3.2. Particle motion in a general electromagnetic potential 157
and the resulting dispersion relation. This is completely equivalent to the postulate-based approach described earlier. Physically,
a close analogy exists with the wave optics of light propagation.
3.2 Particle motion in a general electromagnetic potential
In the preceding sections we have reviewed the conceptual basis of
quantum mechanics, as it relates to the motion of a single particle.
We are now in a position to include electric and magnetic effects.
For the present purpose, we confine our attention to fields which
vary slowly in space and time. This permits a more traditional
approach, based on Schr¨ odinger theory. To this end, we consider
all relevant information about the electric and magnetic effects
to be contained in the electrostatic scalar potential φ(x, t) and
the magnetic vector potential A(x, t), respectively, where these
potentials are functions of position x and time t. Together, these
potentials form the components of a Lorentz-covariant four-vector
(2.5), which we refer to as a general electromagnetic potential.
The central problem is to solve for the wave function ψ(x, t) in the
presence of a general electromagnetic potential, where the absolute
square of this function is the probability density for finding the
particle at position x and time t.
3.2.1 Path integral approach for the timedependent wave function
A great deal of physical insight can be gained from the path integral description of quantum mechanics, originally due to Feynman