. (b) Now we measure

B

first and then

A

immediately afterwards. Find the probability of obtaining a value of

-1

for

B

and a value of

\sqrt{2}

for

A

. (c) Compare the results of (a) and (b). Explain.