The Wavefunction Squared Is Probability
Nature writes where a particle is as a wavefunction ψ(x). But ψ itself can be negative, and you cannot measure it directly. What you actually observe is its square |ψ(x)|², and that is the probability density of finding the particle at x. In the first figure we watch ψ swing above and below zero.
ψ is a new symbol in this course: the wavefunction that records the particle's state. This ψ(x), a cosine rippling inside a Gaussian envelope, goes positive and then dips negative. Slide the center across the axis. The shape follows along, but remember the negative part is not a measured value, just an amplitude. Because ψ can swing into the negative like this, two waves that meet can even cancel each other, a distinctly quantum feature that classical probabilities, always positive, can never imitate.
Now plot |ψ(x)|², the square of that same ψ. Squaring flips every negative valley up into a positive hump. Slide the same center and this curve follows, but it never drops below zero. The sign is gone, and what remains, an always-positive mound, is the probability density. This rule of squaring to obtain probability was proposed by Max Born and became the standard interpretation of quantum mechanics, which is why it is called the Born rule.
The area under the probability-density curve is the actual probability. Two sliders set the left bound a and right bound b, and the shaded area between them reads out as P(a ≤ x ≤ b). A narrow window gives a small probability, a wide one a large probability. The area literally tells you the chance of finding the particle in that interval. If you measure a great many identically prepared particles, the fraction found in that interval converges to exactly this area, so the probability becomes a value confirmed by experiment.
Now add two waves of identical shape, changing only their relative phase with the slider. At phase 0 they reinforce and |ψ|² swells; near π they cancel and it flattens. Each piece looks the same on its own, yet the combined probability shifts with the phase. Phase is invisible in a single |ψ|², but it governs interference. The relative phase of two paths showing up as bright and dark fringes is exactly double-slit interference, so even though phase stays invisible it leaves a clear trace in experiment.
The particle is somewhere for sure, so the total area of |ψ|² must always be 1. Move the slider that multiplies ψ by a factor c, and the total area of |c·ψ|² grows or shrinks as c². At the c that makes the area exactly 1, the figure lights up. Tuning the total probability to 1 like this is normalization. Once it is set to 1, even as time passes and the wave moves and spreads, the total area stays conserved at 1, so a particle never suddenly disappears.