1549380323-Statistical Mechanics Theory and Molecular Simulation

Sum over paths 449

Ultimately, an infinite number of amplitudes must be summed in order toobtain
the overall probability, which we can express suggestively as

P(y) =

∣ ∣ ∣ ∣ ∣ ∣

∑

paths

Apath(y)

∣ ∣ ∣ ∣ ∣ ∣

2

. (12.1.3)

In such an expression, the number of interference terms is infinite. Nevertheless, if
the sum over paths is applied to the double-slit experiment, the correct observed
interference pattern, whose intensityI(y) is proportional toP(y), is obtained.
In their bookQuantum Mechanics and Path Integrals, Feynman and Hibbs (1965)
employ an interesting visual device to help understand the nature of the many paths.
Imagine modifying the double-slit experiment by introducing a large number of in-
termediate gratings, each containing many slits, as shown in Fig. 12.4. The electrons

S

D

.y

Fig. 12.4Passage of electrons through a large number of intermediategratings in the ap-
paratus of the double-slit experiment.

may now pass through any sequence of slits before reaching the detector; the number
of possible paths increases with both the number of intermediate gratings and the
number of slits in each grating. If we now take the limit in which infinitely many
gratings are placed between the source and the detector, each with an infinite number
of slits, there will be an infinite number of possible paths the electrons can follow.
However, when the number of slits in each intermediate grating becomes infinite, the
space between the slits goes to zero, and the gratings disappear,reverting to empty
space. The suggestion of this thought experiment is that empty space allows for an
infinity of possible paths, and since the electrons are not observeduntil they reach
the detector, we must sum over all of these possible paths. Indeed, this sum of path
amplitudes should exactly recover or build up the amplitude pattern ina wave-like
picture of the particles.