Question

An electron (a spin-1/2 particle) sits in a uniform magnetic field pointed in the x-direction:
B = B0xˆ.
a) What is the quantum Hamiltonian for this electron? Express your answer in terms of B0,
other constants, and the spin operators Sx, Sy and Sz, and then also write it as a matrix (in z basis).

b) What are the energy eigenvalues, and what are the associated normalized eigenvectors
(in terms of our usual basis)? You may express the eigenvectors either in Dirac notation or
as column vectors.

c) At t = 0, the electron is pointed down in the z-direction: |ψ(0)> = |−>. What is |ψ(t)>,
written in our usual basis? Again, you may use either Dirac notation or column vectors. In
words, what is the electron spin doing as time goes on?

d) At a later time t, what is the probability that a measurement of Sz will produce Sz = ̄h/2?

Answer 1