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Wave functions encode complex probability amplitudes; measurement probabilities follow from the Born rule (typically |amplitude|^2).
In quantum mechanics, a system’s state is described by a wave function, which assigns a complex probability amplitude to each point in space (or to each basis state). The probability of obtaining a particular measurement outcome is not taken directly from the wave function itself, but from the Born rule: it is given by the square of the absolute value of the relevant complex amplitude (or, more generally, by expectation values/projection operators). This is why quantum mechanics predicts probabilities rather than certain results for quantities like position. The wave function’s time evolution is governed by the Schrödinger equation, which deterministically updates the collection of probability amplitudes from one time to another. Because amplitudes are complex, they can interfere: different paths or components of the wave function can add or cancel, producing observable interference patterns. A key implication is that the same underlying wave function can yield different probability distributions depending on how measurements are arranged, as illustrated by interferometer and double-slit-type setups.
Wave functions encode complex probability amplitudes; measurement probabilities follow from the Born rule (typically |amplitude|^2).
Time evolution of probability amplitudes is deterministic via the Schrödinger equation, while measurement outcomes are probabilistic.
Because amplitudes are complex, they can interfere, leading to characteristic quantum interference patterns.
A complex-valued function (or state vector) that represents a quantum system and encodes probability amplitudes for measurement outcomes.
A complex number associated with a particular outcome or location whose magnitude determines the probability via the Born rule.
The rule stating that measurement probabilities are given by the square of the absolute value of the relevant probability amplitude (or by projection/expectation values in the general case).
The quantity obtained from the wave function (e.g., |ψ(x)|^2 in position space) that gives the likelihood of finding a particle in a small region.
The differential equation that governs how the wave function (and thus the probability amplitudes) evolves over time.
The phenomenon where complex probability amplitudes add and cancel, producing patterns in measurement probabilities that depend on relative phases.
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