heisenberg uncertainty principle - Why do quantum physical properties come in pairs?

Why do quantum physical properties come in pairs, governed by the uncertainty principle (that is, position and momentum?)

Why not in groups of three, four, etc.?

Answer

The duality is a duality because of the notion of canonical conjugation in classical mechanics.

The reason people say that they come in pairs has nothing to do with quantum mechanics, but with the structure of classical mechanics. In classical mechanics, to give you the initial conditions for a system, you need to give the initial position of everything, and also the initial momentum. The classical variables come in pairs. These pairs are called canonically conjugate, because they have the property that their time rate of change of one is given by the derivative of the energy with respect to the other one.

The quantum mechanical description is only for wavefunctions which vary over values of one of the two canonically conjugate pairs. The other one is not freely specified, the wavefunction which gives its quantum description can be derived from the first.

People express the failure of classical mechanics by saying that half of all the initial data is related by uncertainty to the other half. This is the quantum duality. This idea was important historically, becuase it could explain how the classical equations could be taken up in quantum mechanics unchanged, while the predictions became probabilistic. People made analogies with the case where you have a classical particle whose position and momentum are unknown, and obey the Heisenberg relation. Quantum mechanics is completely different in that the presence at different position states is parametrized by probability amplitudes, not probabilities. But the description is always on half the phase space variables.

The uncertainty in position/momentum is directly analogous to the uncertainty in angular position/angular momentum, in the uncertainty of phase of a field mode and particle number of that mode, and in every other canonically cojugate pair. States of definite position are also uncertain energy, because energy and position do not commute, but nobody calls this a duality, because energy and position are not canonically conjugate.

I should also point out the energy/time uncertainty principle, which is hard to think of in terms of canonical pairs in the usual formulations of QM, because particles don't have a time associated to them, but all particles have a global time. In Schwinger/Feynman particle formalisms this is taken care of, but the uncertainty relation can be worked out in any formalism of course. This uncertainty relation might not be called a duality by some, I don't know.