InterferaThe Path Unchosen / Quantum Interference

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.