### D.74 Orbital motion in a magnetic field

This note derives the energy of a charged particle in an external magnetic field. The field is assumed constant.

According to chapter 13.1, the Hamiltonian is

where and are the mass and charge of the particle and the vector potential is related to the magnetic field by . The potential energy is of no particular interest in this note. The first term is, and it can be multiplied out as:

The middle two terms in the right hand side are the changes in the Hamiltonian due to the magnetic field; they will be denoted as:

Now to simplify the analysis, align the -​axis with so that . Then an appropriate vector potential is

The vector potential is not unique, but a check shows that indeed for the one above. Also, the canonical momentum is

Therefore, in the term above,

the latter equality being true because of the definition of angular momentum as . Because the -​axis was aligned with , , so, finally,

Similarly, in the part of the Hamiltonian, substitution of the expression for produces

or writing it so that it is independent of how the -​axis is aligned,