- An electron is bound in a harmonic oscillator potential
. Small electric fields in the
direction are applied to the system.
Find the lowest order nonzero shifts in the energies of the ground
state and the first excited state if a constant field is applied.
Find the same shifts if a field is applied.
- A particle is in a box from to in one dimension.
A small additional potential
is applied. Calculate the energies of the first and second
excited states in this new potential.
- The proton in the hydrogen nucleus is not really a point particle
like the electron is. It has a complicated structure, but, a good
approximation to its charge distribution is a uniform charge density
over a sphere of radius 0.5 fermis. Calculate the effect of this
potential change for the energy of the ground state of hydrogen.
Calculate the effect for the states.
- Consider a two dimensional harmonic oscillator problem described by the Hamiltonian
Calculate the energy shifts of the ground state and the
degenerate first excited states, to first order,
if the additional potential is applied.
Now solve the problem exactly.
Compare the exact result for the ground state to that
from second order perturbation theory.
- Prove that
by starting from the expectation value of the commutator
in the state and summing over all energy eigenstates.
and write in
terms of the commutator to get the result.
- If the general form of the spin-orbit coupling for
a particle of mass and spin moving in a
what is the effect of that coupling on the spectrum of a three
dimensional harmonic oscillator?
Compute the relativistic correction for the ground state
of the three dimensional harmonic oscillator.