In physics, especially in quantum mechanics, conjugate variables are pairs of variables that share an uncertainty relation. The terminology comes from classical Hamiltonian mechanics, but also appears in quantum mechanics and engineering. The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ... Fig. ... In quantum physics, the Heisenberg uncertainty principle or the Heisenberg indeterminacy principle — the latter name given to it by Niels Bohr — states that when measuring conjugate quantities, which are pairs of observables of a single elementary particle, increasing the accuracy of the measurement of one quantity increases the uncertainty of... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... Fig. ...

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.
• Position and momentum: precise measurements of position lead to ambiguity of momentum, and vice versa.
• 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. This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... This long range RADAR antenna, known as ALTAIR, is used to detect and track space objects in conjunction with ABM testing at the Ronald Reagan Test Site on the Kwajalein atoll[1]. RADAR is a system that uses radio waves to determine and map the location, direction, and/or speed... This article or section is in need of attention from an expert on the subject. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ...

A more precise mathematical definition, in the context of Hamiltonian mechanics, is given in the article canonical coordinates. Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ...