Conjugate variables

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Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform duals,[1][2] or more generally are related through Pontryagin duality. The duality relations lead naturally to an uncertainty relation—in physics called the Heisenberg uncertainty principle—between them. In mathematical terms, conjugate variables are part of a symplectic basis, and the uncertainty relation corresponds to the symplectic form.


There are many types of conjugate variables, depending on the type of work a certain system is doing (or is being subjected to). 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 determine its frequency very accurately.[3]
  • 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.
  • Surface energy: γdA (γ = surface tension ; A = surface area).
  • Elastic stretching: FdL (F = elastic force; L length stretched).

Derivatives of action[edit]

In classical physics, the derivatives of action are conjugate variables to the quantity with respect to which one is differentiating. In quantum mechanics, these same pairs of variables are related by the Heisenberg uncertainty principle.

Quantum physics[edit]

In quantum mechanics, two variables are conjugate if the commutation relation is non zero. A standard example is the relation between position (x) and momentum (p), where the quantum mechanical operators involved obey the commutation relation . This can be expressed in terms of an uncertainty relation as .

Fluid Mechanics[edit]

In Hamiltonian fluid mechanics and quantum hydrodynamics, the action itself (or velocity potential) is the conjugate variable of the density (or probability density).

See also[edit]