Multiple Paths, Phase and Amplitude

Our first task is to get straight on multiple paths, phase and amplitude. To do this let's simplify the situation by considering first a single possible arrival point on the screen, rather than all possible arrival locations. Then we assert that:

1. There are lots of paths to get from the source location to any particular location on the screen.
2. Each path has associated with it two quantities, an amplitude and a phase. They can be represented together as a vector where the vector length is the amplitude and the vector orientation is the relative phase of the particular path.
3. In our case, where the likelihood of all paths is the same, all individual paths have the same amplitude.
4. Phase is a cyclic variable whose value changes rapidly for longer paths and more slowly for shorter ones. Therefore, the phase, unlike the amplitude, differs for different paths.
5. The total likelihood of going from the source to a point on the screen is obtained from the total amplitude and phase which is calculated by adding together the individual vectors representing the amplitude and phase for each of the paths.
You can see all of this work in the interactive figure below. The yellow horizontal line is an imaginary line, half-way between the barrier with the opening (bottom) and the screen (top). The paths are defined as starting at the opening, going to some point on the yellow line and from there to the screen. Click anywhere on the yellow line to select one possible path. When you do, a vector appears to the right, showing the amplitude and phase for that path (if the vector is very small, zoom in on it by clicking the appropriate button). Now click on the yellow line to designate a second path to the same arrival point. You will now see two grey arrows, one corresponding to the vector for one path and the second to the vector for the second, and a red arrow, corresponding to their sum. By continuing to click at different locations on the yellow line, you can build up your sum over possible paths (zoom in or out to resize the vectors to fit the display).

You can more quickly add vectors by clicking any of the red buttons. Each time you click these, or the yellow line, vectors will be added to those already present. Click "clear paths" to start a new sum. Play around and get a feeling for what happens when you sun lots of different paths.

Notice that clicking at various locations on the yellow line yields vectors of the same length (amplitude) but differing directions (phase). Notice also that closely space paths going through the peripheral regions of the yellow line have quite different directions and so tend to cancel each other in the sum while closely spaced paths toward the center have similar directions and so tend to reinforce one another in the sum. As a result, the sum over a large number of evenly spaced paths is dominated by the central vectors (those show in black) and the more peripheral vectors (those shown in grey) have less effect on the sum.

To be sure you understand this, do some more playing around. Do nearby paths have vectors that cancel out? Or is it paths symmetrically placed about the center that cancel? Check out the "up paths" and "down paths" to help think about this.

With all this, you should at this point have the feeling that summing across all paths yields a vector (amplitude and phase) whose values are largely determined by a subset of the paths. This is, in fact, rigorously provable, not only for the kinds of paths illustrated here but for ALL conceivable paths, including ones that zig-zag any number of times between the source and the screen and even ones that reverse directions.

You should also be wondering what any of this has to do with what actually appears on a screen when you let electrons fly to it from a source. Is it the amplitude or the phase or what that represents how bright the image is on the screen at any given point? It turns out that both are relevant, as we will begin to see in the next section.

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First off, note that the paths you see are not paths for different electrons, but rather just one single electron is utilizing every single path you select.

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On the other hand, however, there are only a few paths of primary importance. The shortest distance between two points is a straight line, and classically a particle without an outside force acting on it will travel only in a straight line. The paths that are straight lines or very close to straight lines all have about the same length and thus all correspond to about the same direction. In fact, as you can see from the simulation, the paths that contribute the most to the final vector are those highlighted in black, which are the paths that come closest to the classical path in a straight line from the slit to the screen. Additionally, for more massive particles the width of the set of black paths will be thinner, and for objects as massive as say a penny, any primary contributing path is so close to the classical path that we can't tell the difference. Quantum mechanics then explains both behavior at the microscopic scale and events in our everyday lives as the limiting case for large, massive objects.

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