Conjugate variables: Difference between revisions

Revision as of 05:39, 4 December 2024

Latest revision as of 13:39, 24 May 2025

(5 intermediate revisions by 5 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 }}</ref>

* [[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''&nbsp;d''A'' (''γ'' = [[surface tension]]; ''A'' = surface area).

* Elastic stretching: ''F''&nbsp;d''L'' (''F'' = elastic force; ''L'' length stretched).

* Energy and time: Units <math> \Delta E \times \Delta t </math> being ''Kg'' kg&thinsp;m<mathsup> m^2 </sup>&thinsp;s^{-1}<sup>−1</mathsup>.

===Derivatives of action===

Line 19:

* The ''[[angular momentum]]'' of a particle is the derivative of its action with respect to its ''[[orientation (geometry)|orientation]]'' (angular position).

* The ''[[Relativistic angular momentum#Dynamic mass moment|mass-moment]]'' (<math>\mathbf{N}=t\mathbf{p}-E\mathbf{r}</math>) of a particle is the negative of the derivative of its action with respect to its ''[[rapidity]]''.

* The ''[[electric potential]]'' (φ, [[voltage]]) and ''[[electric charge]]'' in a [[quantum LC circuit]].<ref>{{Cite journal |last=Vool |first=Uri |last2=Devoret |first2=Michel |date=2017 |title=Introduction to quantum electromagnetic circuits |url=https://onlinelibrary.wiley.com/doi/abs/10.1002/cta.2359 |journal=International Journal of Circuit Theory and Applications |language=en |volume=45 |issue=7 |pages=897–934 |doi=10.1002/cta.2359 |issn=1097-007X|arxiv=1610.03438 }}</ref>

* The ''[[electric potential]]'' (φ, [[voltage]]) at an event is the negative of the derivative of the action of the electromagnetic field with respect to the density of (free) ''[[electric charge]]'' at that event. {{citation needed|date=April 2013}}

* The ''[[Magnetic vector potential|magnetic potential]]'' ('''A''') at an event is the derivative of the action of the electromagnetic field with respect to the density of (free) ''[[electric current]]'' at that event. {{citation needed|date=April 2013}}

* The ''[[electric field]]'' ('''E''') at an event is the derivative of the action of the electromagnetic field with respect to the ''electric [[polarization density]]'' at that event. {{citation needed|date=April 2013}}

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>