A Single-Slit Experiment

Imagine that we have a particle of mass m traveling along our x-axis in the positive x-direction with linear momentum p=mv. Imagine further that we have put a thin plate as a barrier on our y-z plane and that we have cut into the plate a long, straight slit of width Δy=w. We want to find out what happens if the particle goes through the slit.

We suspect that the particle will not continue its motion in a straight line, because we know Heisenburg's indeterminacy principle. When we shot the particle at the slit we did so in such a way that the particle gained no momentum in the y-direction. Nonetheless, in accordance with our quantum theorem, we must have inflicted upon that particle at least one unit of action associated with the y-direction; that is, we must have

Δp_{y}Δy=h.

(Eq'n 1)

As long as the particle has a fetch in the y-direction that approaches infinity, we can assume that the indeterminacy in its y-ward linear momentum approaches zero. But when a barrier diminishes the fetch, the indeterminacy in the particle's linear momentum in that direction must grow in proportion. When the particle goes through the slit, then, we must have as true to Reality

Δp_{y}=h/w.

(Eq'n 2)

We thus have a finite, if small, indeterminacy in the particle's y-ward linear momentum.

Please note my use of the word indeterminacy
rather than uncertainty: Δp_{y} does not represent a lack of knowledge
on our part, but represents the fact that the particle does not have a definite
linear momentum in the y-direction. To describe the motion of the particle,
then, we must resort to the concept of probability. We won't actually know the
particle's y-ward momentum until we measure it, for example by finding where the
particle strikes a screen set up some distance behind the barrier.

We can now conceive the idea of determining the probabilities associated with the particle's y-ward motion by launching a large number of identical particles at the slit. We give each of them the same linear momentum as we did to the first particle and find where it subsequently strikes the screen. We expect the pattern that we find will display a simple symmetry: the particles will show the greatest density near the point that classical dynamics would put all of them and then we will see the density fall off in the same way on both sides of that point. The narrower we make the slit, the wider that array of particle marks on the screen will become.

But we have tacitly assumed that the particles do not touch the sides of the slit, that they do not have any contact with the barrier. Now I make explicit the statement that none of the particles touched the sides of the slit. We used particles so small that they passed through the slit with room to spare. And yet something deflected them from their initial courses, doing so in order to fulfill the requirements of the quantum theorem.

Like Einstein, I dislike invoking spooky actions at a distance to explain phenomena. If we have a phenomenon in which one body seems to affect the motion of another body whose location does not coincide with that of the first body, then I infer the existence of a forcefield, a continuous thing that emanates from one body and transmits force. In this case, though, we cannot infer the existence of a field of force to account for the motion of the particle. That we must have a field of some kind I don't doubt, but that field does not exert a direct force. Instead, it seems to exert a probability of force.

In this imaginary experiment we see a hint at the new quantum theory, the quantum theory of Louis deBroglie, Erwin Schrödinger, and others, in which theory we describe particles with a wave function that represents probability densities. But, because the experiment does not show us interference, especially self-interference, we cannot take the next step and introduce the wave function to describe the probability densities associated with the particle. In order to take that step we must conduct yet another imaginary experiment, but that is a topic for a separate essay.

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