- A particle of mass in 3 dimensions is in a potential
.
Find the energy eigenstates in terms of
**3 quantum numbers**. What is the energy of the ground state and first excited state? *****N identical fermions are bound (at low temperature) in a potential . Use separation in Cartesian coordinates to find the energy eigenvalues in terms of a set of three quantum numbers (which correspond to 3 mutually commuting operators). Find the Fermi energy of the system. If you are having trouble finding the number of states with energy less than , you may assume that it is .- A particle of mass m is in the potential . Find operators that commute with the Hamiltonian and use them to simplify the Schrödinger equation. Solve this problem in the simplest way possible to find the eigen-energies in terms of a set of "quantum numbers" that describe the system.
- A particle is in a cubic box. That is, the potential is zero inside a cube of side L and infinite outside the cube. Find the 3 lowest allowed energies. Find the number of states (level of degeneracy) at each of these 3 energies.
- A particle of mass m is bound in the 3 dimensional potential
.
- a)
- Find the energy levels for this particle.
- b)
- Determine the number of degenerate states for the first three energy levels.

- A particle of mass is in a cubic box. That is, the potential is
zero inside a cube of side and infinite outside.
- a)
- Find the three lowest allowed energies.
- b)
- Find the number of states (level of degeneracy) at each of these three energies.
- c)
- Find the Fermi Energy for particles in the box. (N is large.)

- A particle is confined in a rectangular box of length , width , and ``tallness'' . Find the energy eigenvalues in terms of a set of three quantum numbers (which correspond to 3 mutually commuting operators). What are the energies of the three lowest energy states if , , and .
- A particle of mass m is bound in the 3 dimensional potential .
- a) Find the energy levels for this particle.
- b) Determine the number of degenerate states for the first three energy levels.
- In 3 dimensions, a particle of mass is bound in a potential
.
- a)
- The definite energy states will, of course, be eigenfunctions of . What other operators can they be eigenfunctions of?
- b)
- Simplify the three dimensional Schrdinger equation by using these operators.

Jim Branson 2013-04-22