Probability Amplitude

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, the probability of measuring that state is . The values taken by a normalized 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.

These probability amplitudes have special significance because they act in quantum mechanics as the equivalent of conventional probabilities, with many analogous laws. For example, in the classic double-slit experiment where electrons are fired randomly at two slits, an intuitive interpretation is that P(hit either slit) = P(hit first slit) + P(hit second slit), where P(event) is the probability of that event. However, it is impossible to observe which slit is passed through without altering the electron. Thus, when not watching the electron, the particle cannot be said to go through either slit and this simplistic explanation does not work. However, the complex amplitudes taken by the two wavefunctions which represent the electron passing each slit do follow a law of precisely the form expected (ψ_total =ψ_first + ψ_second), and the calculations agree with experiment. This is the principle of quantum superposition, and explains the requirement that amplitudes be complex, as a purely real formulation has too few dimensions to describe the system's state when superposition is taken into account.