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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

ψ*ψ

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:

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

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

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

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

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,

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

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.

03-10-2013 05:06:04