Consider an electron at rest in the magnetic field:

B = B₀x^,

where B₀ is constant.

(a) Show that the Hamiltonian matrix in the S_z representation is

b) What are the eigenvalues for energy, and the corresponding eigenvectors?

(c) Assume the electron is in the spin-down state along the z axis at time t = 0:

|ψ(t=0)> = |->.

Determine the quantum state |ψ(t)> of the electron at a later time t. Express your result in terms of the basis for measurement of S_z.

(d) What is the expected value of the energy at time t?

(e) Determine the probability of a spin flip in z, i.e. calculate

P₊ = |<+|ψ(t)>|²

and sketch your result as a function of t.