Two particles of mass m_{1} and m_{2} 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?

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 L^{2}? If so, what is its l
value? If not, what are the possible values of l we may obtain when L^{2}
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).

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).

The wave function *y*(**r**) of a
spinless particle is *y*(**r**)=*Nz*^{2}exp(-*r*^{2}/*b*^{2}),
where *b* is a real constant and *N* is a normalization constant.

(a) If *L*^{2} 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
*L*^{2}
or *L*_{z}?