### A Plasma in a Magnetic Field

An important place where both magnetic terms come into play is in a plasma.There, many electrons are not bound to atoms and external Electric fields are screened out. Let's assume there is a constant (enough) B field in the z direction. We then have cylindrical symmetry and will work in the coordinates, , , and .

The problem clearly has translational symmetry along the z direction and rotational symmetry around the z axis. Given the symmetry, we know that and commute with the Hamiltonian and will give constants of the motion. We therefore will be able to separate variables in the usual way.

In solving the equation in we may reuse the Hydrogen solution ultimately get the energies

and associated LaGuerre polynomials (as in Hydrogen) in (instead of ).

The solution turns out to be simpler using the Hamiltonian written in terms of if we choose the right gauge by setting .

This Hamiltonian does not depend on or and therefore has translational symmetry in both x and y so their conjugate momenta are conserved. We can use this symmetry to write the solution and reduce to a 1D equation in .

Then we actually can use our harmonic oscillator solution instead of hydrogen! The energies come out to be

Neglecting the free particle behavior in , these are called the Landau Levels.This is an example of the equivalence of the two real problems we know how to solve.

Jim Branson 2013-04-22