Electron Fringes, One Hit at a Time

Send electrons through two slits one by one and watch the interference pattern build from random dots.

Check: running...
The sciencelambda = h / p, p c = sqrt(K (K + 2 m c^2)), K = e U (relativistic; 50 kV gives lambda = 5.36 pm) Two-slit intensity (far field): I(y) ~ cos^2(pi d y / (lambda L)) * sinc^2(pi a y / (lambda L)) Fringe spacing: delta_y = lambda L / d (small angles)

The plotted pattern is not taken from that formula. It is a Huygens sum: 16 point sources across each slit, each contributing exp(i k r) to every screen point with exact path lengths, so the single-slit envelope and slight Fresnel departures come out by themselves. Each electron is then placed on the screen by sampling that probability distribution, one at a time: a single hit looks random, and only the accumulation shows fringes.

The check measures the fringe period from the actual hits with a periodogram of the histogram (a Fourier search, no use of lambda L/d) and compares it with the formula. It tests the sampling and the geometry, and needs about 5000 electrons for 3% accuracy. Counting noise falls as 1/sqrt(N).

Simplifications: scalar wave, 1D screen coordinate, no electron-electron interaction (real experiments also send electrons through one at a time), ideal slits and perfect coherence. The formula lambda L/d is a far-field (Fraunhofer) result: it needs the Fresnel number d^2/(lambda L) well below 1, otherwise the true spacing drifts away from it (try the Near field preset); the check only claims a pass in the far field. Screen width is set to 24 fringes for display. Based on the idea of Joensson (1961) and Tonomura (1989) electron double-slit experiments; parameters are of that order, not a copy of either apparatus.