Let {*|f _{n}>*} be an
orthonormal eigenbasis of the Hamiltonian

.

(a) Find , the wave function in momentum space, by first finding the differential equation which satisfies and then finding the solutions of this differential equation.

(b) Compare and .

(c) Consider the operator . Show that is a unitary operator.

(d) Let
be the
unitary transformation that transforms the {*|f _{n}>*}
basis into the {

(e) Solve for the eigenvectors and eigenvalues of a
one-dimensional harmonic oscillator with charge *q* in an external electric field
.

Consider the state space E=E_{1}ÄE_{2}
of two non-identical spin 1/2 particles spanned by the basis vectors {|++>, |+->,
|-+>, |-->}. Use what you know about the common eigenvectors of
*S*^{2}
and *S*_{z}, and find the common eigenvectors of *S*^{2} and
*S*_{x}.
Express these eigenvectors in terms of the basis vectors {|++>, |+->, |-+>,
|-->}.

Two states of a spin 1/2 particle are represented in the eigenbasis of *S*_{z}
by

.

(a) Find their representation in the eigenbasis of *S*_{y}.

(b) Find the amplitude <*y*_{1}|*y*_{2}> in the
*S*_{z} basis and show
that this amplitude remains unchanged when calculated in the *S*_{y
}basis.
(Show your work.)

(c) The Hamiltonian for the particle is *H=w _{0}S_{z}*.
Find |

Consider a system composed of two spin 1/2 particles, **S**_{1}
and **S**_{2}, and the basis of vectors

{|++>, ||+->, |-+>, |-->}.

At t=0 the system is in the state

|y(0)>=½|++> +
½|+-> + ½^{1/2}|-->.

(a) At t=0 S_{1z} is measured. What is the
probability of finding -h/2? What is the
state vector after this measurement? If we then measure S_{1x},
what results can be found and with what probability? Answer the same
questions for the case where S_{1z} yielded +h/2.

(b) When the system is in the stat |y(0)>,
S_{1z} and S_{2z} are measured simultaneously. What is
the probability of finding opposite results? Identical results?

(c) Instead of performing the preceding measurements, we let the
system evolve under the influence of the Hamiltonian H=w_{1}S_{1z}+w_{2}S_{2z}.
What is the state vector |y(t)> at time
t? Calculate at time t the mean values <**S**_{1}>
and <**S**_{2}>. Give a physical interpretation.

(d) Show that the lengths of the vectors <**S**_{1}>
and <**S**_{2}> are less than h/2.
What must be the form of |y(0)> for each of
these lengths to be equal to h/2?