1)The Planck constant is a fixed universal constant, however it is very instructive sometimes " to play" with this constant in order to see, what would be the consequences of changing its values.

2) The Planck constant, together with mass of a particle and parameters of the fields, which act on a quantum particle, determines characteristic length and temporal scales of a wave, which describes the quantum motion. The smaller is the Planck constant, the smaller is such a scale. Roughly speaking, this characteristic spatial scale(the wave-length) determines the distance range, where all quantum phenomena manifest themselve, in particular this is the region in which the probability density of a particle is smeared out around classical trajectory. Thus, the smaller is this scale, more localized is a particle motion around the classical trajectory.

3) The mathematically strict sense has the so called WKB approximation, which shows the relation between quantum and classical mechanics. An artificial procedure of making the Planck constant tend to zerocan be thought of as a trick to make clear the limits of its validity.

1)The Planck constant is a fixed universal constant, however it is very instructive sometimes " to play" with this constant in order to see, what would be the consequences of changing its values.

2) The Planck constant, together with mass of a particle and parameters of the fields, which act on a quantum particle, determines characteristic length and temporal scales of a wave, which describes the quantum motion. The smaller is the Planck constant, the smaller is such a scale. Roughly speaking, this characteristic spatial scale(the wave-length) determines the distance range, where all quantum phenomena manifest themselve, in particular this is the region in which the probability density of a particle is smeared out around classical trajectory. Thus, the smaller is this scale, more localized is a particle motion around the classical trajectory.

3) The mathematically strict sense has the so called WKB approximation, which shows the relation between quantum and classical mechanics. An artificial procedure of making the Planck constant tend to zerocan be thought of as a trick to make clear the limits of its validity.

I consider it as the condition which allows us to apply classical mechanics to the system whenever we may neglect the Planck constant, or equivalently the Broglie wavelength of the particle is much shorter than the size of the particle.

Thank you for the answer. I perfectly agree with point 1 and 2. About point 3, it is iilluminating to see what happens in some hypothetical cases. But in the quantum hydrodynamic analogy (QHA) of Madelung if we put the Plank's constat to zero we immeduiately destroy the stationary states that are the eigenstates of Schroedinger representation and strictly speaking in mathematical sense we alter the nature of the equation. Moreover, putting to zero the Plank's constat the QHA equations becomes the classical (deterministic) equations of motion. But how it borns the everyday realized classical reality that is compatible with the quantum one?

Sorry Piero, I am not familiar with QHA. In my opinion, the classical formulation of a system cannot be obtained by just changing the value of the Planck's constant in the quantum formulation to zero, but by applying the classical mechanics principles to the system. However, it would be very nice if you can realize it in your new approach - combining the deterministic (classical) and stochastic (quantum) into one entity from the beginning.

Dear Johannes Wirjawan, here there is a paper accepted for pubblication on Foundation of Physics:: Deterministic quantum mechanics and stochastic classical (large scale) mechanics from a single theory

Article Can fluctuating quantum states acquire the classical behavio...

Mathematically speaking this weir statement only means that the numerical value of teh constant should be small when when expressed in the units of action characteristic of the class of systems under consideration (Coutinho)

Basically you agree with the answer of Alexei Voronin (and me too) , but there still a problem: The WKB approximation is non-local and the quantum superposition of states does not exist  in the classical limit .

Crudely it means that quantum mechanics is the correct way to describe motion at subatomic scales and classical mechanics is only an approximation. We also sometimes call it just "classical approximation". The reason for doing classical approximation is that certain intuitive concepts such as `exact momentum' or 'exact position' of an object have evolved in physical sciences during pre-quantum mechanics era. In a quantum world these quantities can never be exactly measured, and hence they loose their physical significance in a considerable way.

Instead of taking the limit  h → 0 there is a nicer way to look at the classical correspondence of quantum laws, which is via Ehrenfest's theorem. For example  consider the two equations:   m d/d t=

and d

/dt=-. The first one says that rate of change of the expectation value of position is the expectation value of momentum. This is not exactly Newton's second law, however it shows a classical correspondence. Similarly, the second equation corresponds with the classical idea of force being the negative gradient of a potential function.

Hi, 

Yes it is a common mistake because as Mr. Piero Chiarelli  said " it is a fixed universal constant" !

in fact when the action S grow the value of S/h grow so we go in this case to the classical limit and mathematically the same thing happened when h tends to zero ! and this is the common mistake here !

so the right thing is to said S tends to infinity not h tends to zero !

thank you.

Varying Planck''s constant simply means you are changing units for x and /or p. Without being so extreme as letting h->0 there are interesting issue related to vetry small variations of h. (See the work of the astrophysicist John Webb et al.). Also see my Phys.Lett.A paper

M. de Gosson, Mixed Quantum States with Variable Planck Constant, Phys. Lett. A 381(36), 3033--3037 (2017)

thanks

The Planck constant is adjusted to the geometry of phase space, being indeed fixed if one considers the whole position-momentum plane, which is infinitely degenerated. But one can impose boundary periodicity conditions on the states, which restrict the model to a torus. Now we are in a compact surface, whence of finite degeneration N (number of degrees of freedom). The number of degrees of freedom in a torus is then inverse proportional to the Planck constant. Here, the limit case of N ->infinity is equivalent to h->0. It leads to the collapse of the phase space by violating the finite degeneration condition, providing a rigorous mathematical description of the transition to classical mechanics. The torus model is becoming the most popular phase space in quantum physics, leading, for instance, to the discovery of topological states of matter, awarded with the 2016 Nobel Prize in Physics.

Luis Daniel Abreu. The topology of the torus is reminiscent on the lattice fermion doubling phenomena, -- the so-called Nielsen and Ninomiya no-go theorem https://link.springer.com/content/pdf/10.1007/BF01403500.pdf. After I've read Haldane's paper Article Model for a Quantum Hall Effect without Landau Levels: Conde...

With respect to the scaling behavior of Planck's constant, maybe the best explanation for it is the renormalization approach of Kenneth G. Wilson -- see, for instance, https://www.nobelprize.org/prizes/physics/1982/wilson/lecture/

On the page 23. of Wilson's nobel lecture you may find an intriguing quote:

"

There is a murky connection between scaling ideas in critical phenomena and Mandelbrot’s “fractals” theory - a theory of scaling of irregular geometrical structures (such as coastlines).

"

In more plain terms I think that one must distinguish between the mathematical model and the physical system, which admits more than one mathematical model, with difference conventions for the Planck constant. For instance, a classical Coulomb plasma with N interacting charged-liked particles obeying Fermi statistics can be modelled as a log-gas on the (unbounded) plane (the plane of complex numbers, for instance). Since such a plane allows to accomodate an unbounded number of particles, one can fix the Planck constant without violating the Pauli exclusion principle, where no more than one electron per cell (of size proportional to the Planck constant) is allowed. Now, if we consider the same N interacting fermions in a 2-D torus, which is the compactification of a single cell in the complex plane, then, if the system has more than one particle we would be violating the Pauli exclusion principle by definition! The solution is to consider the single cell as the whole space and adjust the Planck constant so that it is inverse proportional to the number N of particles. This allows to model the same Coulomb plasma with N interacting fermions in a single cell. Thus, to increase the number N of particles one is required to make the Planck constant small. In general, torus models allow to make a mathematical precise interpretation of the limit h->0.