Summing Over All Paths for Two Openings

You now know that when an electron passes through a single, very small opening, it may arrive at different points on the screen with equal amplitudes but varying phase values. In the single opening experiment there is nothing in the observations that reveals this variation of phase along the screen. However, with an electron traveling through two openings, the phase variation becomes important in understanding the observations of where the electron will strike the screen. To see this, we need a new interactive figure which you can get by clicking here.

In this situation of two openings we are actually dealing with two different processes that are linked together. As the electron goes from the source to a final point on the screen it can go through either the left or the right opening. If we block either one we expect the single opening result from before: a roughly flat distribution of electrons along the screen. If we have both open, we might expect that they would add to give again a roughly flat distribution. In fact, we get regions of zero probability interspersed with regions of high probability. The reason for this lies at the heart of quantum mechanics, the phases which we have spoken about so much before. When combining two processes the probability of the two processes put together is not a linear sum of the individual probabilities. The phase dictates how the two amplitudes add to give a total probability.

If the phases for passing through each slit are the same, then the two amplitude vectors line up and the total probability is at a maximum. If, however, the phases correspond to vectors in opposite directions, the vectors will cancel out in the sum and we get a region of zero probability. Even though there may be several paths to that particular point on the screen, the sum over all the paths perfectly cancels out to leave a zero probability for that point. This is how we know about phase and why it is important in quantum mechanics.

Additionally, we note that we can't say which slit the electron actually travels through. In fact, it is essentially as if the electron simultaneously travels through both. If we look at the electron to see which opening it travels though, the quantum mechanical interference breaks down, and we get a flat distribution corresponding to blocking one of the slits. In essence, if we force the electron to actually pick an opening, it can no longer interact with paths through the other opening.

These thoughts lead us to two of the major ways in which quantum mechanics violates our physical intuition. First, we can't actually know along what particular path a process actually occurs, the reason being that in fact all of the paths occur simultaneously.  Second, by looking at or observing something we actually interact with it and thus change it. That is, we can't separate ourselves and our measuring instruments from the phenonmena we are trying to study.   Thus, the existance of an independent physical reality should be regarded as a good working hypothesis, rather than a demonstrated "Truth". 

You can see that the phase interaction results in a varying total amplitude on the screen by focussing on the length of the black arrow in the simulation below. Notice that it increases and decreases as one sweeps across the screen from one side to the other. By looking at the red and green arrows you can see that the variation in total amplitude is due to the varying orientation (phase) of the red and green arrows. When they point in the same direction they tend to add constructively and when they point in opposite direction they tend to cancel each other.

You can also try to vary the distance between the slits to see what effect this has. As you make the slits farther apart, does the length of the total amplitude vector vary slower or faster as points on the screen are scanned? What do you think this corresponds to in terms of observations of where the electron strikes the screen?