Summing Over All Paths for One Opening

You now know how summing over paths yields the amplitude and phase for a specific final point on the screen. Here you will be able to see how the probability and phase varies for the electron traveling through a single slit and ending at all possible points on the screen. To do this, you need a new interactive figure which you can get by clicking here.

Click at a particular location on the final screen (rather than on the yellow midpoint line as you did in the last exhibit). When you do so, the computer will automatically generate 100 regularly spaced paths to that point. These paths are similar to those you generated by clicking on the "100 Paths" button previously. Like there, a vector sum over these paths (shown in grey and black) appears as a red arrow on the right. The black paths and arrows again represent those which contrubute the most to the final red arrow. Note again that the paths drawn do not reperesent the paths of single electrons traveling through the slit but rather they give the possible paths for a single electron arriving at the screen at that point. Each electron explores all paths when it goes from the slit to the screen.

The red circle represents the average length for the total vector (the amplitude) corresponding to each final point on the screen. By clicking on other screen points you can see the total amplitude and phase for any final point on the screen. The length of the vector is closely related to the likelihood that a electron will be found at that point on the screen; the angle of the vector is the "phase". Notice that the amplitudes change rather little (in fact, the small variation in amplitude is only an artifact of the simulation, and can be ignored entirely) while the phases change dramatically.

You can more easily see how the vector changes by clicking either the "left" or "right" buttons. Notice that just behind the opening the rate of change of the vector direction becomes small. For adjacent points in that area of the screen the summed phase for final points does not change very much, whereas it changes more rapidly for peripheral areas. While this difference does not show up as any observable variation on the screen for this case of a single opening (remember, previously we saw that the likelihood of the electron striking the screen was the same at all points on the screen), it will be very important for cases with multiple openings.

Notice that the region of slow phase change also corresponds to the location where you would expect the electron to hit the screen were it behaving "classically". In fact, as a particle increases in mass, there is a change in the vector amplitude with screen position much like the change that occurs with increasing the opening size as we saw earlier.  Thus by varying either particle mass or opening size we can move smoothly from the classical to the quantum world.

Although at this stage the phase of the processes is irrelevant, note how the direction of the final vector changes as the endpoint for the electron on the screen varies. Unlike the length, the phase is not a constant, and this will be important in the next lesson where we examine a double slit experiment.