conjugate variables
Conjugate variables are pairs of physical quantities that are bound together by an uncertainty relation, so that knowing one sharply forces the other to be uncertain. The textbook pair is position and momentum, but energy and time, and angle and angular momentum, behave the same way. What ties such a pair together is a deep mathematical relationship inherited from classical mechanics and carried into the quantum theory.
In quantum mechanics this pairing shows up as a nonzero commutator between the two quantities' operators. Position and momentum satisfy the canonical commutation relation, and it is precisely this failure to commute that places them on opposite sides of an uncertainty trade-off. Variables that are not conjugate — whose operators do commute — can in principle be known together with no built-in limit at all.
There is also a beautiful way to see the pairing without operators at all. Conjugate variables are linked by a Fourier transform: the wavefunction written in terms of position and the same state written in terms of momentum are Fourier transforms of one another. A function and its Fourier transform cannot both be narrow, which is the wave-mathematics root of why conjugate quantities resist being sharp together.
Three classic conjugate pairs, each bound by its own uncertainty trade-off.
Energy and time are conjugate, but time is not an operator in ordinary quantum mechanics the way position is. The energy–time relation therefore has a subtly different meaning and must be read more carefully than the position–momentum one.