Problem 1:

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 {|f_n’>} basis.  Express the new basis vectors in terms of the old basis vectors.  Find the operator 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=E₁ÄE₂ of two non-identical spin 1/2 particles spanned by the basis vectors {|++>, |+->, |-+>, |-->}.  Use what you know about the common eigenvectors of S² and S_z, and find the common eigenvectors of S² 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 <y₁|y₂> 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=w₀S_z.  Find |y₁(t)>.  At what times t is |y₁(t)> an eigenvector of S_x?

Problem 4:

Consider a system composed of two spin 1/2 particles, S₁ and S₂, 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=w₁S_1z+w₂S_2z.  What is the state vector |y(t)> at time t?  Calculate at time t the mean values <S₁> and <S₂>.  Give a physical interpretation.

(d)  Show that the lengths of the vectors <S₁> and <S₂> are less than h/2.  What must be the form of |y(0)> for each of these lengths to be equal to h/2?