Here is a question with no answer: where will the next electron land? Quantum mechanics can tell you, with superb accuracy, the pattern that a million electrons will make. It cannot tell you where any single one will go. According to the standard theory, this is not ignorance waiting to be cured. It is built into the rules.

A wave of possibilities

In 1926 Erwin Schrödinger wrote an equation describing how matter waves change over time. The wave it describes is called the wavefunction. It spreads through space, passes through both slits, reflects and interferes. Max Born then supplied the interpretation that physicists still use: the wavefunction tells you the chances of finding the particle in each place.

The precise rule matters. The wavefunction at each point has a size, called its amplitude. The probability of finding the particle there is the square of the amplitude, a recipe known as the Born rule. Born received a share of the Nobel Prize for this work in 1954, nearly three decades later.

Amplitudes can cancel; probabilities cannot

Why not just work with probabilities directly? Because probabilities are never negative, so they cannot cancel. If an electron could reach a spot through slit A with some chance and through slit B with some more chance, ordinary reasoning says the total chance must be larger than either alone. Opening a second slit should only add more electrons to every spot.

Yet at the dark stripes, opening the second slit makes electrons stop arriving. The explanation is that amplitudes, unlike probabilities, can cancel each other when added together. Amplitudes can be positive or negative; more precisely, they behave like little arrows that can point in any direction. The amplitude for the path through A and the amplitude for the path through B are added first. Where they point in opposite directions, they cancel, and the square of zero is zero. Only after adding do we square to get a probability.

This one rule, add the amplitudes for each path, then square, is the core recipe of quantum mechanics. It is what makes quantum chance different from the chance in a coin toss, where you would square first and add afterward, and so never get cancellation.

Smooth change, sudden results

There is something odd about the rules. Schrödinger's equation is deterministic: given the wavefunction now, it fixes the wavefunction at every later time, with no randomness at all. In the standard rules, chance enters only at one point, when a measurement produces an outcome. Between measurements, the quantum world evolves as predictably as a planet in its orbit. Why the moment of measurement should be special is the subject of the next level.

Is the randomness real?

Einstein suspected that the chance was only apparent, as it is with a tossed coin, whose outcome is fixed by forces we fail to track. On that view, hidden details would decide each result. Most physicists today accept that the randomness is genuine, partly because of experiments on entangled particles that rule out the simplest kinds of hidden details. That story belongs to QUAN 201. Some interpretations, though, keep a fully determined reality underneath, so the question is not closed for everyone.

Predictable crowds, unpredictable individuals

Quantum chance does not make quantum mechanics vague. Its predictions about large numbers of events are extremely precise, even though single events are unpredictable. Its calculation of the electron's magnetic strength agrees with experiment to better than one part in a billion, among the most accurate predictions in all of science.

The chance is also memoryless. For an unstable atom, one that has survived a long time is no more likely to decay soon than a fresh one. There is no internal clock running down, only the same fixed odds at every moment.

Worked example: radioactive iodine

Take a sample of iodine 131, a radioactive form of iodine used in medicine to treat an overactive thyroid and some thyroid cancers. Its half life is about eight days: after eight days, half the atoms in the sample will have decayed; after sixteen, three quarters; after twenty four, seven eighths. Hospitals plan doses and safety rules around this number with complete confidence.

Now pick out one single iodine atom. When will it decay? Quantum mechanics gives only a probability: a fifty percent chance within eight days, the same as for every other atom in the sample. An atom that has already survived a month is no more "due" than one made this morning. Nothing in the atom, so far as anyone has found, marks it as the one that will go next.

This is the Born rule in action. The wavefunction of each nucleus fixes the odds, not the outcome. Add up billions of atoms and the odds become a smooth, reliable curve, which is why a doctor can calculate a dose to the day while a physicist cannot predict a single click of the Geiger counter. The same pattern, certainty for crowds and chance for individuals, runs through all of quantum physics.