Conjugate variables: Difference between revisions

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A pair of conjugate variables aremathematically defined in such a way that they oftenbecome [[Fourier transform]] duals of one-another, or more generally are related through [[Pontryagin duality]]. The duality relations lead naturally to an uncertainty ([[Heisenberg uncertainty principle]]) relation between them.

In [[physics]], especially in [[quantum mechanics]], '''conjugate variables''' are pairs of variables that share an [[uncertainty principle|uncertainty relation]]. The terminology comes from classical [[Hamiltonian mechanics]], but also appears in [[quantum mechanics]] and engineering.

A more precise [[mathematical]] definition, in the context of [[Hamiltonian mechanics]], is given in the article [[canonical coordinates]].

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 more time). Conversely, a very short musical note becomes just a click, and so one can't know its frequency very accurately.

* [[Time]] and [[energy]] - as energy and frequency in QM are proportional to each other.

* Position and momentum: precise measurements of position lead to ambiguity of momentum, and vice versa.

* Angle[[Position]] (angularand [[momentum]]: precise definition of position) andlead angularto ambiguity of momentum;, and vice versa.

* [[Angle]] (angular position) and [[angular momentum]];

* [[Doppler]] and 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'''.

A pair of conjugate variables are often [[Fourier transform]] duals of one-another, or more generally are related through [[Pontryagin duality]]. The duality relations lead naturally to an uncertainty relation between them.

A more precise [[mathematical]] definition, in the context of [[Hamiltonian mechanics]], is given in the article [[canonical coordinates]].

[[Category:Classical mechanics]]