Conjugate variables

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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.

Examples of canonically conjugate variables include the following:

• Time and frequency: the longer a musical note is sustained, the more precise 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.

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