In quantum mechanics, a probability amplitude is a complex number-valued function which describes an uncertain or unknown quantity. For example, each particle has a probability amplitude describing its position.

For a probability amplitude ψ, the associated probability density function is

<math>\psi*\psi<math>

which is equal to |ψ|². This is sometimes called just probability density, and may be found used without normalisation (to have the total 1).

If |ψ|² has a finite integral over the whole of three-dimensional space, then it is possible to choose a normalising constant, c, so that by replacing ψ by cψ the integral becomes 1. Then the probability that a particle is within a particular region V is the integral over V of |ψ|².

The change over time of this probability (in our example, this corresponds to a description of how the particle moves) is expressed in terms of ψ itself, not just the probability function |ψ|². See Schrödinger equation.

In order to describe the change over time of the probability density it is acceptable to define the probability flux (also called probability current). The probability flux j is defined as:

<math> \mathbf{j} = {\hbar \over m} \cdot {1 \over {2 i}} \left( \psi ^{*} \nabla \psi - \psi \nabla \psi^{*} \right) = {\hbar \over m} Im \left( \psi ^{*} \nabla \psi \right) <math>

and measured in units of (probability)/(area*time) = r^-2t^-1.

The probability flux satisfies a quantum continuity equation, i.e.:

<math> \nabla \cdot \mathbf{j} = { \partial \over \partial t} P(x,t) <math>

where P(x,t) is the probability density and measured in units of (probability)/(volume) = r^-3. This equation is the mathematical equivalent of probability conservation law.

It is easy to show that for a plain wave function,

<math> | \psi \rang = A \exp{\left( i k x - i \omega t \right)} <math>

the probability flux is given by

<math> j(x,t) = |A|^2 {k \hbar \over m} <math>

The bi-linear form of the axiom has interesting consequences as well.