Conjugate variables: Difference between revisions

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* 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> \hat{x} </math> and <math> \hat{p} </math>, which necessarily satisfy the canonical commutation relation:

: <math>[\hat{x},\hat{p}]=\hat{x}\hat{p}-\hat{p}\hat{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> \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> \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 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 Mechanicsmechanics===

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]]).