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Interfera

The Path Unchosen / Quantum Interference Physics

\[P = \lvert\psi_1 + \psi_2\rvert^{2} = \lvert\psi_1\rvert^{2} + \lvert\psi_2\rvert^{2} + 2\,\mathrm{Re}\!\left(\psi_1^{*}\psi_2\right)\]
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Size guide S–5XL · inches

Send particles through two paths, one at a time, and stripes still appear. Quantum interference: probability amplitudes add or cancel, so even single particles build wave patterns. Interfera wears it in pixel art on a unisex tee. The physics panel below has the equation, every symbol and the sources.

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Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

Para ter uma escala: o tamanho M tem 20 in (50.80 cm) de largura, cerca de 3.143 × 1034 comprimentos de Planck (lP = 1.616255 × 10−35 m). Uma fita métrica é mais rápida.

Measurements in inches, ±1.5 in tolerance.

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The physics

When a particle can reach the screen by two paths, you add the amplitudes before squaring them, and the cross term creates bright and dark fringes. The fringe spacing is set by the particle's de Broglie wavelength, even when particles arrive one at a time.

Interfera is haunted by a path not taken. The haunting has a formula.

Fire particles one at a time at a wall with two slits and a screen behind it. Close one slit: a smooth hump of arrivals. Close the other: another hump. Common sense says both open should give the humps added together. You do not. You get stripes.

The first formula on this page says why. Quantum mechanics gives each route an amplitude, \(\psi_1\) and \(\psi_2\), and the probability is the square of their sum, not the sum of their squares. Squaring a sum leaves a third piece, the cross term, and that term can be negative. At some places on the screen, opening the second slit makes particles arrive less often than with one slit alone. No story of particles going through one slit or the other produces that. Feynman, opening the third volume of his Lectures with this experiment, said it contains the only mystery.

The second formula sets the spacing of the stripes: wavelength times the distance to the screen, divided by the slit separation. For a particle the wavelength is h over momentum, de Broglie's relation, so faster electrons make finer stripes.

It has been done literally. In 1976 Pier Giorgio Merli, Gian Franco Missiroli and Giulio Pozzi in Bologna filmed electron interference building up from single arrivals. In 1989 Akira Tonomura's team at Hitachi published a cleaner version, each electron landing as a single dot, the stripes appearing only as thousands accumulated, with essentially never two electrons in the apparatus at once.

Learn which slit each particle used, by any means, and the stripes vanish. Interference needs the two paths to stay indistinguishable, even in principle.

Interfera takes one step. The steps she did not take are in the pattern.

Equations

\[P = \lvert\psi_1 + \psi_2\rvert^{2} = \lvert\psi_1\rvert^{2} + \lvert\psi_2\rvert^{2} + 2\,\mathrm{Re}\!\left(\psi_1^{*}\psi_2\right)\]
\[\Delta y = \frac{\lambda L}{d}, \qquad \lambda = \frac{h}{p}\]
SymbolMeaningUnit
\(\psi_1, \psi_2\) amplitudes for the two paths (unit for a 1-D wavefunction; dimensionless if normalized over discrete outcomes) \(\mathrm{m^{-1/2}}\)
\(P\) detection probability density \(\mathrm{m⁻¹}\)
\(\Delta y\) fringe spacing \(\mathrm{m}\)
\(\lambda\) de Broglie wavelength \(\mathrm{m}\)
\(L\) slit-to-screen distance \(\mathrm{m}\)
\(d\) slit separation \(\mathrm{m}\)
\(h\) Planck constant \(\mathrm{J·s}\)
\(p\) momentum \(\mathrm{kg·m/s}\)

Sources

  1. Feynman Lectures on Physics Vol. III Ch. 1: Quantum Behavior
  2. Merli, Missiroli, Pozzi (1976) On the statistical aspect of electron interference phenomena, Am. J. Phys. 44, 306
  3. Tonomura et al. (1989) Demonstration of single-electron buildup of an interference pattern, Am. J. Phys. 57, 117
  4. Encyclopaedia Britannica — Interference

Source Merli, P. G.; Missiroli, G. F.; Pozzi, G. (1976). On the statistical aspect of electron interference phenomena. American Journal of Physics 44, 306–307.doi.org/10.1119/1.10184

Field notes

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