Homework Problems

- Show that the Hamiltonian
yields the Lorentz force law for an electron. Note that the fields must
be evaluated at the position of the electron. This means that the total
time derivative of must also account for the motion of the electron.
- Calculate the wavelengths of the three Zeeman lines in the
transition in Hydrogen atoms in a gauss field.
- Show that the probability flux for system described by the Hamiltonian

is given by

Remember the flux satisfies the equations . - Consider the problem of a charged particle in an external magnetic
field
with the gauge chosen so that
.
What are the constants of the motion?
Go as far as you can in solving the equations of motion
and obtain the energy spectrum.
Can you explain why the same problem in the gauges
and
can represent the same physical situation?
Why do the solutions look so different?
- Calculate the top left corner of the matrix representation of
for the harmonic oscillator. Use the energy eigenstates as the
basis states.
- The Hamiltonian for an electron in a electromagnetic field can be written as
.
Show that this can be written as the Pauli Hamiltonian

Jim Branson 2013-04-22