A beam of electrons in an eigenstate of S_z with eigenvalue ½ħ is fed into a Stern-Gerlach apparatus, which measures the component of spin along an axis at an angle θ to the z-axis and separates the particles into distinct beams according to the value of this component.  Find the ratio of the intensities of the emerging beams.

A quantum system can exist in two states |ψ₁> and |ψ₂>, which are eigenstates of the Hamiltonian with eigenvalues E₁ and E₂. 
An observable A has eigenvalues ±1 and eigenstates |ψ_±>= (1/√2)(|ψ₁> ± |ψ₂>).
This observable is measured at times t = 0, T, 2T, ... .  The normalized state of the system at t = 0, just before the first measurement, is c₁|ψ₁>+  c₂|ψ₂>.  If p_n denotes the probability that the measurement at t = nT gives the result A = 1, show that p_n+1 = ½(1 - cosα) + p_n cosα, where
α/T = (E₁ - E₂)/ħ
and deduce that  p_n = ½ (1 - cosⁿα) + ½|c₁ + c₂|² cosⁿα.
What happens in the limit as n --> ∞ with nT = t fixed?

Consider a spin ½ particle with magnetic moment m = γS.  Let |+> and |-> denote the eigenvectors of S_z and let the state of the system at t = 0 be |ψ(0)> = |+>.
(a)  At t = 0 we measure S_y and find +½ħ.  What is the state vector |ψ(0)> immediately after the measurement?
(b)  Immediately after this measurement we apply a uniform, time-dependent field parallel to the z-axis.
The Hamiltonian operator becomes H(t) = ω₀(t)S_z.
Assume ω₀(t) = 0 for t < 0 and for t > T, and increases linearly from 0 to ω₀ when 0 < t < T.  Show that at time t the state vector can be written as
|ψ(t)> = 2^-½[exp(iθ(t))|+> + iexp(-iθ(t))|->]
and calculate the real function θ(t).
(c)  At time t = τ > T, we measure S_y.  What results can we find and with what probability? 
Determine the relation that must exist between ω₀ and T in order for us to be sure of the result.  Give a physical interpretation.