Conjugate variables: Difference between revisions

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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]],<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>[http{{cite journal | url=https://wwwdoi.springerlinkorg/10.com/content/r40472577250313r1007%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).

{{For|conjugate[[Conjugate variables in context of (thermodynamics)|Conjugate variables (in thermodynamics)}}]] are widely used.

==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>[{{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 "The Chirplet Transform", |journal=IEEE Transactions on Signal Processing, |volume=43( |issue=11), November|pages=2745–2761 1995, p2745-2761]|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: γdA''γ''&nbsp;d''A'' (''γ'' = [[surface tension]] ; ''A'' = surface area).

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

* Energy and time: Units <math> \Delta E \times \Delta t </math> being kg&thinsp;m²&thinsp;s⁻¹.

===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 ''[[electricRelativistic potentialangular momentum#Dynamic mass moment|mass-moment]]'' (φ, [[voltage]]<math>\mathbf{N}=t\mathbf{p}-E\mathbf{r}</math>) atof ana eventparticle is the negative of the derivative of theits action of the electromagnetic field with respect to the density of (free)its ''[[electric chargerapidity]]'' at that event. {{citation needed|date=April 2013}}

* 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. {{citation needed|date=April 2013}}

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

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

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

* The Newtonian ''[[gravitational potential]]'' at an event is the negative of the derivative of the action of the Newtonian gravitation field with respect to the ''[[mass density]]'' at that event. {{citation needed|date=April 2013}}

===Quantum Theorytheory===

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> \hatwidehat{x} </math> and <math> \hatwidehat{p\,} </math>, which necessarily satisfy the [[canonical commutation relation]]:

<math display="block">[\hatwidehat{x},\hatwidehat{p\,}]=\hatwidehat{x}\hatwidehat{p\,}-\hatwidehat{p\,}\hatwidehat{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>[\hat{x},\hat{p}]=\hat{x}\hat{p}-\hat{p}\hat{x}=i \hbar</math>

<math display="block"> \Delta x \, \Delta p \geq \hbar/2 </math>

ForIn everythis nonill-zerodefined commutatornotation, of<math> two\Delta operators,x there</math> existsand an<math> \Delta p </math> denote "uncertainty principle", which in ourthe presentsimultaneous examplespecification of <math> x </math> and <math> p </math>. mayA bemore expressedprecise, inand statistically complete, statement involving the form: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> \hatwidehat{A} </math> and <math> \hatwidehat{B} </math>, the [[generalized uncertainty principle]] is given by:

<math> \Delta x \Delta p \geq \hbar/2 </math>

<math display="block"> {\sigma_A}^2 {\sigma_B}^2 \geq \left (\frac{1}{2i} \left \langle \left [ \hatwidehat{A},\hatwidehat{B} \right ] \right \rangle \right)^2 </math>

Now suppose we were to explicitly define two particular operators, assigning each of them a ''specific'' mathematical form, such that the pair satisfies the aforementioned canonical 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:

===Fluid Mechanicsmechanics===

<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> \hat{A} </math> and <math> \hat{B} </math>, the generalized uncertainty principle is given by:

<math> {\sigma_A}^2 {\sigma_B}^2 \geq \left (\frac{1}{2i} \left \langle \left [ \hat{A},\hat{B} \right ] \right \rangle \right)^2 </math>

Now suppose we were to explicitly define two particular operators, assigning each of them a ''specific'' mathematical form, such that the pair satisfies the aforementioned canonical 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]]).