Born rule

Born rule is the postulate in quantum mechanics that gives the probability of obtaining a particular measurement outcome from a given quantum state. Formulated by Max Born in 1926, the rule states that the probability of measuring an eigenvalue corresponding to eigenstate |φ⟩ is |⟨φ|ψ⟩|², the squared modulus of the inner product between the state vector |ψ⟩ and the eigenstate |φ⟩. For position measurements, this becomes |ψ(x)|², the probability density of finding the particle at position x.

The Born rule bridges the deterministic formalism of quantum mechanics — the Schrödinger equation governing continuous evolution — and the probabilistic outcomes of actual measurements. It is the rule that extracts predictions from the state. Yet it is not derived from the other postulates of quantum mechanics; it is added as an independent axiom. Numerous attempts have been made to derive the Born rule from more fundamental principles, including decision-theoretic arguments and many-worlds counting, but none has achieved consensus. The rule remains a postulate, not a theorem, and its status as an independent axiom suggests that probability in quantum mechanics is not a derived consequence but a fundamental feature of the theory's relationship to observation.