Problem 1:

Let A and B be two observables of a system with a two dimensional state space, and suppose measurements are made of A, B, and A again in quick succession.  Show that the probability that the second measurement of A gives the same result as the first is independent of the initial state of the system.

Problem 2:

Find <x> and Dx for the nth stationary state of a free particle in one dimension restricted to the interval 0<x<a.  Show that as n®¥ these become the classical values.

Problem 3:

A system makes transitions between eigenstates of H₀ under the action of the time dependent Hamiltonian H₀+eV₀cosw t. Find an expression for the probability of transition from |y₁> to |y₂>, where |y₁> and |y₂> are eigenstates of H₀ with eigenvalues E₁ and E₂.  Show that this probability is small unless .
[This shows that a charged particle in an oscillating electric field with angular frequency w will exchange energy with the field only in multiples of .]

Problem 4:

Consider a one-dimensional system, with momentum operator p and position operator q.

(a) Show that

(b) Show that if the function F(p) can be defined by a finite polynomial or convergent power series in the operator p.

(c) Show that if F(q) is some function of q only.

Problem 5:

If baryon number were not conserved in nature, the wave functions for neutrons (n) and anti-neutrons (n) would not be stationary mass-energy eigenfunctions, rather the wave functions of such particles would be oscillating, time-dependent superposition of neutron and anti-neutron components, given by

 .

Suppose that such oscillations occur and the Hamiltonian of the system, neglecting all degrees of freedom except for the one corresponding to the oscillations, is

,

where E_n=m+V_n  and E_n=m+V_n are energies for n and n individually, and a is a real mixing amplitude.  In an external magnetic field B, the potential energies are V_n=m ×B and V_n=-m ×B where m=m_n=-m_n.  Suppose that at time t=0 the initial state is that of a neutron.

(a) Calculate the time-dependent probability for observing an anti-neutron. Determine the period of oscillation.