How *do* we explain the double-slit experiment in quantum mechanics’ native language, mathematics? Quantum mechanics predicts only the *probability* of physical phenomena. In the case of the double-slit experiment, it can predict the probability of the photon going through either hole. If you send a single photon, you cannot predict where it will land on the photographic plate. However, if you send many photons, the theory *predicts the aggregate* result, or pattern. This is the content of the *law of large numbers* in probability theory, which is behind the very accurate predictions of pollsters after they interview a *large* number of people. With both holes open, the figure on the right shows the photographic plate with increasing exposure times. Parts (a) and (b) look completely random, because the number of photons landing on the plate is not sufficient to reveal the pattern. Part (c) begins to show the pattern, and (d) has enough photons to completely display the interference pattern.

Why does the two-slit pattern differ so enormously from a single slit? Quantum mechanics says that there is a *probability amplitude* associated with the photon going through each slit. The total probability amplitude is the sum of individual amplitudes. This is the *superposition principle*, which is one of the most fundamental principles of the quantum theory. The probability itself is the *square* of the amplitude. That’s it! Everything follows from this! The probability amplitude for each slit is identical to the other one. When squared, they give the same result. Therefore if either of the slits is open, one observes this probability, which is represented by a single bright image. But when both slits are open, the amplitude is the sum of two amplitudes and the square of this sum – the probability of the position of each photon on the photographic plate – is not the sum of the squares of the summands, and could be zero at some points of the plate! (If you are comfortable with math, look at this to be convinced beyond the shadow of a doubt that mathematics explains this weird behavior beautifully.)

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