Starting with a problem

Let's start with a particle, an opening in a barrier, and a screen on the other side. If we send a particle though the opening, we would expect it to strike the screen directly behind the opening. If the opening is much bigger than the particle, things work out pretty much as we would expect. On the other hand, if the opening is very small, things get odd. Instead of always finding the particle just behind the opening, I may find it off to either side. Is there a way to keep our intuition about particles and add to it some new concepts that allow us to make sense of these observations? The multiple paths interpretation indeed makes this possible. What one needs to add to our intuition of particles are the concepts of uncertainty, phase and amplitudes. The uncertainty arises from not knowing what path the particle actually takes is some sense; phase and amplitude allow us to get past this uncertainty by combining the information from all the possible paths to make a prediction of where we will find the particle. For relatively large openings, the prediction is the familiar one of the particle being pretty much directly behind the openning. What is surprising is that for small openings, the prediction is that the particle may strike the screen almost anywhere.

You can see this for yourself using the interactive figure below. Imagine a stream of electrons moving from a source (bottom) through a screen with an opening of variable width (middle) to a screen (top). The curve shows the number of electrons hitting each point along the screen. You can vary the width of the opening by clicking on the "thinner" or "thicker" buttons. Notice that with a very thick (wide) opening, most of the electrons hit the screen in a localized area directly opposite the opening. With a thin opening, more or less the same number of electrons hit the screen at all locations along the screen.

The difference in behavior for thick and thin openings is an experimental observation, which has to be accounted for. You probably noticed some more buttons and an arrow off to the right. These help to understand how the experimental observations can be made sense of using the "explore all paths" idea. Don't worry about them right now. We'll return to this figure at the end of the exhibit, and you'll know then what the buttons and arrows represent.

The Road Map from here:

  1. Paths Have Amplitude and Phase. Explore how the idea of multiple paths can be used to predict the likelyhood that the electron has arrived at a particular point on the screen. To do this you will need to understand that each path has associated with it an "amplitude" and a "phase" and appreciate how these properties of the paths interact.

  2. Summing Over All Paths for One Opening. Once you know how to sum over all possible paths for a particular point on the screen, you will next need to explore how that sum varies for different points accross the screen. While the total amplitude does not change, there is a prominant variation in the summed phases which does not show up in observations of the electron striking the screen, but has observable effects in situations where there is more than one opening for the electron to pass through.

  3. Summing Over All Paths for Two Openings. Investigate what happens when there are two openings present for the electron to travel through. At this point the summed phases resulting from each opening interact destructively and constructively to alter the likelihood of finding an electron at particular points accross the screen.

  4. Summing Over All Paths for Multiple Openings. Explore multiple openings varying both the starting phase and amplitude for each set of possible paths from each opening.

  5. Thinking Again About Large and Small Openings. Return to varying the size of a single opening to appreciate the role that total phase plays in localizing the area over which electrons strike the screen.