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 amplitude. The uncertainty arises from not knowing what path
the particle actually takes in 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 opening. 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.
Instructions Click the "Thinner Slit" and "Thicker Slit" buttons to vary the size
of the opening through which the electrons travel. Click the "Left" or
"Right" buttons to watch the black arrow on the right change as a function
of the position along the screen. Press "Stop" to stop the movement.
The difference in behavior for thick and thin openings is an experimental
observation, which has to be accounted for. The "Left" and "Right" buttons
and the black arrow on the right will help explain 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 these buttons and arrows represent.
The Road Map from here:
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.
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 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 will not show up in any
observations of the electron striking the screen, but will have observable
effects in situations where there is more than one opening for the electron
to pass through.
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.