# 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.  Note that these paths are not the paths for different electrons, but the possible paths for a single electron.
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 variable that repeatedly cycles through the same set of values.  Its final value at the screen depends on the length of the path. 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 an interactive figure by clicking here. 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).

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 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.

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.

To be sure you understand this, do some more playing around. 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 sum a lot of different paths.

Do nearby paths have vectors that cancel out? Or is it the paths symmetrically placed about the center that cancel? Check out the "Up Paths" and "Down Paths" butttons o help think about this.  A selection of "Up Path" vectors add to produce a total "up vector" shown in red.  A selection of "Down Path" vectors yield a total "down vector".   Notice that the up and down path vectors are interleaved.  Hence, it is vectors for nearby paths, rather than vectors from paths symmetrically placed about the opening, that cancel each other.

With all this, you should at this point have the feeling that summing over all paths yields a vector (amplitude and phase) whose value is 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 the screen when you let electrons fly to it from a source. Is it the amplitude or the phase that represents how likely we are to find an electron at any given point on the screen? It turns out that both are relevant, as we will begin to see in the next section.