## Problem 1:Let { . (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} basis.
Express
the new basis vectors in terms of the old basis vectors. Find the operator _{n’}>H’
that does to the new basis vectors what H does to the old basis vectors, i.e. that
has the new basis vectors as eigenvectors with eigenvalues .(e) Solve for the eigenvectors and eigenvalues of a
one-dimensional harmonic oscillator with charge | |

## Problem 2:Consider the state space E=E | |

## Problem 3:Two states of a spin 1/2 particle are represented in the eigenbasis ofS_{z}
by. (a) Find their representation in the eigenbasis ofS_{y}.
(b) Find the amplitude < (c) The Hamiltonian for the particle is y_{1}(t)>. At what times t
is |y_{1}(t)> an eigenvector of S_{x}? | |

## Problem 4:Consider a system composed of two spin 1/2 particles, {|++>, ||+->, |-+>, |-->}. At t=0 the system is in the state |y(0)>=(1/2)|++> +
(1/2)|+-> + (1/2) (a) At t=0 S (b) When the system is in the stat |y(0)>,
S (c) Instead of performing the preceding measurements, we let the
system evolve under the influence of the Hamiltonian H=w (d) Show that the lengths of the vectors < |