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 ∆x 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 H_{0}
under the action of the time dependent Hamiltonian H_{0 }+ Wcosωt, W << H_{0}.

Find an expression for the probability of transition from
||ψ_{1}> to ||ψ_{2}>,
where ||ψ_{1}> and ||ψ_{2}>
are eigenstates of H_{0} with eigenvalues E_{1} and E_{2}.

Show that this probability is small unless E_{2} - E_{1} ≈ ħω.

[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
E ≈ ħω.]

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

(a) Show that
[q,p^{n}] = iħ n p^{n-1}.

(b) Show that
[q,F(p)] = iħ ∂F/∂p, if
the function F(p) can be defined by a finite polynomial or convergent power series
in the operator p.

(c) Show that
[q,p^{2}F(q)] = 2iħ pF(q) 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_{1} = m + U_{1} and E_{2} = m +
U_{2} are energies for n and
n
individually, and α
is a real mixing amplitude.

In
an external magnetic field **B**, the potential energies are U_{1}
= **μ**∙**B**
and U_{2}
= -**μ**∙**B**
where **μ**
≈ **μ**_{n }= -**μ**__ _{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.