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?
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 ml 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
Sz(Sz+h)(Sz-h) and (Sx(Sx+h)(Sx-h).
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
(b) If Lz is measured, what results can be obtained and with what probabilities?
(c) Is y(r) an eigenfunction of L2 or Lz?