Quantum Mechanics for Engineers 

© Leon van Dommelen 

D.73 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,