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 |ψ_{1}>
and |ψ_{2}>, which are eigenstates of the
Hamiltonian with eigenvalues E_{1} and E_{2}.

An observable A
has eigenvalues ±1 and eigenstates |ψ_{±}>= (1/√2)(|ψ_{1}>
± |ψ_{2}>).

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 c_{1}|ψ_{1}>+ c_{2}|ψ_{2}>.
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 = (E_{1} - E_{2})/ħ

and deduce that p_{n} = ½ (1 - cos^{n}α)
+ ½|c_{1 }+ c_{2}|^{2} cos^{n}α.

What happens in the limit as n --> ∞ with nT = t
fixed?

Consider a spin ½ particle with magnetic moment **m **= γ**S**. 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) = ω_{0}(t)S_{z}.

Assume ω_{0}(t) = 0 for t < 0 and for t > T, and increases linearly from
0 to ω_{0} 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 ω_{0} and T in order for
us to be sure of the result. Give a physical interpretation.