Summing Over All Paths for Multiple Openings

You now know that phase is important when there are two openings because the phases at a screen point determine how amplitudes associated with paths through each opening add to give a total probability. You can see this again in another interactive figure by clicking at two different locations in the region between the barrier and the screen so as to create two openings. The resulting black curve shows the probability for observing an electron at various locations on the screen. Note the minima and maxima due to the constructive and destructive addition of the amplitudes to give the total probability. You can also see here the effects of varying the distance between the openings.

We have made a simplifying assumption in this simulation that we have not made before. To find an amplitude for a given opening and final point on the screen, we previously summed over an explicit set of 100 paths. Here we have used the mathematical result from when one does a full sum over all possible paths. Hence, we have not drawn any specific paths as we did earlier. By doing this, we obtain accurate results and significantly reduce the computation time.

This also makes it possible to ask the question of what happens if the amplitude and phase of the electron at each opening are varied. By clicking in the circle coresponding to a particular opening you can set the amplitude (length) and phase (orientation) at each opening. Notice that varying the phase at one opening shifts the probability pattern on the screen right and left, while changing the amplitude at one opening alters the minimum probability you can obtain. See how this is also displayed in the vector addition of the arrows on the right.

By clicking multiple times in the region between the barrier and the screen you can create up to five openings spaced any way you want. For each, you can vary the phase and amplitude of the electron. As you'll see, quite complicated and intriguing patterns can result from the phase interactions.