Probability amplitude

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 $\backslash \backslash psi$

• \\psi

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: ::$\backslash \backslash mathbf\{j\}\; =\; \{\backslash \backslash hbar\; \backslash \backslash over\; m\}\; \backslash \backslash cdot\; \{1\; \backslash \backslash over\; \{2\; i\}\}\; \backslash \backslash left(\; \backslash \backslash psi\; ^\{$

• } \

abla \\psi - \\psi \ abla \\psi^{

• } \ight) = {\\hbar \\over m} Im \\left( \\psi ^{*} \

abla \\psi \ight) and measured in units of (probability)/(area

• time) = r^-2t^-1.

The probability flux satisfies a quantum continuity equation, i.e.: ::$\backslash \; abla\; \backslash \backslash cdot\; \backslash \backslash mathbf\{j\}\; =\; \{\; \backslash \backslash partial\; \backslash \backslash over\; \backslash \backslash partial\; t\}\; P(x,t)$ 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, :$|\; \backslash \backslash psi\; \backslash ang\; =\; A\; \backslash \backslash exp\{\backslash \backslash left(\; i\; k\; x\; -\; i\; \backslash \backslash omega\; t\; \backslash ight)\}$ the probability flux is given by :$j(x,t)\; =\; |A|^2\; \{k\; \backslash \backslash hbar\; \backslash \backslash over\; m\}$ The bi-linear form of the axiom has interesting consequences as well.

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