In physics, conjugate variables are 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
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 Quantum Mechanics are directly proportional to each other.
- Position and momentum: precise definition of position lead 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.
