A three-level system has an orthonormal basis

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · The formalism of quantum mechanics

The Structure of Quantum Mechanics, Ch Compatible observables and uncertainty as a theorem

Problem

\{|1\rangle,|2\rangle,|3\rangle\}

. In this basis, three self-adjoint operators are represented by

\hat A = \begin{pmatrix}1&0&0\\0&1&0\\0&0&2\end{pmatrix}, \qquad \hat B = \begin{pmatrix}0&1&0\\1&0&0\\0&0&3\end{pmatrix}, \qquad \hat C = \begin{pmatrix}0&i&0\\-i&0&0\\0&0&4\end{pmatrix}.

((a)) Compute

[\hat A,\hat B]

and

[\hat A,\hat C]

. State exactly what the compatibility theorem then guarantees about a common eigenbasis for

\hat A

with

\hat B

, and separately for

\hat A

with

\hat C

. ((b)) Find the common eigenbasis of

\hat A

and

\hat B

explicitly, by diagonalising

\hat B

within the two-dimensional eigenspace of

\hat A

where

\hat A

is degenerate. Give each basis vector's pair of eigenvalues

(a,b)

. ((c)) Compute

[\hat B,\hat C]

. Decide, from this commutator alone, whether measuring

\hat B

then

\hat C

on a general state of the system gives the same statistics as measuring

\hat C

then

\hat B

. ((d)) Reconcile parts (a) and (c): explain why

\hat B

and

\hat C

each being compatible with

\hat A

does not make

\hat B

and

\hat C

compatible with each other.