The questions

Q1. What are simple ways to think mathematically about the physical meanings of the Planck constant?

Q2. How does the Planck constant appears in mathematics of quantum mechanics. In particular, quantization is an important notion in mathematical physics and there are various forms of quantization for classical Hamiltonian systems. What is the role of the Planck constant in mathematical quantization.

Q3. How does the Planck constant relate to the uncertainty principle and to mathematical formulations of the uncertainty principle.

Q4. What is the mathematical and physical meaning of letting the Planck constant tending to zero. (Or to infinity, if this ever happens.)

###Motivation:

One purpose of this question is for me to try to get better early intuition towards a seminar we are running in the fall. Another purpose is that the Planck constant plays almost no role (and, in fact, is hardly mentioned) in the literature of quantum computation and quantum information and I am curious about it.

Related MO questions: Does quantum mechanics ever really quantize classical mechanics? ;