Conjugate variables: Difference between revisions

Revision as of 04:10, 15 October 2010

Revision as of 04:11, 15 October 2010

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{{For|conjugate variables in context of thermodynamics|Conjugate variables (thermodynamics)}}

'''Conjugate variables''' are pair of variables mathematically defined in such a way that they become [[Fourier transform]] [[dual (mathematics)|duals]] of one-another<ref>[http://www.aip.org/history/heisenberg/p08a.htm Heisenberg -– Quantum Mechanics, 1925-19271925–1927: The Uncertainty Relations]</ref><ref>[http://www.springerlink.com/content/r40472577250313r/ Some remarks on time and energy as conjugate variables]</ref>, or more generally are related through [[Pontryagin duality]]<ref>[http://mathworld.wolfram.com/PontryaginDuality.html Pontryagin Duality --– from Wolfram MathWorld]</ref>. The duality relations lead naturally to an uncertainty in [[physics]] called the [[Heisenberg uncertainty principle]] relation between them. In mathematical terms, conjugate variables are part of a symplectic basis, and the uncertainty principle corresponds to the [[symplectic form]]<ref>[http://mathworld.wolfram.com/SymplecticForm.html Symplectic Form --– from Wolfram MathWorld]</ref>.

==Examples==

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* [[Time]] and [[frequency]]: the longer a musical note is sustained, the more precisely we know its frequency (but it spans more time). Conversely, a very short musical note becomes just a click, and so one can't know its frequency very accurately.

* [[Time]] and [[energy]] -– as energy and frequency in [[quantum mechanics]] are directly proportional to each other.

* [[Position (vector)|Position]] and [[linear momentum]]: a precise definition of position leads to ambiguity of momentum, and vice versa.

* [[Angle]] (angular position) and [[angular momentum]];