Are superposition and uncertainty principles logically dependent?

Question

If we assume superposition and define an Hilbert space with canonical commutation relations we can derive uncertainty relations. So it seems the uncertainty principle isn't required, or should be called the "canonical commutators principle".

On the other direction, if we would like to start assuming the uncertainty relations, it would be tricky even to give them a precise meaning, not having yet a phase space.

So is it safe to assert that uncertainty is called a principle maybe for some historical reason but it's hard to give it that role from a logical point of view?

Answer 1

The uncertainty principle states that for any two Hermitian operators $\hat{A}$ and $\hat{B}$, the variances $\sigma_A$ and $\sigma_B$ that is measured in the corresponding observables $A$ and $B$ are related by the following inequality:

$$\sigma_A\sigma_B\geq\frac{1}{2}\left\vert\langle[\hat{A},\hat{B}]\rangle\right\vert$$

The proof of this does not depend on the canonical commutation relations; rather, it only depends on the existence of the Hermitian operators involved, the definition of variance, and the Cauchy-Schwarz inequality. The proof can be found in Griffiths's quantum mechanics textbook, as well as on Wikipedia here: https://en.wikipedia.org/wiki/Uncertainty_principle#Robertson%E2%80%93Schr%C3%B6dinger_uncertainty_relations.

In order to prove the Heisenberg uncertainty principle with this approach (which is a special case of the uncertainty principle where $\hat{A}=\hat{x}$ and $\hat{B}=\hat{p}$), you do need to provide extra information. This information can either be provided by directly assuming the canonical commutation relations, or by defining the action of the position and momentum operators themselves on a wavefunction in any basis, by which the canonical commutation relations can be derived, as is shown, for example, here: Derivation of canonical position-momentum commutator relation.

The uncertainty principle is not an assumption, but it is required; it's a direct mathematical consequence of the definition of operators and the definition of variance. In addition, the Heisenberg uncertainty principle is not necessarily dependent on assuming the canonical commutation relations. There are other choices of assumptions one can make in which the canonical commutation relations are derived from other axioms (I specified one such choice here: defining the position and momentum operators), so it should be clear that even for the Heisenberg uncertainty principle, the canonical commutation relations do not necessarily need to be assumed. They often are in practice, but there is nothing in principle that elevates that set of axioms above any other self-consistent set, except for aesthetic elegance and a later connection to the assumptions of QFT.

Answer 2

Yes, uncertainty relations are a theorem in nowadays approach to quantum mechanics, and the word principle is the remnant of a different perspective at some stage of the construction of the conceptual framework of QM.

However, for the aim of historical correctness, one should stress that the meaning of the Heisenberg's uncertainty principle was not exactly the same as the present uncertainty relations. The former was a statement about the uncertainty on the value of non commuting observable at the level of an individual measurement, while the latter refers to the statistical spread of values in an ensemble of measurements. The two things are correlated, but are not the same. Some interesting debate about the relations between the two problems has gone on in the last two decades, starting from Ozawa's work.