The Uncertainty Principle and Probability Waves

With extremely faint light, G. I. Taylor showed as early as 1909 that interference fringes still form on a long-exposure photograph. Electrons make it even clearer. In a famous 1989 experiment at Hitachi, Akira Tonomura's team sent electrons through an electron version of the double slit at a rate so low that each one crossed alone. Every electron arrived as a single, sharp dot at one place on the detector, just like a particle. But the dots did not land at random. After a few hundred, a pattern emerged, and after tens of thousands the bright and dark fringes of two-slit interference were unmistakable. Each electron, on its own, behaved as if it went through both slits and interfered with itself. Stranger still, if a detector is placed at the slits to find out which one each electron uses, the fringes vanish and the screen shows two plain bands.

In 1926 Max Born proposed the interpretation that physics still uses. Each particle is described by a wave function, written $\psi$ (psi), that spreads out, passes through both slits and interferes exactly like a wave. But the wave is not a smeared-out electron. The square of its amplitude, $|\psi {|}^2$, at any point gives the probability of finding the particle there if you look. Bright fringes are places of high probability, dark fringes places of zero probability. A single electron's landing spot cannot be predicted, but the pattern made by many electrons can be predicted exactly. The Copenhagen interpretation, developed mainly by Niels Bohr and Werner Heisenberg, adds that a measurement forces a definite result: before detection the electron has no single position, and the act of detecting it, at the screen or at a slit, produces one. Wave and particle are complementary descriptions, and which one appears depends on what you measure. Einstein was never comfortable with this built-in randomness, and debates about the interpretation continue, but every prediction of the theory has been confirmed.

Because a particle is described by a wave, it cannot have both a sharply defined position and a sharply defined momentum. A wave with one exact wavelength, and so one exact momentum $p=h/\lambda$, stretches on forever, so its position is completely unknown. To build a wave packet confined to a small region, you must add together many wavelengths, and then the momentum is spread out. Squeezing $\Delta x$ forces $\Delta p$ to grow, and Heisenberg showed in 1927 that their product can never be smaller than $h/4\pi$. You can see it in a single slit: narrowing the slit pins down the particle's sideways position more tightly, and the diffraction pattern spreads wider, meaning the sideways momentum is less certain. This is not a flaw in our instruments. It is a property of nature, and it is why an electron cannot sit motionless at a point inside an atom.

A ball rolling toward a hill too high for its energy always rolls back. A quantum particle does not quite: its wave function does not stop dead at a barrier but fades away inside it, and if the barrier is thin enough a little of the wave reaches the other side. That gives a small but real probability that the particle appears beyond the barrier, a process called quantum tunnelling. The probability falls off extremely steeply as the barrier gets thicker, which turns out to be useful. Tunnelling explains how alpha particles escape from nuclei in radioactive decay, how charge is stored and erased in flash memory, and how the scanning tunnelling microscope works.