#### Problem 1:

Let {

*|f*} be an orthonormal eigenbasis of the Hamiltonian_{n}>.

(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 into the {_{n}>*|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

*q*in an external electric field .#### Problem 2:

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 {|++>, |+->, |-+>, |-->}.-
#### Problem 3:

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*. Find |_{0}S_{z}*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,

**S**_{1}and**S**_{2}, and the basis of vectors{|++>, ||+->, |-+>, |-->}.

At t=0 the system is in the state

|y(0)>=(1/2)|++> + (1/2)|+-> + (1/2)

^{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?