Wavefunctions and probability

In classical mechanics a particle has a position and you either know it or you don't. Quantum mechanics replaces that position with a wavefunction, ψ, spread over space. The wavefunction is not itself observable and is not a probability: it takes complex values, and can be negative or imaginary in places. What you measure comes from |ψ|², its squared magnitude, which gives the probability density of finding the particle at each point.

It is worth being careful about what the spread means. The electron is not smeared out like butter, with a bit of its mass here and a bit there — every measurement finds one whole electron at one place. Nor is the spread a statement about our ignorance of a position the electron secretly has. Interference experiments rule that out: send electrons one at a time through a double slit and the pattern that builds up depends on both slits being open, which no particle taking one definite path could produce.

The honest reading is that position is not a property the particle carries between measurements at all. The wavefunction is the complete description, it evolves smoothly and deterministically while nothing is measured, and it is only when a position measurement is made that a single definite outcome appears — drawn at random, with probabilities set by |ψ|². The squaring is also why the sign of ψ matters even though it never shows up directly: two contributions can cancel where they overlap with opposite signs, and that cancellation is exactly what interference is.