Problem 1:

Two particles of mass m₁ and m₂ are separated by a fixed distance r.  Their center of mass is fixed at the origin of the coordinate system and they are free to rotate about their center of mass.  (The system is a "rigid rotator".)
(a)  Write down the Hamiltonian of the system.
(b)  Find the eigenvalues and eigenfunctions of this Hamiltonian.  What is the separation between adjacent levels?  What is the degeneracy of the eigenvalues?

Problem 2:

The wave function of a particle subjected to a spherically symmetric potential U(r) is given by ψ(r) = (x + y + 3z)f(r).
(a)  Is ψ an eigenfunction of L²?  If so, what is its l value?  If not, what are the possible values of l we may obtain when L² is measured?
(b)  What are the probabilities for the particle to be found in various m_l states?  
(c)  Suppose it is known somehow that ψ(r) is an energy eigenfunction with eigenvalue E.  Indicate how we may find U(r).

Problem 3:

Consider a spin 1 particle.  Evaluate the matrix elements of

S_z(S_z+h)(S_z-h) and (S_x(S_x+h)(S_x-h).

Problem 4:

The wave function y(r) of a spinless particle is y(r)=Nz²exp(-r²/b²), where b is a real constant and N is a normalization constant.

(a)  If L² is measured, what results can be obtained and with what probabilities?
(b)  If L_z is measured, what results can be obtained and with what probabilities?
(c)  Is y(r) an eigenfunction of L² or L_z?