Problem 1:

Let r, f, z be the cylindrical coordinates of a spinless particle.  Assume the potential energy of this particle depends only on r and not on f and z.

(a)  Write in cylindrical coordinates the differential operator associated with the Hamiltonian.  Show that H commutes with L_z and P_z.  Show from this that the wave function associated with the stationary states of the particle can be written as

y_nmk(r, f, z) = f_nm(r)exp(imf)exp(ikz),

where the values that can be taken on by the indices m and k are to be specified.

(b)  Write in cylindrical coordinates the eigenvalue equation of the Hamiltonian of the particle.  Derive from it the differential equation which yields f_nm(r).

(c)  Let O_y be the operator whose action, in the {|r>} representation is to change y to -y (reflection with respect to the x-z plane).  Does O_y commute with H?  Show that O_y anti-commutes with L_z and show that Oy|y_nmk>  is an eigenvector of L_z.  What is the corresponding eigenvalue?  What can be concluded concerning the degeneracy of the energy levels of the particle?  Could this result be predicted directly from the differential equation established in (b)?

Problem 2:

Consider a particle of mass m trapped in a spherical box.

V(r)=0 for r<a,  V(r)=¥ for r>a.

(a) Wrire down the radial equation for R(r).
(b) Define k²=8p²mE/h² and change from the variable r to the variable r=kr.
(c) Write down solutions to the radial equation.  You do not have to find the normalization constant N.
(d) Find the lowest eigenvalues for l=0, l=1, and l=2.