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In quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density. For example, if the probability amplitude of a quantum state is α, the probability of measuring that state is | α | 2. The values taken by a normalised wave function ψ at each point x are probability amplitudes, since |ψ(x)|2 gives the probability density at position x.
The principal use of probability amplitudes is as the physical meaning of the wavefunction, a link first proposed by Max Born and a pillar of the Copenhagen interpretation of quantum mechanics. In fact, the properties of the wave function were being used to make physical predictions (such as emissions from atoms being at certain discrete energies) before any physical interpretation was offered. Born was awarded half of the 1954 Nobel Prize in physics for this understanding, though it was vigorously contested at the time by the original physicists working on the theory, such as Schrödinger and Einstein. Therefore, the probability thus calculated is sometimes called the "Born probability", and the relationship used to calculate probability from the wavefunction is sometimes called the Born rule.