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## Homework Statement

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.