the Born rule
/ born (Max Born) /
The Schrodinger equation tells you how the wavefunction moves, but by itself psi is just complex numbers. What connects those numbers to the clicks of a real detector? The Born rule is that bridge. It is the single postulate that turns a quantum amplitude into a probability, and it is the reason quantum mechanics is a predictive science rather than abstract art.
The statement: for a state |psi> and an observable with eigenstates |a_n> of eigenvalues a_n, the probability that a measurement yields a_n is P(a_n) = |<a_n|psi>|^2, the squared magnitude of the amplitude <a_n|psi>. For a continuous variable like position it becomes a density, P(x) dx = |psi(x)|^2 dx. Equivalently the average of many measurements is the expectation value <A> = <psi|A|psi>. The defining move is squaring the modulus of a complex amplitude — probabilities go as amplitude squared, which is exactly why interfering amplitudes (not probabilities) produce fringes.
Be honest about its status: the Born rule is a postulate, not a theorem derived from the rest of the formalism. Gleason's theorem shows the squared-modulus form is essentially forced once you accept the Hilbert-space structure, and decoherence explains why we perceive definite outcomes, but why probability equals amplitude-squared remains an input to standard quantum theory. Its deepest signature is that you add amplitudes first and square second: that ordering is the whole of quantum interference.
In the double-slit experiment the amplitude to reach a point is the sum psi = psi_1 + psi_2 from the two slits; the Born rule gives brightness |psi_1 + psi_2|^2, whose cross term produces the interference fringes.
Add amplitudes, then square: the cross term |psi_1||psi_2| is the interference no classical particle picture can give.
You square the amplitude, not the individual probabilities; squaring after summing amplitudes is what makes quantum interference, and it is postulated, not derived.