Popular accounts love to say that a quantum particle can be "in two places at once." The phrase is catchy and misleading. The real idea, superposition, is both stranger and more precise, and it leads straight to the deepest open puzzle in physics.

Adding states together

In quantum mechanics, the possible states of a system can be added together, like waves. If "electron goes through slit A" is one possible state and "electron goes through slit B" is another, then a combination of the two, with some amplitude for each, is also a perfectly good state. This is superposition: a system's state can be a weighted combination of other possible states.

A superposition is not the same as not knowing. If a coin is hidden under a cup, it is heads or tails and you simply do not know which. A superposition of A and B is different, because its parts can interfere with each other, which a hidden but definite fact never could. The stripes of the double slit are the evidence that the electron was not secretly going through one slit all along.

Nor is it simply "both at once," as if two electrons were present. Only one electron is ever detected. A better description is that the electron is in a definite state, but one that has no definite answer to the question "which slit?"

It depends on the question

Light offers a clean example. Light waves vibrate in a direction called polarization. A polarizing filter, like the lens of good sunglasses, lets through light vibrating in one direction and blocks light vibrating at right angles to it. Each photon either passes or is absorbed.

Light polarized diagonally can be described as a superposition of horizontal and vertical. Yet relative to a diagonal filter, it is not a superposition at all; it simply passes. Whether a state counts as a superposition depends on which question, or measurement, is being asked. A superposition is not a special kind of fuzziness: every state is definite for some questions and undecided for others.

Measurement

When we measure a superposition, we get one outcome, with odds set by the Born rule. Afterward, the system behaves as if it is in the state that matches the outcome: measure the photon's polarization again in the same way and you get the same answer every time. The standard textbook account, set out by the Hungarian American mathematician John von Neumann in 1932, says that the wavefunction collapses onto the result found.

The trouble is that measuring devices are made of atoms too, and atoms obey Schrödinger's smooth equation. Followed strictly, that equation would put the device itself into a superposition of showing different results. So when, and why, does one result become real? This is the measurement problem: the theory's smooth rule for change and its rule for outcomes do not obviously fit together. Nobody disputes the predictions; the dispute is over what is really happening.

Schrödinger's cat

In 1935 Schrödinger sharpened the problem with a deliberately absurd scene. A cat is sealed in a box with a radioactive atom, a Geiger counter and a vial of poison. If the atom decays, the counter triggers a hammer that breaks the vial. Applied literally, the theory would leave the cat in a superposition of alive and dead until someone opens the box. Schrödinger meant the cat as a criticism, to show how strange it would be to apply superposition to everyday objects, not as a claim that cats really exist in such states. The rival answers to his challenge are the interpretations of quantum mechanics. The final level of this course explains why we never actually see such cats.

Worked example: three filters

Take two polarizing filters, one horizontal and one vertical, and stack them. No light gets through: whatever passes the first is blocked by the second. Now slide a third filter, set diagonally, between them. Common sense says that adding a filter can only block more light. Instead, light now comes through, about an eighth of the original if it started out unpolarized. Anyone with three pairs of cheap polarizing sunglasses can try this at home.

Superposition explains it. Light leaving the horizontal filter is horizontal for sure. But horizontal is a superposition of the two diagonal directions, so half of these photons pass the diagonal filter. Those that pass are now definitely diagonal, and diagonal is a superposition of horizontal and vertical, so half of them pass the vertical filter. No photon was "secretly vertical" all along; the middle filter asked a different question, and asking it wiped out the answer to the first. In quantum mechanics, measurement is not the passive reading of a fixed fact. It is an act that changes what can happen next.