1. (Commute)

   Consider a three-dimensional state space. In the basis defined by three orthonormal kets $\vert 1\rangle$, $\vert 2\rangle$, and $\vert 3 \rangle$, the operators $A$ and $B$ are represented by:

   $$A\doteq\pmatrix{a_1&0&0\cr 0&a_2&0\cr 0&0&a_3} \qquad\qquad\qquad B\doteq\pmatrix{b_1&0&0\cr 0&0&b_2\cr 0&b_2&0}$$ where all the matrix elements are real.

   1. Do the operators $A$ and $B$ commute?

   2. Find the eigenvalues and normalized eigenvectors of both operators.

   3. Assume the system is initially in the state $\vert 2\rangle$. Then the observable corresponding to the operator $B$ is measured. What are the possible results of this measurement and the probabilities of each result?

      After this measurement, the observable corresponding to the operator $A$ is measured. What are the possible results of this measurement and the probabilities of each result?

   4. Interpret the Mathematical Model How are questions (a) and (c) above related?

See Time Evolution Homework for problems involving the expectation value and uncertainties.