Problem 1:

Consider the matrices A and B.

(a)  Argue why both matrices are diagonalizable.  No calculation allowed.
(b)  Find the eigenvalues and eigenvectors of A.
(c)  Find [A,B].
(d)  The similarity transformation B' = Q^-1BQ diagonalizes matrix B.  Find Q and Q^-1.

Problem 2:

You prepare an ensemble of identical experiments at t = 0.  You measure the values of a physical observable A at time t = t₁ > 0 in all of the experiments of the ensemble.  At time t₂ > t₁ you measure observable A again in all of the experiments, i.e. you follow the measurement at t = t₁ in each experiment with a subsequent measurement of A in each experiment at t = t₂.

True or False: (Explain your choice in one or two sentences.)
(a)  The values you obtain for A across all of the experiments at time t = t₁ are the same given you have made all of the measurements at the same time t = t₁.
(b)  The value you obtain for A in each experiment at t = t₂ may be different relative to the value obtained for A in the same experiment at t = t₁.
(c)  The expectation value of A is the same at both t = t₁ and t = t₂.
(d)  The values you obtain for A in any of the experiments at either t = t₁ or t = t₂ must be one of the eigenvalues of its associated Hermitian operator.

At t = t₃ > t₂ you measure a second physical observable B across all of the experiments, i.e. you follow the measurement of A at t = t₂ in each experiment with a measurement of B at t = t₃ in each experiment.  The commutator of the two Hermitian operators associated with A and B satisfies [A, B] ≠ 0.

True or False:
(e)  The values you obtained for A at t = t₂ in any of these experiments will remain the same at t = t₃ given you are measuring a different physical quantity B, not A.

Now you reset all of the experiments to what they were at t = 0, i.e., you start all over again with the same initial set of identical experiments.  At time t = t₄, you measure A again in all of your experiments.  Moreover, the commutator of the operator associated with A and the Hamiltonian operator satisfies
[H, A] =0 .

True or False:
(f)  The value you obtain for A in each experiment at t = t₄ is the same as the value you obtained for A in the corresponding experiment at t = t₁.
(g)  The expectation values of A at t = t and t = t₄ are the same.

Problem 3:

Consider a two-state system governed by the Hamiltonian H with energy eigenstates |E₁> and |E₂>,  where H|E₁> = E₁|E₁> and H|E₂> = E₂ |E₂>.  Consider also two other states,
|x> = (|E₁> + |E₂>)/√2,  and  |y> = (|E₁> - |E₂>)/√2.
At time t = 0 the system is in state |x>.  At what subsequent times is the probability of finding the system in state |y> the largest, and what is that probability?

Problem 4:

Consider the one-dimensional time-independent Schroedinger equation for some arbitrary potential U(x).  Prove that if a solution ψ(x) has the property that ψ(x) → 0 as x → ±∞, then the solution must be non-degenerate and therefore real, apart from a possible overall phase factor.
Hint:  Show that the contrary assumption leads to a contradiction.

Problem 5:

A particle in one dimension is in a stationary state.
(a)  Show that for any time-independent operator Q, the expectation or mean value <Q> is independent of time.
(b)  Show that the mean value of the particles momentum, <p>, is zero.