Sum over paths 447individual amplitudes. This picture of quantum mechanics is known as thesum over
pathsorpath integralformulation. How we calculate the amplitude for a path will be
made explicit in the next section.
Before delving into the mathematics of path integrals, let us apply the Feynman
picture heuristically to a concrete example. Consider an experimentin which electrons
from a sourceSimpinge on a double-slit system and are registered on a detection screen
placed atD(see Fig. 12.1). Invoking a wave-like picture, the interference of electron
SDyInterference
patternWa ve frontsFig. 12.1 Interference pattern observed in the electron double-slitexperiment.waves emanating from the two slits leads to the well-known interference pattern at the
detector, as shown in the figure. This pattern is actually observedwhen the experiment
is performed (Merliet al., 1976). Next, consider Feynman’s sum over paths formulation.
If the electrons followed definite paths, as they would if they were classical particles,
each path would have a separate probability and no interference pattern would be
seen between the paths. We would, therefore, expect two brightspots on the screen
directly opposite the slits as shown in Fig. 12.2. However, the quantum sum over
paths requires that we consider all possible paths from the sourceSthrough the
double-slit apparatus and finally to the detectorD. Several of these paths are shown
in Fig. 12.3. Let each path have a corresponding amplitudeAi(y). Specifically,Ai(y) is
the amplitude that an electron following pathiis detected at a pointyon the screen
at timet. The total amplitudeA(y) for observing an electron atyis, therefore, the
sumA(y) =A 1 (y) +A 2 (y) +A 3 (y) +···, and the corresponding probabilityP(y) is
given byP(y) =|A(y)|^2 =|A 1 (y) +A 2 (y) +A 3 (y) +···|^2. Suppose there were only
two such paths. ThenP(y) =|A 1 (y) +A 2 (y)|^2. Since each amplitude is complex, we
can write
A 1 (y) =|A 1 (y)|eiφ^1 (y), A 2 (y) =|A 2 (y)|eiφ^2 (y), (12.1.1)
and it can be easily shown that