https://www.wikidoc.org/index.php?title=Probability_amplitude&feed=atom&action=historyProbability amplitude - Revision history2024-03-29T07:27:06ZRevision history for this page on the wikiMediaWiki 1.40.0https://www.wikidoc.org/index.php?title=Probability_amplitude&diff=724116&oldid=prevWikiBot: Robot: Automated text replacement (-{{reflist}} +{{reflist|2}}, -<references /> +{{reflist|2}}, -{{WikiDoc Cardiology Network Infobox}} +)2012-09-06T14:02:56Z<p>Robot: Automated text replacement (-{{reflist}} +{{reflist|2}}, -<references /> +{{reflist|2}}, -{{WikiDoc Cardiology Network Infobox}} +)</p>
<p><b>New page</b></p><div>In [[quantum mechanics]], a '''probability amplitude''' is a [[complex number|complex]]-valued [[function (mathematics)|function]] that describes an [[uncertainty principle|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.<br />
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For a state ψ, the associated [[probability density function]] is ψ*ψ, which is equal to |ψ|<sup>2</sup>. This is sometimes called just '''probability density'''<ref>[[Max Born]] was awarded part of the 1954 [[Nobel Prize]] in Physics for this work.</ref>, and may be found and used without [[Normalizable wavefunction|normalization]].<br />
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Probability ''amplitude'': <math>\psi (x) \,</math><br />
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Probability ''density'': <math>|\psi (x)|^2 = \psi (x)^* \psi (x) \,</math><br />
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If |ψ|<sup>2</sup> 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 ''c''ψ the integral becomes 1. Then the probability that a particle is within a particular region ''V'' is the integral over ''V'' of |ψ|<sup>2</sup>. 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<br />
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:<math> P(\varepsilon)=\int_\varepsilon^{} |\psi(x)|^2\, dx. </math><br />
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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 <math>t\to\infty</math> 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.<br />
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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|<sup>2</sup>. See [[Schrödinger equation]].<br />
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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<br />
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:<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><br />
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and measured in units of (probability)/(area&nbsp;&times;&nbsp;time) = ''r''<sup>&minus;2</sup>''t''<sup>&minus;1</sup>.<br />
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The probability flux satisfies a quantum [[continuity equation]], i.e.:<br />
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:<math> \nabla \cdot \mathbf{j} + { \partial \over \partial t} P(x,t) = 0</math><br />
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where ''P''(''x'', ''t'') is the [[probability density function|probability density]] and measured in units of (probability)/(volume) = ''r''<sup>&minus;3</sup>.<br />
This equation is the mathematical equivalent of [[probability]] [[conservation law]]. <br />
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It is easy to show that for a plane [[wave function]],<br />
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:<math> | \psi \rang = A \exp{\left( i k x - i \omega t \right)} </math><br />
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the probability flux is given by<br />
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:<math> j(x,t) = |A|^2 {k \hbar \over m}. </math><br />
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The bi-linear form of the axiom has interesting consequences as well.<br />
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==Notes==<br />
{{reflist|2}}<br />
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[[Category:Quantum mechanics]]<br />
[[Category:Fundamental physics concepts]]<br />
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[[cs:Amplituda pravděpodobnosti]]<br />
[[fr:Amplitude de probabilité]]<br />
[[hu:Valószínűségi amplitúdó]]<br />
[[it:Ampiezza di probabilità]]<br />
[[sk:Amplitúda pravdepodobnosti]]<br />
[[uk:Амплітуда ймовірності]]<br />
[[zh:機率幅]]</div>WikiBot