Problem 1:

A beam of electrons in an eigenstate of S_z with eigenvalue is fed into a Stern-Gerlach apparatus, which measures the component of spin along an axis at an angle q 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.

Problem 2:

A quantum system can exist in two states |y₁> and |y₂>, which are eigenstates of the Hamiltonian with eigenvalues E₁ and E₂.  An observable A has eigenvalues and eigenstates 

.  

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|y₁>+c₂|y₂>.  If p_n denotes the probability that the measurement at t=nT gives the result A=1, show that p_n+1 = ½ (1-cosa) + p_n cosa, where

,

and deduce that  p_n = ½ (1-cosⁿa) + ½ |c₁+c₂|² cosⁿa.

What happens in the limit as with nT=t fixed?

Problem 3:

Consider a particle in a harmonic oscillator potential with Hamiltonian .

Its state vector at t=0 is

,

where the |f_n> are the orthonormal eigenstates of H.

(a)  Show that |y(0)> is an eigenstate of the lowering operator a and find the eigenvalue.

(b)  If the energy of the particle is measured at t=0, what values of E can be found and with what probability?

(c)  .  Find <H> and the root mean square deviation DH at t=0.

(d) Find |y(t)> and show that it is still an eigenstate of a.  Find the eigenvalue.

(e) Find <x>(t).  Why is the state |y(0)> called a quasi classical state?

Problem 4:

Consider a spin 1/2 particle with magnetic moment m=gS.  Let |+> and |-> denote the eigenvectors of S_z and let the state of the system at t=0 be |y(0)>=|+>.

(a)  At t=0 we measure S_y and find +h/2.  What is the state vector |y(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)=w₀(t)S_z.  Assume w₀(t)=0 for t<0 and for t>T, and increases linearly from 0 to w₀ when 0<t<T.  Show that at time t the state vector can be written as

|y(t)>=2^-1/2[exp(iq(t))|+> + iexp(-iq(t))|->]

and calculate the real function q(t).