Introduction to Quantum Mechanics · Problem 3.46

Formalism — David J. Griffiths

Physics — Quantum Mechanics · three_level_systems

Problem

The Hamiltonian for a certain three-level system is represented by the matrix

\mathsf{H} = \hbar\omega \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{pmatrix}.

Two other observables,

A

and

B

, are represented by the matrices

\mathsf{A} = \lambda \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 2 \end{pmatrix}, \quad \mathsf{B} = \mu \begin{pmatrix} 2 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix},

where

\omega

\lambda

, and

\mu

are positive real numbers. (a) Find the eigenvalues and (normalized) eigenvectors of

\mathsf{H}

\mathsf{A}

, and

\mathsf{B}

. (b) Suppose the system starts out in the generic state

|S(0)\rangle = \begin{pmatrix} c_1 \\ c_2 \\ c_3 \end{pmatrix},

with

|c_1|^2 + |c_2|^2 + |c_3|^2 = 1

. Find the expectation values (at

t = 0

) of

H

A

, and

B

. (c) What is

|S(t)\rangle

? If you measured the energy of this state (at time

t

), what values might you get, and what is the probability of each? Answer the same questions for observables

A

and for

B