Quantum Mechanics: Concepts and Applications · Problem 3.19

One-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · sequential-measurements-csco

Problem

Consider a system whose state

|\psi(t)\rangle

and two observables

A

and

B

are given by

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

(a) Are

A

and

B

compatible? Which among the sets of operators

\{\hat{A}\}

\{\hat{B}\}

, and

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

form a complete set of commuting operators? (b) Measuring

A

first and then

B

immediately afterwards, find the probability of obtaining a value of

-1

for

A

and a value of

3

for

B

. (c) Now, measuring

B

first then

A

immediately afterwards, find the probability of obtaining

3

for

B

and

-1

for

A

. Compare this result with the probability obtained in (b).