In Two Places at Once — Superposition and Probability

This is the double-slit experiment. Feynman said the whole mystery of quantum mechanics is in it — that nothing else contributes a new mystery, only variations on this one.

Overlapping can subtract

Start with what happens when waves overlap.

Drop two stones into still water and the ripples cross. Where crest meets crest the water rises higher; where crest meets trough they erase each other and the surface goes flat. That is interference.

In 1801 Thomas Young showed it with light. Light through two slits made alternating bright and dark bands on the wall. Each dark band is where light met light and cancelled. That is how light became a wave.

One at a time

Run the same experiment with electrons, one at a time.

A single electron makes one dot on the screen. It does not smear. Like a grain, one place, one dot.

And yet, as the dots accumulate, stripes appear.

So people say the electron went through both slits at once. That it was in two places at the same time.

The phrase is convenient and inaccurate, and most of the confusion starts there.

A little further in

The second article ended with Bohr's seats explained by treating the electron as a matter wave. Then there ought to be an equation that writes that wave properly.

Schrödinger's equation

Early in 1926 Erwin Schrödinger wrote it. Treating the electron as a wave and solving, the seats Bohr had assumed came out of the solutions on their own. Not put in by hand — produced.

The equation also worked for atoms beyond hydrogen, where Bohr's model had failed. And the picture changed: the electron does not circle in an orbit but sits as a spread cloud around the nucleus. The orbital shapes in textbooks are the solutions of this equation.

But what is waving

The equation worked. The question was what it was describing.

A wave on a guitar string is the string moving; a ripple is water moving. So in an electron's wave function, what moves?

Schrödinger wanted the electron to be something genuinely spread through space — a natural picture. Two things stood in the way.

1. If it were spread out it should smear on the screen; instead it always lands as one dot.
2. Write a two-electron system and the wave lives in six dimensions, not three. No real substance does that.

A footnote

In the summer of 1926 Max Born gave the answer in a footnote. The wave function itself is nothing; its square is the probability of finding the electron there.

Schrödinger disliked the reading and never accepted his equation being taken for a table of odds. Neither did Einstein, and one line from his letter to Born became famous.

He does not play dice.Einstein to Born, 1926

Why probability alone will not do

Something must be underlined here. Saying the outcome is probabilistic does not explain the double slit.

Dice are probabilistic too. But dice probabilities only add. Open a second route and the chance of arriving goes up; it cannot go down.

So the calculation runs in two stages. Each route is assigned an amplitude — a quantity that can cancel when added — the amplitudes are summed, and only then is the total squared into a probability.

Feynman pushed the method to its limit, assigning an amplitude to every possible path and summing them all. Two slits became infinitely many routes.

On 'observation'

Now the most misread part.

Fit the slits with apparatus that reveals which one the electron took, and the stripes vanish. Two piles remain, as with baseballs.

This is usually followed by claims that the electron changed because a person looked, that consciousness creates reality, that the observer decides the world.

The word observation carries much of the blame. Here it does not mean looking; it means becoming entangled with something else and leaving a trace. Once the electron is entangled with a detector, the wave is no longer the electron's alone, and the electron's part on its own no longer shows interference.

Those who actually did it

For a long time this was a thought experiment, since handling electrons one at a time was impossible.

1. 1801Young — stripes made with light
2. 1927Davisson and Germer — electrons make patterns too
3. 1961Jönsson — the double slit actually run with electrons
4. 1974Merli and colleagues — one at a time, the pattern photographed as it builds
5. 1989Tonomura and colleagues — the build-up recorded on video

When a physics magazine asked its readers in 2002 to name the most beautiful experiment, this one came first — plain apparatus, unambiguous result, and a century on, nobody has settled what it means.

What is left

The electron is described as a spread wave and caught as one dot. So what happens if the spread wave is forced into a single point?

Waves have one more property: squeeze one into a narrow space and its wavelength blurs, the way a very short pulse has no clear pitch. And for an electron, wavelength is momentum (article two).

The question that remainsNarrow the position and the wavelength blurs; sharpen the wavelength and the position spreads. Is that clumsiness in how we measure — or does the electron simply not hold both at once?