Born ruleprinciple
In 1926, Max Born published a paper proposing a probabilistic interpretation of the wave function introduced by Schrödinger. Before that, the Schrödinger equation predicted waves, but it was unclear what they described. Born guessed: the intensity of the wave is the probability density. For this idea he received the Nobel Prize only almost three decades later, it seemed so radical. In physics, it became established: the quantum world is fundamentally random, but this randomness is strictly mathematically predictable.
How it works
Determines how electrons are distributed around the atomic nucleus, why molecules have certain shapes, and how lasers work. For example, in the double slit, the interference pattern is a direct consequence of the square of the wave function.
💡 Born himself initially doubted the probabilistic interpretation and even sent the paper to press with reservations. Only later, seeing its success, did he accept his own guess.
P(x) = |\psi(x)|^2
P(x) — probability density of finding a particle at coordinate x; ψ(x) — wave function of the particle in coordinate representation (complex amplitude); |·|² — squared modulus of a complex number.
P(m) = \operatorname{Tr}(M_m \rho)
P(m) — probability of obtaining measurement result m; ρ — density matrix of the quantum system (Hermitian operator with trace 1); M_m — positive operator-valued measure for outcome m (for example, projector |ψ_m⟩⟨ψ_m|); Tr — trace operation of a matrix.