Distinction between a Theory and an Equation

We criticized the simple theory of the two-slit interference pattern which is based on the physically incorrect assumption that empty space (slits) is a source of waves. Why, then, does this theory give correct results (sometimes)? Consider first an opaque screen with no slits and illuminated on one side. No light is transmitted. Why? If you accept the superposition principle for electromagnetic waves, you cannot believe that the incident wave is destroyed. It exists everywhere just as it did without the screen in place. But the screen gives rise to secondary waves excited by the primary (incident) wave, and the superposition (interference) of all these waves is what is observed. With the screen in place, interference is destructive everywhere behind it: no net wave is transmitted. If it bothers you that the incident wave still exists in the space on the dark side of the screen, consider a standard problem in electrostatics. A conducting shell is placed in an external electric field. Inside the shell the total field is zero (electric shielding). Why? The external field induces a charge distribution on the outer surface of the shell (with zero total charge), within which the total field is the sum of the field of this charge distribution and the external field. These two fields are equal and opposite inside the shell so their sum is zero. But no one, to our knowledge, claims that the external field ceases to exist within the shell. To do so would contradict the superposition principle. What is true for electrostatics must also be true for electrodynamics.

An important distinction here (and throughout science), but not made often enough, is that between a theory and an equation. They are not the same. A theory of the two-slit interference pattern based on the literal assumption that empty space is the source of waves is incorrect, but the resulting equation (sometimes) is correct. Who cares as long as "you get the right answer"? Sometimes you don't get the right answer, and if your theory is wrong and you don't know why you won't be able to fix it.

Suppose that we remove material from the screen by punching two holes in it. If the interference pattern without the holes results in destructive interference everywhere on the dark side of the screen because of secondary waves excited in the screen material, removing a bit of this material ought to change the pattern. And this is what happens: light is observed at points on the dark side of the screen where previously there was none. To good approximation, it is as if the holes themselves were the mathematical sources of waves, whereas their physical source is the entire screen with two holes in it. This approximation cannot be universally valid because it is insensitive to the screen thickness and material and polarization state (Ch. 7) of the incident light. A diffraction grating is, in essence, a periodic array of many slits, and it has been long known that the elementary theory of the grating, which is fundamentally no different from two-slit theory, fails to account for what are often called Rayleigh-Wood anomalies.

In the photon language our description of illumination of an opaque screen is quite different. We would say that incident photons do not exist on the dark side of the screen, which does not contradict the assertion that the incident wave does. Where confusion may arise is when we ask if there is any incident light on the dark side. The answer depends on the language we choose, and hence what we mean by light. And this answer becomes garbled when we speak and think in both languages simultaneously (possibly without realizing it). If it seems enigmatic that an incident wave can penetrate solid matter, it is just as enigmatic that incident photons cannot. The size of a nucleus, where most of an atom's mass is concentrated, is smaller than the atom by at least a factor of 10~4, which in turn means that the (volume) fraction of an atom occupied by mass is less than 10~12. Despite its apparent solidity, matter is almost entirely empty space very sparsely populated by charges (electrons and protons). Thus it is not difficult to understand - or at least accept - why waves can penetrate such nearly empty space but not why photons cannot.

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