wavefunction and probability density
Nature writes where a particle is as the wave function ψ(x). ψ itself can go negative and cannot be read off directly. What we see is the square |ψ(x)|², and that is the probability density of finding the particle at that place. The first picture starts from ψ waving up and down. The waving comes first.
ψ is the symbol that first appears in this lesson, the wave function that writes the state of a particle. Inside a Gaussian envelope a cosine ripples, and looking at ψ(x) the value is positive and then drops negative. Shift the center and the shape follows, while the negative stretch is an amplitude, not a measured value. ψ has to reach the negative side so that two waves can meet and cancel, and a classical probability that is always positive cannot imitate that cancellation. A negative amplitude is what makes the cancellation possible.
Now draw the square of the same ψ, |ψ(x)|². Squaring flips every valley that was negative into a positive peak. Move the same center and this curve follows, and it never drops below 0. The positive lump left after the sign disappears is the probability density. The rule that gets a probability by squaring was proposed by Max Born, became the standard reading of quantum mechanics, and so it is called the Born rule. The peak is the place of finding.
The area under the probability-density curve is the actual probability. Set the left edge of the interval and the right edge, and the shaded area between them reads as P(a ≤ x ≤ b). A narrow interval is a small probability and a wide interval is a large one. The area says the chance of finding the particle inside that interval. Measure many particles prepared the same way and the fraction found inside the interval gathers onto this area, so the probability is a value that an experiment can check. The area is that chance.
Add two waves of the same shape and change only the relative phase. When the phase is 0 the two reinforce and |ψ|² rises, and near π they cancel and the curve goes flat. The two pieces look the same when seen apart, yet the combined probability changes with the phase. The phase is invisible if you look at only one |ψ|², but it steers the interference. The relative phase of two paths showing up as bright and dark stripes is the double-slit interference, so the phase leaves a clear trace in the experiment even though it is not seen.
A particle has to be somewhere, so the whole area of |ψ|² must always be 1. Multiply ψ by a scale c and the total area of |c·ψ|² grows or shrinks by c². The picture is marked where the area is exactly 1. Making the total probability 1 is normalization. Once it is set to 1, the whole area stays 1 even as time passes and the wave moves and spreads, so the particle does not suddenly vanish. The area is kept, and the particle stays somewhere in the picture the whole time.