Conjugate variables: Difference between revisions

Revision as of 16:51, 5 June 2007

Revision as of 01:39, 16 June 2007

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A pair of variables mathematically defined in such a way that they become [[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. 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 longermore precisely we know how long a musical note is sustainedheld, the moreless precisely we know its frequency (butat itany spansgiven morepoint in 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 [[Quantum Mechanics]] are directly proportional to each other.

* [[Position]] and [[momentum]]: precise definition of position leadleads to 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'''.