TITLE: The Planck constant for mathematicians
QUESTION [69 upvotes]: The questions
Q1. What are simple ways to think mathematically about the physical meanings of the Planck constant?
Q2. How does the Planck constant appear 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 tend 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 on quantum computation and quantum information, and I am curious about it.
Related MO questions: Does quantum mechanics ever really quantize classical mechanics? ;

REPLY [9 votes]: The previous answers all give good discussions of the physical significance of $\hbar$ and the classical limit "$\hbar \to 0$" (in large quotation marks), but few of them discuss your "motivation" comment that $\hbar$ rarely appears in the study of quantum computing and information. David Mermin dedicates an entire section of his great paper "From Cbits to Qbits" to that question:

Like my disapproving colleague, some physicists may be appalled to have finished what purports to be an exposition of quantum mechanics — indeed, of applied (well, gedanken applied) quantum mechanics — without ever having run into Planck’s constant. How can this be?
The answer goes back to my first reason why enough quantum mechanics to understand quantum computation can be taught in a mere four hours. We are interested in discrete (2-state) systems and discrete (unitary) transformations. But Planck’s constant only appears in the context of continuously infinite systems (position eigenstates) and continuous families of transformations (time development) that act on them. Its role is to relate the conventional units in which we measure space and time, to the units in which it is quantum-mechanically natural to take the generators of the unitary transformations that produce translations in space or time.
If we are not interested in location in continuous space and are only interested in global rather than infinitesimal unitary transformations, then $\hbar$ need never enter the story. The engineer, who must figure out how to implement unitary transformations acting over time on Qbits located in different regions of physical space, must indeed deal with $\hbar$ and with Hamiltonians that generate the unitary transformations out of which the computation is built. But the designer of algorithms for the finished machine need only deal with the resulting unitary transformations, from which $\hbar$ has disappeared as a result, for example, of judicious choices by the engineers of the times over which the interactions that produce the unitary transformations act.
Deploring the absence of $\hbar$ from expositions of quantum computer science is rather like complaining that the $I$-$V$ curve for a $p$-$n$ junction never appears in expositions of classical computer science. It is to confuse computer science with computer engineering.

So basically, quantum algorithms researchers are implicitly setting $\hbar = 1$, or more precisely they're sweeping it up in  the various fixed time constants and so on that determine the gates' operation at the microscopic level, and only considering the value of the dimensionless ratios $(\Delta E) T/\hbar$ that arise from the Schrodinger equation.