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{{Short description|
{{For|conjugate variables in context of thermodynamics|Conjugate variables (thermodynamics)}}▼
'''Conjugate variables''' are pairs of variables mathematically defined in such a way that they become [[Fourier transform]] [[dual (mathematics)|duals]],<ref>
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==Examples==
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>
* [[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 kg m<sup>2</sup> s<sup>−1</sup>.
===Derivatives of action===
In [[classical physics]], the derivatives of [[action (physics)|action]] are conjugate variables to the quantity with respect to which one is differentiating. In quantum mechanics, these same pairs of variables are related by the Heisenberg [[uncertainty principle]].
* The ''[[energy]]'' of a particle at a certain [[event (relativity)|event]] is the negative of the derivative of the action along a trajectory of that particle ending at that event with respect to the ''[[time]]'' of the event.
* The ''[[linear momentum]]'' of a particle is the derivative of its action with respect to its ''[[position (vector)|position]]''.
* 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 ''[[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}}
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===Quantum theory===
In [[quantum mechanics]], conjugate variables are realized as pairs of observables whose operators do not commute. In conventional terminology, they are said to be ''incompatible observables''. Consider, as an example, the measurable quantities given by position
For every non-zero commutator of two operators, there exists an "uncertainty principle", which in our present example may be expressed in the form:
▲: <math>[\widehat{x},\widehat{p\,}]=\widehat{x}\widehat{p\,}-\widehat{p\,}\widehat{x}=i \hbar</math>
In this ill-defined notation,
▲For every non-zero commutator of two operators, there exists an "uncertainty principle", which in our present example may be expressed in the form:
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> \Delta x \, \Delta p \geq \hbar/2 </math>
Now suppose we were to explicitly define two particular operators, assigning each a ''specific'' mathematical form, such that the pair satisfies the aforementioned commutation relation. It's important to remember that our particular "choice" of operators would merely reflect one of many equivalent, or isomorphic, representations of the general algebraic structure that fundamentally characterizes quantum mechanics. The generalization is provided formally by the [[Heisenberg Lie algebra]] <math>\mathfrak h_3</math>, with a corresponding group called the Heisenberg group <math> H_3 </math>.▼
▲In this ill-defined notation, <math> \Delta x </math> and <math> \Delta p </math> denote "uncertainty" in the simultaneous specification of <math> x </math> and <math> p </math>. A more precise, and statistically complete, statement involving the standard deviation <math> \sigma </math> reads:
▲: <math> \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> {\sigma_A}^2 {\sigma_B}^2 \geq \left (\frac{1}{2i} \left \langle \left [ \widehat{A},\widehat{B} \right ] \right \rangle \right)^2 </math>
▲Now suppose we were to explicitly define two particular operators, assigning each a ''specific'' mathematical form, such that the pair satisfies the aforementioned commutation relation. It's important to remember that our particular "choice" of operators would merely reflect one of many equivalent, or isomorphic, representations of the general algebraic structure that fundamentally characterizes quantum mechanics. The generalization is provided formally by the Heisenberg Lie algebra <math>\mathfrak h_3</math>, with a corresponding group called the Heisenberg group <math> H_3 </math>.
===Fluid mechanics===
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