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{{Short description|Variables that are Fourier transform duals}}
{{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]]
▲
==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:
*
* [[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:
* Elastic stretching:
* 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 ''[[
* The ''[[
* 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 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.▼
* The ''[[
* The ''[[
* The
▲* The Newtonian ''[[
===Quantum theory===
=See also=▼
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 <math> \left (x \right) </math> and momentum <math> \left (p \right) </math>. In the quantum-mechanical formalism, the two observables <math> x </math> and <math> p </math> correspond to operators <math> \widehat{x} </math> and <math> \widehat{p\,} </math>, which necessarily satisfy the [[canonical commutation relation]]:
[[Canonical coordinates]]▼
<math display="block">[\widehat{x},\widehat{p\,}]=\widehat{x}\widehat{p\,}-\widehat{p\,}\widehat{x}=i \hbar</math>
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 display="block"> \Delta x \, \Delta p \geq \hbar/2 </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 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>
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===
In [[Hamiltonian fluid mechanics]] and [[quantum hydrodynamics]], the ''[[action (physics)|action]]'' itself (or ''[[velocity potential]]'') is the conjugate variable of the ''[[density]]'' (or ''[[probability density]]).
▲==See also==
▲*[[Canonical coordinates]]
==Notes==
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[[Category:Classical mechanics]]
[[Category:Quantum mechanics]]
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