Introduction to Quantum Mechanics · Problem 3.33

Formalism — David J. Griffiths

Physics — Quantum Mechanics · sequential_measurements

Problem

Sequential measurements. An operator

\hat{A}

, representing observable

A

, has two (normalized) eigenstates

\psi_1

and

\psi_2

, with eigenvalues

a_1

and

a_2

, respectively. Operator

\hat{B}

, representing observable

B

, has two (normalized) eigenstates

\phi_1

and

\phi_2

, with eigenvalues

b_1

and

b_2

. The eigenstates are related by

\psi_1 = (3\phi_1 + 4\phi_2)/5, \quad \psi_2 = (4\phi_1 - 3\phi_2)/5.

(a) Observable

A

is measured, and the value

a_1

is obtained. What is the state of the system (immediately) after this measurement? (b) If

B

is now measured, what are the possible results, and what are their probabilities? (c) Right after the measurement of

B

A

is measured again. What is the probability of getting

a_1

? (Note that the answer would be quite different if I had told you the outcome of the

B

measurement.)