The Hamiltonian for a spin-half particle is
H = 2a/ħ (Sx + Sy)
where a is a positive constant and Sx , Sy are the x and y components of the spin. Initially (at time t=0) the particle is in the state
|ψ> = (1/√2) (|↑>+|↓>)
where up and down arrows denote eigenstates of the z component of the spin.
a) Calculate the expectation value of the energy <E> and of the three components of the spin <Sx>, <Sy> and <Sz> at t=0.
b) What is the probability that a measurement of the x component of the spin at t=0 will give the value +ħ/2 ?
c) Find the eigenstates and eigenvalues of the Hamiltonian.
d) Find the state |ψ(t)> at a later time t.
e) What is the probability that a measurement of the x component of the spin at time t will give the value +ħ/2 ?
This is a practice exam question that I'm using to prepare for my final. Any help would be greatly appreciated. It's tough being the only Mechanical Engineer in a group full of physics majors. Thanks to anyone who can help me out.