3.2. Particle motion in a general electromagnetic potential 159

Figure 3.2: Possible paths between two fixed points in one spatial
dimension.
Feynman and Hibbs, we will refer to this overall amplitude as a
kernel, and the summation over all possible paths as a path integral.
The absolute square of the kernel represents the probability density P ba for finding the particle at position x b at time t b , having
started at position x a at an earlier time t a . Equivalently,
P ba = | K(x b , t b ; x a , t a ) |
2 .
(3.134)
For a charged particle optics system of macroscopic dimensions,
the action integral S ba is very large compared to ¯
h. A small variation in path therefore results in a large variation in the action
integral divided by h ¯, or equivalently in the phase of ϕ ba . The
classical path of motion is labeled 1 in the figure, with the path
shown as bold. According to Hamilton’s principle of least action
(2.8), the action integral S ba is stationary with respect to firstorder variations about this classical path. Consequently, all paths