13.3 Sample Test Problems
- A particle of massmin 3 dimensions is in a potentialV(x,y,z) =^12 k(x^2 + 2y^2 + 3z^2 ). Find
the energy eigenstates in terms of3 quantum numbers. What is the energy of the ground
state and first excited state?
2.*N identical fermions are bound (at low temperature) in a potentialV(r) =^12 mω^2 r^2. Use
separation in Cartesian coordinates to find the energy eigenvaluesin 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 numberof states with energy less
thanEF, you may assume that it isα(EF/ ̄hω)^3.
- A particle of mass m is in the potentialV(r) =^12 mω^2 (x^2 +y^2 ). Find operators that commute
with the Hamiltonian and use them to simplify the Schr ̈odinger equation. Solve this problem
in the simplest way possible to find the eigen-energies in terms of a setof ”quantum numbers”
that describe the system.
- A particle is in a cubic box. That is, the potential is zero inside a cubeof side L and infinite
outside the cube. Find the 3 lowest allowed energies. Find the numberof states (level of
degeneracy) at each of these 3 energies.
- A particle of mass m is bound in the 3 dimensional potentialV(r) =kr^2.
a) Find the energy levels for this particle.
b) Determine the number of degenerate states for the first three energy levels.
- A particle of massmis in a cubic box. That is, the potential is zero inside a cube of sideL
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 EnergyEFforNparticles in the box. (N is large.)
- A particle is confined in a rectangular box of lengthL, widthW, and “tallness”T. 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
L= 2a,W= 1a, andT= 0. 5 a.
- A particle of mass m is bound in the 3 dimensional potentialV(r) =kr^2.
- 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 massmis bound in a potentialV(r) = −e
2
√
x^2 +z^2.
a) The definite energy states will, of course, be eigenfunctions ofH. What other operators
can they be eigenfunctions of?
b) Simplify the three dimensional Schr ̈odinger equation by using these operators.
