Content deleted Content added
m Open access bot: arxiv updated in citation with #oabot. |
m Open access bot: url-access updated in citation with #oabot. |
||
| (3 intermediate revisions by 3 users not shown) | |||
Line 1:
{{Short description|Variables that are Fourier transform duals}}
'''Conjugate variables''' are pairs of variables mathematically defined in such a way that they become [[Fourier transform]] [[dual (mathematics)|duals]],<ref>{{Cite web |url=http://www.aip.org/history/heisenberg/p08a.htm |title=Heisenberg – Quantum Mechanics, 1925–1927: The Uncertainty Relations |access-date=2010-08-07 |archive-date=2015-12-22 |archive-url=https://web.archive.org/web/20151222204440/https://www.aip.org/history/heisenberg/p08a.htm |url-status=dead }}</ref><ref>{{cite journal | url=https://doi.org/10.1007%2FBF02731451 | doi=10.1007/BF02731451 | title=Some remarks on time and energy as conjugate variables | year=1962 | last1=Hjalmars | first1=S. | journal=Il Nuovo Cimento | volume=25 | issue=2 | pages=355–364 | bibcode=1962NCim...25..355H | s2cid=120008951 | url-access=subscription }}</ref> or more generally are related through [[Pontryagin duality]]. The duality relations lead naturally to an uncertainty relation—in [[physics]] called the [[Heisenberg uncertainty principle]]—between them. In mathematical terms, conjugate variables are part of a [[symplectic basis]], and the uncertainty relation corresponds to the [[symplectic form]]. Also, conjugate variables are related by [[Noether's theorem]], which states that if the laws of physics are invariant with respect to a change in one of the conjugate variables, then the other conjugate variable will not change with time (i.e. it will be conserved).
[[Conjugate variables (thermodynamics)|Conjugate variables in thermodynamics]] are widely used.
Line 7:
There are many types of conjugate variables, depending on the type of work a certain system is doing (or is being subjected to). Examples of canonically conjugate variables include the following:
* Time and [[frequency]]: the longer a musical note is sustained, the more precisely we know its frequency, but it spans a longer duration and is thus a more-distributed event or 'instant' in time. Conversely, a very short musical note becomes just a click, and so is more temporally-localized, but one can't determine its frequency very accurately.<ref>{{Cite journal |last1=Mann |first1=S. |last2=Haykin |first2=S. |date=November 1995 |title=The chirplet transform: physical considerations |url=http://wearcam.org/chirplet.pdf |journal=IEEE Transactions on Signal Processing |volume=43 |issue=11 |pages=2745–2761 |doi=10.1109/78.482123|bibcode=1995ITSP...43.2745M
* [[Doppler effect|Doppler]] and [[slant range|range]]: the more we know about how far away a [[radar]] target is, the less we can know about the exact velocity of approach or retreat, and vice versa. In this case, the two dimensional function of doppler and range is known as a [[radar ambiguity function]] or '''radar ambiguity diagram'''.
* Surface energy: ''γ'' d''A'' (''γ'' = [[surface tension]]; ''A'' = surface area).
* Elastic stretching: ''F'' d''L'' (''F'' = elastic force; ''L'' length stretched).
* Energy and time: Units <math> \Delta E \times \Delta t </math> being
===Derivatives of action===
Line 35:
<math display="block"> \sigma_x \sigma_p \geq \hbar/2 </math>
More generally, for any two observables <math> A </math> and <math> B </math> corresponding to operators <math> \widehat{A} </math> and <math> \widehat{B} </math>, the [[generalized uncertainty principle]] is given by:
<math display="block"> {\sigma_A}^2 {\sigma_B}^2 \geq \left (\frac{1}{2i} \left \langle \left [ \widehat{A},\widehat{B} \right ] \right \rangle \right)^2 </math>
| |||