The relation between commutation and quanta

This question discusses discretization in some sense, and this question talks about how quantization and Hilbert Spaces are related (the answer seems to to be not at all), but what I'm curious about is how the uncertainty principle, quantization (as in quantized values for observables), and non-commutation are related- mathematically and physically. I'm not far enough along in physics to see the `obvious' answers to these questions, and I'm not far enough along in math to read through some of the more dense articles talking about Lie Algebras.

However, I feel as if there must be some relation between the generally quantization of values in quantum mechanics, and the non-commutativity of operators; when people discuss moving from the quantum realm to the classical realm they talk about the commutator going to zero. So what is it about the commutation/non-commutation of operators that makes a system quantized mathematically? The answer here attempts to give some link between the two, but I did not find it very cohesive or convincing.

I also understand that there may be a different meaning to the phrase quantization, but what I'm referring to is the appearance of quantized values of angular momentum, energy levels, etc.

• It might be worth noting that free particles still have non-computing conjugate operator pairs (say $\hat{x}$ and $\hat{p}$), but don't experience the quantization of energy levels that appears for bound particles. This is because the quantization fundamentally caused by the boundary condition to which the particle state is subject. Until you are clear on that I'm not sure there is any point in even asking the question. – dmckee --- ex-moderator kitten Jun 25 '15 at 15:25
• Related post by OP: physics.stackexchange.com/q/190303/2451 – Qmechanic Jun 25 '15 at 15:33
• @dmckee That makes sense- but there should still be a relation between non-commutation and quantizations, right? Because non-commutation is related to the uncertainty principle, which should somehow be related to quantizations, correct? – user83928 Jun 25 '15 at 15:34
• – Qmechanic Jun 25 '15 at 15:36