Conjugate variables

For conjugate variables in context of thermodynamics, seeConjugate variables (thermodynamics).

Conjugate variablesare pairs of variables mathematically defined in such a way that they becomeFourier transformduals,or more generally are related throughPontryagin duality.The duality relations lead naturally to an uncertainty relation—inphysicscalled theHeisenberg uncertainty principle—between them. In mathematical terms, conjugate variables are part of asymplectic basis,and the uncertainty relation corresponds to thesymplectic form.Also, conjugate variables are related byNoether'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).

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 andfrequency: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.
• Dopplerandrange:the more we know about how far away aradartarget 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 aradar ambiguity functionorradar ambiguity diagram.
• Surface energy:γdA(γ=surface tension;A= surface area).
• Elastic stretching:FdL(F= elastic force;Llength stretched).
• Energy and Time: Units$\Delta E\times \Delta t$beingKg$m^{2}s^{-1}$

Derivatives of action

Inclassical physics,the derivatives ofactionare conjugate variables to the quantity with respect to which one is differentiating. In quantum mechanics, these same pairs of variables are related by the Heisenberguncertainty principle.

• Theenergyof a particle at a certaineventis the negative of the derivative of the action along a trajectory of that particle ending at that event with respect to thetimeof the event.
• Thelinear momentumof a particle is the derivative of its action with respect to itsposition.
• Theangular momentumof a particle is the derivative of its action with respect to itsorientation(angular position).
• Themass-moment($\mathbf {N} =t\mathbf {p} -E\mathbf {r}$) of a particle is the negative of the derivative of its action with respect to itsrapidity.
• Theelectric 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 chargeat that event.
• Themagnetic potential(A) at an event is the derivative of the action of the electromagnetic field with respect to the density of (free)electric currentat that event.
• Theelectric field(E) at an event is the derivative of the action of the electromagnetic field with respect to theelectricpolarization densityat that event.
• Themagnetic induction(B) at an event is the derivative of the action of the electromagnetic field with respect to themagnetizationat that event.
• The Newtoniangravitational potentialat an event is the negative of the derivative of the action of the Newtonian gravitation field with respect to themass densityat that event.

Quantum theory

Inquantum mechanics,conjugate variables are realized as pairs of observables whose operators do not commute. In conventional terminology, they are said to beincompatible observables.Consider, as an example, the measurable quantities given by position$\left(x\right)$and momentum$\left(p\right)$.In the quantum-mechanical formalism, the two observables$x$and$p$correspond to operators${\widehat {x}}$and${\widehat {p\,}}$,which necessarily satisfy thecanonical commutation relation: $$[{\widehat {x}},{\widehat {p\,}}]={\widehat {x}}{\widehat {p\,}}-{\widehat {p\,}}{\widehat {x}}=i\hbar$$

For every non-zero commutator of two operators, there exists an "uncertainty principle", which in our present example may be expressed in the form: $$\Delta x\,\Delta p\geq \hbar /2$$

In this ill-defined notation,$\Delta x$and$\Delta p$denote "uncertainty" in the simultaneous specification of$x$and$p$.A more precise, and statistically complete, statement involving the standard deviation$\sigma$reads: $$\sigma _{x}\sigma _{p}\geq \hbar /2$$

More generally, for any two observables$A$and$B$corresponding to operators${\widehat {A}}$and${\widehat {B}}$,the generalized uncertainty principle is given by: $${\sigma _{A}}^{2}{\sigma _{B}}^{2}\geq \left({\frac {1}{2i}}\left\langle \left[{\widehat {A}},{\widehat {B}}\right]\right\rangle \right)^{2}$$

Now suppose we were to explicitly define two particular operators, assigning each aspecificmathematical 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 theHeisenberg Lie algebra${\mathfrak {h}}_{3}$,with a corresponding group called the Heisenberg group$H_{3}$.

Fluid mechanics

InHamiltonian fluid mechanicsandquantum hydrodynamics,theactionitself (orvelocity potential) is the conjugate variable of thedensity(orprobability density).

See also

• Canonical coordinates