Born rule

The Born rule assigns probabilities to quantum measurement outcomes by taking squared magnitudes of probability amplitudes. For a normalized state, the probabilities of all possible outcomes sum to one.

Probability amplitude: A complex number whose squared magnitude contributes to the probability of a quantum measurement outcome. The rule takes each outcome's amplitude and squares its magnitude.
Hilbert space: A complete inner-product vector space used to represent quantum states and observables. Quantum states and measurement vectors are represented within this space.
Max Born: A German physicist who formulated the probabilistic interpretation of the quantum wavefunction. Born introduced the probability interpretation that gives the rule its name.
Classical probability theory: A mathematical framework for assigning probabilities to events using nonnegative numbers that sum to one. The rule resembles ordinary probability but derives its values from quantum amplitudes.
Quantum probability problem: The problem of explaining why quantum theory assigns probabilities rather than definite measurement outcomes. The rule gives probabilities, but its physical status remains interpretively disputed.
Quantum state: A mathematical description encoding the possible outcomes and probabilities of a quantum system. The rule extracts measurement probabilities from a system's state.
Inner product: A mathematical operation that assigns a scalar to a pair of vectors and defines their lengths and angles. Measurement amplitudes are inner products between states and outcome vectors.
Schrödinger equation: A differential equation governing how a nonrelativistic quantum state evolves over time. Born's interpretation assigns probabilities to solutions of wave mechanics.
Many-worlds interpretation: An interpretation of quantum mechanics in which all outcomes persist in branching components of the universal state. Its account of observed probabilities must explain why branch weights follow the rule.
Everettian decision theory: An approach deriving Born-rule behavior from rational choices in the many-worlds interpretation. It seeks to justify the rule without adding collapse to quantum dynamics.

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