Matrix quantum mechanics: The Hamiltonian of a two-state system is given by H =

(a) Write down the energy eigenvalues, and the energy eigenvectors U(1) and U(2) of H.

()

[2]

(c) Assume the state of the system at t = 0 is given by C(0) = V. Find the expectation values (H) and (H²). [3]

(b) Show that V =

is a normalised eigenvector of A. What is the corresponding eigenvalue?

[1]

(d) Assuming the system is in the state C(0) = V at t = 0 find the state C(t) at later times t > 0 using the

expansion theorem. Hint: write V as a linear combination of U(1) and U(2) and put "wiggle factors".

(e) Calculate the expectation values (A) and (A²) using the time dependent state C(t). Hence, find the un-

certainty AA. For which values of time t does AA vanish?

Hint: recall the expectation values of operators in matrix quantum mechanics are defined as (O) = cioc

where O is the matrix form of the operator and C is the state; either C(0) or C(t) in the examples above.