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 (ψtotal =ψfirst + ψsecond), 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.
A basic example 
Take a quantum system that can be in two possible states: for example, the polarisation of a photon. When the polarisation is measured, it could be horizontal, labelled as state , or vertical, state . Until its polarisation is measured the photon can be in a superposition of both these states, so its wavefunction, , would be written:
The probability amplitudes of states and are α and β respectively. When the photon's polarisation is measured, it has probability of being horizontally polarised, and probability of being vertically polarised.
Therefore, a photon with wavefunction whose polarisation was measured would have a probability of 1/3 to be horizontally polarised, and a probability of 2/3 to be vertically polarised.
The measurement must give either or , so the total probability of measuring or must be 1. This leads to a constraint that ; more generally the sum of the squared moduli of the probability amplitudes of all the possible states is equal to one. Wavefunctions that fulfill this constraint are called normalised wavefunctions.
Wavefunctions as probability amplitudes 
Normalizable states 
The Schrödinger wave equation, describing states of quantum particles, has solutions that describe a system and determine precisely how the state changes with time. Suppose a wavefunction ψ0(x, t) is a solution of the wave equation, giving a description of the particle (position x, for time t). If the wavefunction is square integrable, i.e.
for some t0, then ψ = ψ0/k is called the normalized wave function. Under the standard Copenhagen interpretation, the normalized wavefunction gives probability amplitudes for the position of the particle. Hence, at a given time t0, ρ(x) = |ψ(x, t0)|2 is the probability density function of the particle's position. Thus the probability that the particle is in the volume V at t0 is
Note that if any solution ψ0 to the wave equation is normalisable at some time t0, then the ψ defined above is always normalised, so that
is always a probability density function for all t. This is key to understanding the importance of this interpretation, because for a given initial ψ(x, 0), the Schrödinger equation fully determines subsequent wavefunction, and the above then gives the probable location of the particle at all subsequent times.
Non-normalizable states 
Probability amplitudes which are not square integrable are usually interpreted as the limit of a series of functions which are square integrable. For example, the plane wave solution to the wave equation is not normalisable, so it is not possible to give a physical interpretation of it for a single particle. Instead, one way to interpret this solution is as an infinite stream of monochromatic (identical) particles, in this case the limit of the series giving the wavefunction for increasingly many particles. The definition of P(V) given above is still valid, however since now many particles are involved, there could therefore be a high probability everywhere of finding a particle nearby.
Conservation relationship between probability amplitudes and probabilities 
- For more details on this topic and the proof, see probability current.
Intuitively, since a normalised wave function stays normalised while evolving according to the wave equation, there will be a relation between the change in the probability density of the particle's position and the change in the amplitude at these positions.
Define the probability current (or flux) j as
measured in units of (probability)/(area × time).
Then the current satisfies the quantum continuity equation
Discrete amplitudes 
While the wave function describes the state of a system for the continuous variable position, there are also many discrete variables to which probabilities may also be attached, which in quantum mechanics are found from complex amplitudes.
Example: One-dimensional quantum tunnelling 
- For more details on this example, see finite potential barrier.
In the one-dimensional case of particles with energy less than U > 0 in the square potential
the steady-state solutions to the wave equation have the form (for some constants )
The standard interpretation of this is as a stream of particles being fired at the step from the left (the direction of negative x): setting A =1 corresponds to firing particles singly; the terms containing A, C, and E signify motion to the right, while B, D, and F to the left. Under this beam interpretation, put F = 0 since no particles are coming from the right. By applying continuity of wave functions and their derivatives at the boundaries, it is hence possible to determine the constants above.
The conclusion is that the complex value B is a probability amplitude, with a real interpretation in the problem. The corresponding probability |B|2 describes the probability of a particle fired from the left being reflected by the potential barrier. Note that, very neatly, |B|2 +|E|2 =1 just as expected.
Probability frequency 
A discrete probability amplitude may be considered as a fundamental frequency in the Probability Frequency domain (spherical harmonics) for the purposes of simplifying M-theory transformation calculations.
See also 
- Double-slit experiment
- Electron shell
- Normal distribution
- Quantum chemistry
- Resonance (particle physics)
- Uncertainty principle
- Wave packet
- Example taken from Raymond, David (2006-04-07). "Sense and Nonsense in Quantum Mechanics". Retrieved 2011-02-23.