What is a wavefunction?

In 1926 Erwin Schrödinger wrote down an equation for how electrons behave inside atoms. Its answers spread out through space and rise and fall like waves, so he called them wave functions.

The name stuck, and physicists now use it for any quantum state, even one with no wave in sight, such as a photon's polarisation or a qubit.

In chapter 2 a photon's state is an arrow, and in chapter 7 a qubit's state is an arrow inside a ball. That arrow is a wavefunction in its simplest form.

An electron in an atom needs much more: a value at every point in space. Each value has a size and a direction of its own, which is why parts can add up or cancel each other out (chapter 4).

It doesn't tell you where the particle is. It tells you the odds.

Also in 1926, Max Born proposed the rule: square the size of the value at a place and you get the chance of finding the particle there. That's the Born rule from chapter 2. One measurement gives one result. Prepare the same state many times, measure each one, and the pattern of results matches the odds.

Not directly. A single measurement gives one answer and changes the state (chapter 3). You can't copy an unknown state and measure the copies either (page 5a).

So labs work a wavefunction out the slow way. They prepare the same state thousands of times, ask different questions of different copies, and piece the state together from all the answers.

Here physicists disagree (chapter 6). In Many-Worlds, the wavefunction is the whole of reality. QBism treats it as a person's betting odds (page 6b).

In 2012 Matthew Pusey, Jonathan Barrett and Terry Rudolph proved that, under a few reasonable assumptions, the wavefunction can't just be our ignorance about some deeper, definite state (page 10c). Views that reject one of those assumptions survive. A 2025 Nature survey of physicists found no agreement on the question.