Double-Slit

Fire one quantum at the wall. It goes through two slits at once, lands as a single dot, and over thousands of shots the dots pile into bright and dark bands. Nothing crossed the gap but lone particles, yet they arranged themselves as if each had interfered with a copy of itself. It had.

The concept

A particle leaving the source has no single path. Its amplitude reaches the screen by way of both slits, and the two contributions add as complex numbers before you ever see anything. Probability is the square of that sum, not the sum of the squares, and that one fact is the whole show: cross terms survive, and cross terms are the fringes.

Look at which slit it took and the story collapses. Not because your detector shoved the particle, but because a which-path record makes the two routes distinguishable, and distinguishable routes do not interfere. The bands do not dim. They vanish.

The math

Let $y$ be the position on the screen, normalised to $[-1, 1]$. With no detector, the landing density is a single-slit diffraction envelope times a two-slit fringe comb:

$P(y) = \operatorname{sinc}^2(\alpha y)\,\cos^2\!\left(\frac{\pi \beta y}{2}\right), \qquad \operatorname{sinc}(x) = \frac{\sin(\pi x)}{\pi x}$

The $\cos^2$ term is interference, its period set by the slit separation $\beta$; the $\operatorname{sinc}^2$ term is the diffraction envelope of a single slit of finite width $\alpha$. Their product is the banded pattern you build up dot by dot.

Switch on a which-path detector and the coherence between the two routes is gone. The fringe term drops out and only the envelope remains:

$P(y) = \operatorname{sinc}^2(\alpha y)$

One broad smear, no bands. The experiment lets you slide $\alpha$ and $\beta$ and toggle the detector, sampling each quantum from whichever density is live, so the fringes assemble (or refuse to) one event at a time.

Why it stays strange

The usual dodge is that measurement "disturbs" the particle. But the pattern dies the moment which-path information exists, recoverable or not, even when the nudge is arbitrarily gentle. What kills interference is distinguishability, not force. The particle was never a tiny ball taking one slit; it was an amplitude over both, and looking is the act that decides which question reality answers.

Further reading

• Feynman, The Feynman Lectures on Physics, Vol. III, ch. 1 - the canonical telling.
• Tonomura et al. (1989), the single-electron buildup that shows the pattern emerging dot by dot.

@book{feynman1965vol3,
  author    = {Feynman, Richard P. and Leighton, Robert B. and Sands, Matthew},
  title     = {The Feynman Lectures on Physics, Vol. III: Quantum Mechanics},
  publisher = {Addison-Wesley},
  year      = {1965}
}
@article{tonomura1989,
  author  = {Tonomura, A. and Endo, J. and Matsuda, T. and Kawasaki, T. and Ezawa, H.},
  title   = {Demonstration of single-electron buildup of an interference pattern},
  journal = {American Journal of Physics},
  volume  = {57},
  number  = {2},
  pages   = {117--120},
  year    = {1989}
}