Problems

Problem:

Find the condition that must be satisfied by the spherically symmetric square well potential
V(r) = -|V0|   for r < a,  V(r) = 0   for r > a,
if it is just barely deep enough to contain one bound state.

Solution:


Problem:

A particle of mass m is bound in a 2-dimensional isotropic oscillator potential with a spring constant k.  Use the fact that

 ,

and write the Schroedinger equation for this system in both Cartesian and polar coordinates.  Separate the equation in polar coordinates and solve the resulting equation in θ.  Write the resulting radial equation utilizing the θ solution, but do not solve it.  Demonstrate the connection between the θ solution and its classical analog.

the θ solution and its classical analog.

Solution:

 

Problem:

A particle of charge -e and mass m is under the influence of two stationary heavy nuclei, each with charge Ze positioned at  z = a.  We shall further assume that the particle is spinless and non relativistic. 
(a)  What is the Hamiltonian and the Schroedinger equation of the system.
(b)  Define the angular momentum operator along the z - direction, Lz, and show that its eigenvalues are good quantum numbers for all of the non-degenerate energy eigenstates.  What are the possible eigenvalues of Lz?
(c)  Define the parity operator P and show that parity is a good quantum number for all the non-degenerate energy eigenstates.  What are the possible eigenvalues of P?
(d)  Define the total angular momentum operator L and show that the eigenvalues of L2 are not good quantum numbers for the energy eigenstates.

Solution:


Problem:

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?

Solution: