Probability amplitude

In quantum mechanics, a probability amplitude is a complex-valued function that describes an uncertain or unknown quantity. For example, each particle has a probability amplitude describing its position. This amplitude is the wave function, expressed as a function of position. The wave function is a complex-valued function of a continuous variable.

For a state ψ, the associated probability density function is ψ*ψ, which is equal to |ψ|2. This is sometimes called just probability density, and may be found and used without normalization.

Probability amplitude: $$\psi (x) \,$$

Probability density: $$|\psi (x)|^2 = \psi (x)^* \psi (x) \,$$

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. Which means, according to the Copenhagen interpretation of quantum mechanics, that, if some observer tries to measure the quantity associated with this probability amplitude, the result of the measurement will lie within ε with a probability P(ε) given by


 * $$ P(\varepsilon)=\int_\varepsilon^{} |\psi(x)|^2\, dx.  $$

Probability amplitudes which are not square integrable are usually interpreted as the limit of a series of functions which are square integrable. For instance the probability amplitude corresponding to a plane wave corresponds to the 'non physical' limit of a monochromatic source of particles. Another example: The Siegert wave functions describing a resonance are the limit for $$t\to\infty$$ of a time-dependent wave packet scattered at an energy close to a resonance. In these cases, the definition of P(ε) given above is still valid.

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


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

and measured in units of (probability)/(area &times; time) = r&minus;2t&minus;1.

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


 * $$ \nabla \cdot \mathbf{j} + { \partial \over \partial t} P(x,t) = 0$$

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

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


 * $$ | \psi \rang = A \exp{\left( i k x - i \omega t \right)} $$

the probability flux is given by


 * $$ j(x,t) = |A|^2 {k \hbar \over m}. $$

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