Example 5.2 considers a particular class of paths near a straight-line path. A different class of paths consists of a set of parabolas of the form \(y=n \alpha\left(1-x / x_{0}ight)^{2}\) fit to the endpoints of the straight line at \((x, y)=(0,0)\) and \((x, y)=\left(x_{0}, 0ight)\). Here \(\alpha\) is a (small) constant, and \(n=0, \pm 1, \pm 2, \ldots\) Let \(\alpha=0.1 x_{0}\), and draw a careful phasor diagram including enough integers \(n\) to see the Cornu spiral behavior and obtain a good estimate of the sum of all these phasors.

Data from Example 5.2

Transcribed Image Text:

Let so, be the shortest distance between the source and the detector, corresponding to a straight-line path. We will sum the probability amplitudes for a certain subset of all paths near this path. These particular alternate paths consist of a straight line with a kink in the middle, where the kink is a distance D = |n|Do (n= 1,2,...) from the straight line, as shown in Figure 5.18. If the paths have lengths,, then by the Pythagorean theorem (Sn/2)=(so/2)2 + (nDo). (5.48)