  The two-dimensional hydrogen-like atom in a constant magnetic field is
considered. It is found that this is actually two separate problems. One for
which the magnetic field causes an effective attraction between the nucleus and
the electron and one for which it causes an effective repulsion. Each of the
two problems has three separate cases depending on the sign of a shifted energy
eigenvalue. For two of the six possibilities (shifted energy eigenvalue that is
negative) it is shown that the first four solutions can be obtained exactly.
For another two of the six possibilities (shifted energy eigenvalue that is
positive) it is shown that the first eight solutions can be obtained exactly.
For higher order states the energy eigenvalue is the root of a fifth or higher
order polynomial, hence, the eigenvalue must be obtained numerically. Once the
energy eigenvalue is known the solution to the radial wave equation is also
known. Exact solutions for the radial wave equation, for the remaining two
possibilities (shifted energy eigenvalue that is zero), are given to any
desired order by means of a recursion relation.
