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

See: "Determination of the Planck constant using a watt balance with a superconducting magnet system at the National Institute of Standards and Technology" (Apr 24 2014), by Stephan Schlamminger, Darine Haddad, Frank Seifert, Leon S Chao, David B Newell, Ruimin Liu, Richard L Steiner, and Jon R Pratt along with this APS Physics article: "Living with the New SI" (March 25, 2019, Physics 12, 33):

"Over the past several decades, researchers have developed two kilogram-to-Planck experiments. The first, called the Kibble balance, works by offsetting the downward force of gravity on a chunk of metal with an upward magnetic force on a coil held in a magnetic field. Researchers tune the magnetic force by running current through the coil, and that current is measured in terms of Planck’s constant. The second experiment, conceived by the International Avogadro Project, involves fabrication of a near-perfect sphere of silicon. Using a combination of x-ray crystallography and optical interferometry, researchers can count the number of atoms in the sphere and connect its mass to Planck’s constant

In 2017, both of these methods returned values of Planck’s constant — based on the standard kilogram — having a precision of 10 parts per billion (ppb). These highly precise demonstrations have now allowed the metrology community to “turn the tables” by making Planck’s constant the defined quantity rather than the kilogram. As a result, the kilogram inherits the uncertainty that previously appeared in the Planck measurement.".

The Planck becomes: $h = 6.626 \, 069 \, 79(30) × 10^{−34} \;\text{J} s.$ That increases the uncertainty of the mass of the kilogram by 10 micrograms, a change at the 10 ppb level.

A better physical definition of the Planck permits normalization. A simple explanation without too much overlap is on the Wikipedia webpage "Natural Units":

In physics, natural units are physical units of measurement based only on universal physical constants. For example, the elementary charge $e$ is a natural unit of electric charge, and the speed of light $c$ is a natural unit of speed. A purely natural system of units has all of its units defined in this way, and usually such that the numerical values of the selected physical constants in terms of these units are exactly $1$.

If you nondimensionalize your variables and "normalize" the numerical values of certain fundamental constants to $1$ you can simplify and speed up your calculations. You need to be careful not to mix dimensioned with dimensionless quantities when using Planck base units and Hartree atomic units. You also can not normalize all your constants and need to choose your set carefully. There can also be a loss of precision, Planck units use the gravitational constant $G$, which is measurable in a laboratory only to four significant digits.

These are the Planck units based on Lorentz–Heaviside units (instead of on the more conventional Gaussian units), the rationalized Planck units are defined so that $c$ $=4\pi$$G$ = $\hbar$ = $\epsilon_{0}$ = $k_{\text{B}}$=$\,1$.

In Wikipedia's explanation of derived units it mentions:

- A speed of 1 Planck length per Planck time is the speed of light in a vacuum, the maximum possible physical speed in special relativity; 1 nano-(Planck length per Planck time) is about 1.079 km/h.
- Our understanding of the Big Bang begins with the Planck epoch, when the universe was 1 Planck time old and 1 Planck length in diameter, and had a Planck temperature of 1. At that moment, quantum theory as presently understood becomes applicable. Understanding the universe when it was less than 1 Planck time old requires a theory of quantum gravity that would incorporate quantum effects into general relativity. Such a theory does not yet exist.

With Planck units, the units are defined by properties of quantum mechanics and gravity. Not coincidentally, the Planck unit of length is approximately the distance at which quantum gravity effects become important. Likewise, atomic units are based on the mass and charge of an electron, and not coincidentally the atomic unit of length is the Bohr radius describing the "orbit" of the electron in a hydrogen atom.

In the Hartree system the numerical values of the following four fundamental physical constants are all unity by definition:

In Hartree atomic units, the speed of light is approximately 137 atomic units of velocity. See the webpage: "8.1: Atomic and Molecular Calculations are Expressed in Atomic Units" for an example of how setting the numerical values of four fundamental physical constants to unity permits simplification of the Hamiltonian.

Cosmology

_{Main article: Chronology of the Universe}

In Big Bang cosmology, the Planck epoch or Planck era is the earliest stage of the Big Bang, before the time passed was equal to the Planck time, $t_P$, or approximately $10^{-43}$ seconds. There is no currently available physical theory to describe such short times, and it is not clear in what sense the concept of time is meaningful for values smaller than the Planck time. It is generally assumed that quantum effects of gravity dominate physical interactions at this time scale. At this scale, the unified force of the Standard Model is assumed to be unified with gravitation. Inconceivably hot and dense, the state of the Planck epoch was succeeded by the Grand unification epoch, where gravitation is separated from the unified force of the Standard Model, in turn followed by the Inflationary epoch, which ended after about $10^{-32}$ seconds (or about $10^{10} \; t_P$).

**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.

See above, the Planck is a physical unit of “action” which sets the scale at which effects of quantum physics are genuinely important and physics is no longer well approximated by classical mechanics/classical field theory. See: "Planck's constant in geometric quantization".

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

Any calculation of the position and momentum of an object (at the quantum level) involves some uncertainty and Heisenberg's uncertainty principle states that complementary properties cannot be observed or measured simultaneously. Knowing the value of a Planck reduces the error in the equation:
$$\sigma _{x}\sigma _{p}\geq {\frac {\hbar }{2}}~~\text{where ħ is the reduced Planck constant, h/(2π).}$$

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

I'll leave you with this answer: When does ℏ→0
provide a valid transition from quantum to classcial mechanics? When and why does it fail?

Instituto de Física Teórica IFTwith titleGravedad y Mecánica Cuántica (César Gómez), posted at December 3rd, 2014. I believe that it doesn't answer all your questions, and you will need a friend knowing Spanish. The ponent César Gómez has other talks in this channel about what is the energy, or also the talk¿Por qué los agujeros negros NO son negros?January 29th, 2019. Are the best talks that I know. $\endgroup$ – user142929 Sep 11 at 11:02in their simplest form. It is an artifact of the fact that, historically, humans didn’t know that energy and frequency are “really the same thing”, much like energy and temperature ($E = k_B T$) and length and duration ($x = ct$) are “really the same thing” (the latter in particular mixing under Lorentz boosts). It’s as if we’d decided to use different units for different forms of energy, and had to introduce arbitrary conversion factors in every equation. $\endgroup$ – user76284 Sep 11 at 20:56