Crisis in Physics vs Computational Physics

The May14 issue of Scientific American asks the following questions:

• Does Physics Have a Problem?
• Crisis in Physics?

These questions naturally present themselves because modern theoretical physicists have driven themselves to search for the truth on scales which are either too small (string theory) or too big (cosmology) to be assessed experimentally. But theory without experiment may well be empty theory and that may be the meaning of the crisis. Of course, advocates of string theory like Lubos, forcefully denies that there is a crisis in physics.

But modern physicists have a new tool to use and that is computational physics, which offers an experimental laboratory without the scale limits of a physical laboratory. 

Computational physics needs computable models, but both quantum mechanics and general relativity are based on models which are not computable, and so there is a lot of work to be done. The question is if modern theoretical physicists have the right training to do this work.     

Wanted: Constructive Physics

                                     Wanted: Constructive version of Schrödinger's equation!

The book Constructive Physics by Y.I. Oshigov has an important message:

• Only in the rebuilding of the gigantic construction of the modern physics in the constructive manner can open doors to the understanding of the complex processes in the sense of exact sciences.
• The modern situation in physics looks like a crisis, and the genealogy of this crisis is the same as for the crisis in mathematics in the first third of the 20th century: this is the crisis in the axiomatic method.
• Today we possess the more exact kit of instruments of the constructive mathematics: algorithms must replace formulas.
• (The multidimensional wave function) harbors serious defects….it does not allow the computation of such functions already for a small number of particles, for example 10, let alone for the more complex systems.
• This complexity barrier is principal. We should not think then that the quantum theory for many bodies gives such reliable answers to questions as it was the case in one particle case.

In short, quantum mechanics based on Schrödinger's equation for a wave function in $3N$ space dimensions for $N$ particles (electrons or kernels) must be given a new constructive form. A real challenge! My answer is given as Many-Minds Quantum Mechanics. 

Popper: Realism vs Quantum Muddle vs Statistics

Karl Popper starts out Quantum Theory and the Schism of Physics, as Vol III of Postscript to Logic of Scientific Discovery, with the following declaration:

• Realism is the message of this book. 
• It is linked with objectivity…with rationalism, with the reality of the human mind, of human creativity, and human suffering.

In Preface 1982: On a Realistic and Commonsense Interpretation of Quantum Mechanics, Popper gives his verdict:

• Today, physics is in a crisis….This crisis is roughly as old as the Copenhagen interpretation of quantum mechanics.
• In my view, the crisis is, essentially, due to two things: (a) the intrusion of subjectivism into physics; and (b) the victory of the idea that quantum theory has reached a complete and final truth.
• Subjectivism in physics can be traced to several great mistakes. One is the positivism or idealism of Mach. Another is the subjectivist interpretation of the calculus of probability.
• The central issue here is realism. That is to say, the reality of the physical world we live in: the fact that this world exists independently of ourselves; that it existed before life existed,…and that it will continue to exist long after we have all been swept away.
• The subjectivist dogma was too deeply entrenched within the ruling interpretation of quantum mechanics, the so-called Copenhagen interpretation… this is how the great quantum muddle started….and the whole terminology, introduced in the early period of the theory, conspired to make the muddle worse and worse.
• Another source of the crisis in physics is the persistence of the belief that quantum mechanics is final and complete.
• Philosophers and physicists have been all too prone under the direct influence of Machian positivism, to take up idealist positions…
• One of the things that this volume of the Postscript tries to to is to review many of the past arguments for idealism - which many current physicists still simply take for granted - and to show their error.  

But Popper, one of the greatest philosophers of science of the 20th century, talked to deaf ears and the crisis in physics is deepening every year…

Another thing is that Popper deepened the crisis be dwelling deeper into the statistical interpretation of the wave function of Born as the basis of the Copenhagen interpretation. Popper thus identified the crisis but then himself got drowned by the muddle...

Quantum Theory: Flight from Realism

The book Quantum Theory and the Flight from Realism by Christopher Norris is introduced by:

• Norris examines the premises of orthodox quantum theory as formulated most influentially by Bohr and Heisenberg….as requiring a drastic revision of principles which had hitherto defined the very nature of scientific method, casual explanation and rational enquiry.
• Putting the case for a realist approach which adheres to well-tried scientific principles of casual reasoning and interference to the best explanation, Norris clarifies the debate…

Norris continues:  

• In this book I examine various aspects of the near century-lonh debate concerning the conceptual foundation of quantum mechanics (QM) and the problems it has posed for physicists and philosophers from Einstein to the present. They include the issue of wave-particle dualism; the uncertainty attaching to measurements of particle location or momentum, the (supposedly) observer-induced "collapse of the wave-function"; and the evidence of remote superluminal interaction between widely separated particles.
• It is important to grasp exactly how the problems arose and exactly why - on what scientific or philosophical grounds - any alternative (realist) contrual should have been so often and routinely ruled out as a matter of orthodox QM wisdom. 

This is an important book with the important mission of bringing realism back to physics after a century  of anti-realist confusion ultimately corrupting all of science and with the adoption of climate alarmism by the American Physical Society as the tragic anti-realist irrational expression. 

Today digital computation opens new roads to bring realism back to physics viewed as analog computation.  See The World as Computation.

Essence of Quantum Mechanics: Energy vs Frequency in Wave Models

In Schrödinger's Equation: Smoothed Particle Dynamics we observed that Schrödinger's equation for Hydrogen atom with one electron (normalized to unit mass and charge) reads

• $i\bar h\dot\psi + H\psi =0$,
• $H\psi =\frac{\bar h^2}{2}\Delta\psi +\frac{1}{\vert x\vert}\psi$,

where $\psi (x,t)$ the complex-valued wave function depending on coordinates of space $x$ and time $t$ with the dot denoting differentiation with respect to time, $H$ is the Hamiltonian operator and $\bar h$ Planck's (reduced) constant.

In terms of the real part $\phi$ and imaginary part $\chi$ of $\psi =\phi +i\chi$, Schrödinger's equation takes the system form

1. $\bar h\dot\phi +H\chi =0$,
2. $\bar h\dot\chi - H\phi =0$.

If $\phi_E(x)$ is an eigenfunction of the Hamiltonian satisfying $H\phi_E =E\phi_E$ with $E$ the corresponding eigenvalue, then the solution can be represented as

• $\phi (x,t)=\cos(\omega t)\phi_E(x)$,     $\chi (x,t)=\sin(\omega t)\phi_E(x)$, 

with $\bar h\omega =E$, which expresses a periodic exchange between the two real-valued wave functions $\phi$ and $\chi$ mediated by the Hamiltonian $H$. The parallel to a harmonic oscillator (with $H$ the identity) is obvious.

We see that the effect of the time derivative term is to connect energy $E$ to (angular) frequency $\omega$ by

• $\bar h\omega = E$, 
• or $h\nu =E$, 

where $h=2\pi\bar h$ and $\nu =\frac{\omega}{2\pi}$ is frequency in Hertz, where $h$ acts as scale factor.

Schrödinger's equation thus sets up a connection between frequency $\nu$, which can be observed as atomic emission lines, and a model of internal atomic energy $E$ as the sum of kinetic and potential energies of eigenfunctions of the Hamiltonian with the connection $\bar h\omega =h\nu = E$. Observations of atomic emission then show to fit with energy levels of the model, which gives support to the functionality of the model. 

The basic connection $\nu \sim E$ can also be seen in Planck's radiation law (with simplified high-frequency cut-off)

• $R(\nu ,T)=\gamma T\nu^2$ for $\frac{h\nu}{kT} < 1$,

where $R(\nu ,T)$ is normalized radiance as energy per unit time, with $\gamma =\frac{2k}{c^2}$, $T$ is temperature and $k$ is Boltzmann's constant, which gives an energy per cycle scaling with $\nu$ and a high frequency cut-off $h\nu$ scaling with atomic energy $kT$.

The connection $h\nu =E$ also occurs in the law of photoelectricity

• $h\nu = P + K$,

where $P$ is the release energy and $K=eU$ is the kinetic energy of a released electron with $e$ the electron charge and $U$ the stopping potential. 

The atomic connection $h\nu =E$ between frequency and energy thus has both theoretical and experimental support,  but it does not say that energy is "quantized" into discrete packets of energy $h\nu$ carried by particles named photons of frequency $\nu$. 

The relation $h\nu =E$ is compatible with wave models of both emission from atoms and radiation from clusters of atoms and if so by Ockham's razor particle models have no role to play.

Atomic emission and radiation is a resonance phenomenon much like the resonance in a musical instrument, both connecting frequency to matter.

Text books state that 

1. Blackbody radiation and the photoelectric effect cannot be explained by wave models.
2. Hence discrete quanta and particles must exist. 
3. Hence there is particle-wave duality.    

I give on Computational Blackbody Radiation evidence that 1 is incorrect, and therefore also 2 and 3. Without particles a lot of the mysticism of quantum mechanics can be eliminated and progress made. 

The Strange Story of The Quantum: Physics as Mysticism

The Strange Story of The Quantum by Banesh Hoffman bears witness to the general public about modern physics as mysticism:  

• This book is designed to serve as a guide to those who would explore the theories by which the scientist seeks to comprehend the mysterious world of the atom.
• The story of the quantum is the story of a confused and groping search for knowledge…enlivened by coincidences such as one would expect to find only in fiction.
• It is a story about turbulent revolution…and of the tempesteous emergence of a much chastened regime - Quantum Mechanics.
• The magnificent rise of the quantum to a dominant position in modern science and philosophy is a story of drama and high adventure often well-nigh incredible. It is a chaotic tale…apparent chaos…nonsensical…intricate jagsaw…major discovery of the human mind.
• Planck called his bundle or quota a QUANTUM of energy…This business of bundles of energy was unpardonable heresy, frightening to even the bravest physicist. Prandtl was by no means happy... But all was to no avail….to Max Planck had fallen the immortal honor of discovering them.
• Einstein insisted...that each quantum of energy  somehow must behave like a particle: a particle of light; what we call a photon…But how could a particle theory possibly hope to duplicate the indisputable triumphs of the wave theory? To go back to anything like the particle theory would be tantamount to admitting that the elaborately confirmed theory of electromagnetic phenomena was fundamentally false. Yet Einstein...was actually proposing such a step.
• It is difficult to decide where science ends and mysticism begins….In talking of the meaning of quantum mechanics, physicists indulge in more or less mysticism according to their individual tastes.
• Perhaps it is this which makes it seem so paradoxical.
• Perhaps there is after all some innate logic in quantum theory.
• The message of the quantum suddenly becomes clear: space and time are not fundamental.
• Out of it someday will spring a new and far more potent theory…what will then survive of our present ideas no one can say…
• Already we have seen waves an particles and causality and space and time all undermined.
• Let us hasten to bring the curtain down in a rush lest something really serious should happen...

Hoffman's book was first published in 1947. Since then the mysticism of modern physics has only become deeper...

Schrödinger's Equation: Smoothed Particle Dynamics

Eigenfunctions of the Hamiltonian for the Hydrogen atom with eigenvalues representing the sum of kinetic and potential energies, with Schrödinger's equation as a smoothed version of the particle dynamics of a harmonic oscillator.  

This is continuation of the previous post How to Make Schrödinger's Equation Physically Meaningful + Computable. Consider the basic case of the Hydrogen atom with one electron (normalized to unit mass and charge):

• $ih\dot\psi + H\psi =0$,
• $H\psi =\frac{h^2}{2}\Delta\psi +\frac{1}{\vert x\vert}\psi$,

where $\psi (x,t)$ the complex-valued wave function depending on coordinates of space $x$ and time $t$ with the dot denoting differentiation with respect to time, $H$ is the Hamiltonian operator and $h$ Planck's constant.

In terms of the real part $\phi$ and imaginary part $\chi$ of $\psi =\phi +i\chi$, Schrödinger's equation takes the system form

1. $h\dot\phi +H\chi =0$,
2. $h\dot\chi - H\phi =0$.

If $\phi_E(x)$ is an eigenfunction of the Hamiltonian satisfying $H\phi_E =E\phi_E$ with $E$ the corresponding eigenvalue, then the solution can be represented as

• $\phi (x,t)=\cos(\omega t)\phi_E(x)$,     $\chi (x,t)=\sin(\omega t)\phi_E(x)$, 

with $h\omega =E$, which expresses a periodic exchange between the two real-valued wave functions $\phi$ and $\chi$ mediated by the Hamiltonian $H$.

We can see 1- 2 as an analog of the equation for a harmonic oscillator $\ddot u+\omega^2u=0$ written in system form (with $h=1$)

• $\dot\phi  + \omega\chi =0$
• $\dot\chi  - \omega \phi = 0$,

where $\phi =\dot u$ and $\chi =\omega u$, with solution

• $\phi (x,t)=\cos(\omega t)$,     $\chi (x,t)=\sin(\omega t)$.  

Here the velocity $\phi =\dot u$ connects to kinetic energy $\phi^2 =\dot u^2$ and $\chi =\omega u$ to potential energy $\chi^2 =\omega^2u^2$ and the dynamics of the harmonic oscillation consists of periodic transfer back and forth between kinetic and potential energy with their sum being constant.

Returning now to the Hydrogen atom, we obtain multiplying 1 by $\phi$ and 2 by $\chi$ and integrating in space the following the energy balance

• $h\frac{d}{2dt}\int\phi^2\, dx + \int \phi H\chi \, dx =0$
• $h\frac{d}{2dt}\int\chi^2\, dx - \int \chi H\phi\, dx =0$,    

where 

• $ \int \phi H\chi \, dx = \int \chi H\phi\, dx =\frac{h^2}{2}\int\nabla\phi\cdot\nabla\chi\, dx +\int\frac{\phi\chi}{\vert x\vert}\, dx$,

which shows upon summation (by the symmetry of $H$) that

• $\frac{d}{2dt}\int\phi^2\, dx =\frac{d}{2dt}\int\chi^2\, dx =0$, 

which allows normalization to  

• $\int\phi^2\, dx = \int\chi^2\, dx = \frac{1}{2}$,
• $\int\vert\psi\vert^2\, dx = 1$, for all time. 

Further, multiplying 1 by $\dot\chi$ and 2 by $\dot\phi$ and subtracting the resulting equations shows that

• $\int (\phi H\phi + \chi H\chi)\, dx$ is constant in time. 

We can now summarize as follows:

A. We see that the solution pair $(\phi ,\chi )$ of 1 - 2 as the real and imaginary part of Schrödinger's wave function $\phi$, represents a periodic exchange mediated by the Hamiltonian $H$ with balancing associated total energies 

• $\int \phi H\phi (x,t)\, dx = \frac{h^2}{2}\int\vert\nabla\phi (x,t)\vert^2dx +\int\frac{\phi^2(x,t)}{\vert x\vert}\, dx$,
• $\int \chi H\chi (x,t)\, dx = \frac{h^2}{2}\int\vert\nabla\chi (x,t)\vert^2dx +\int\frac{\chi^2(x,t)}{\vert x\vert}\, dx$    

as the sum of kinetic and potential energies.

B. We see that Schrödinger's equation for the Hydrogen atom can be viewed as a smoothed version of a harmonic oscillator with the smoothing effectuated by the Laplacian and with $h$ acting as a smoothing parameter.

C. We see that the system form 1- 2 combines the spatial eigenfunction $\phi_E$ with a periodic time dependence without introducing energy beyond the kinetic and potential energies defined by the Hamiltonian, thus associating these energies to frequency as the essence of quantum mechanics.

D. We see that quantum mechanics and Schrödinger's equation can be given an interpretation which closely connects to classical mechanics, as smoothed particle mechanics, which avoids the common mystifications of particle-wave duality, complementarity, wave function collapse and statistics forced by insistence to use a multidimensional wave function defying a direct physical meaning.

Extension to several electrons can be naturally be made following the idea of smoothed particle dynamics. For details see Many-Minds Quantum Mechanics.

Comparing Blackbody Radiation Spectrum to Atomic Emission Spectrum

Planck's constant $h$ appears with different roles in a blackbody radiation spectrum and an atomic emission spectrum.  Blackbody radiation can be described as a near-resonance phenomenon in a forced harmonic oscillator with small damping in a mathematical model of the form

• $\ddot u (t) +\omega^2u(t) -\gamma\dddot u = f(t)\approx \sin(\omega t)$, 

where $u(t)$ is displacement as function of time $t$, $\omega$ is angular velocity, $\gamma$ is a small damping constant, $f(t)$ is forcing in near-resonance with $\omega$ and the dot signifies time differentiation. Here the oscillator described by $\ddot u (t) +\omega^2u(t)$ carries energy as background temperature and the dissipative term $-\gamma\dddot u$ gives off radiation balancing forcing $f(t)$.

The dynamics of near-resonance is quite subtle as explained in detail on Computational Blackbody Radiation showing that Planck's constant enters as a parameter in a high-frequency cut-off reflecting Wien's displacement law.   

Atomic emission can be described as an eigenvalue problem for Schrödinger's equation of the form

• $ih\dot\psi = E\psi$,

where $E$ is a real eigenvalue of an atomic Hamiltonian, with solution

• $\psi (t) =\exp(i\omega t) =\cos(\omega t)+i\sin(\omega t)$, 

which can be seen as a periodic exchange of two forms of energy represented by the real part $\cos(\omega t)$ and the complex part $\sin(\omega t)$ reflecting incoming-outgoing radiation. Atomic emission is thus a direct resonance phenomenon without background temperature. Planck's constant serves to convert angular velocity (angular momentum) $\omega$ to atomic energy $E$ as $\bar h\omega$ with $\bar h=\frac{h}{2\pi}$ with $E$ the sum of kinetic and potential energy. 

We conclude:

1. Blackbody radiation is a near-resonance phenomenon of molecules or collections of atoms modeled as a forced harmonic oscillator with small damping. Collections of atoms vibrate without electron configurations changing energy.   
2. Atomic radiation is a direct resonance phenomenon which can be modeled by a harmonic oscillator. Electrons oscillate between two energy levels representing eigenstates of an atom.

In both cases $h$ enters combined with frequency $\nu$ in the form $h\nu =\bar h\omega$ as quantity of energy serving in a threshold condition in blackbody radiation, and as an energy eigenvalue in atomic emission.

The value of $h$ as setting a conversion scale between light energy and electronic energy can be determined by the photoelectric effect and can then be used by definition in blackbody radiation and Schrödinger's equation. 

  

Water Dam Analog of Photoelectric Effect

                               Open sluice gates in the Three Gorges Dam in the Yangtze River.

Einstein was awarded the 1921 Nobel Prize in Physics for his "discovery of the law of the photoelectric effect", connecting frequency $\nu$ of light shining on a metallic surface with measured potential $U$:

• $h\nu = h\nu_0 + e\, U$ or $h(\nu -\nu_0) = e\, U$,

where $h$ is Planck's constant with dimension $eVs = electronvolt\,\times second$,  $\nu_0$ is the smallest frequency releasing electrons and $U$ in Volts $V$ is the stopping potential bringing the current to zero for $\nu >\nu_0$ and $e$ is the charge of an electron. Observing $U$ for different $\nu$ in a macroscopic experiment shows a linear relationship between $\nu -\nu_0$ and $U$ with $h$ as scale factor with reference value 

• $h = 4.135667516(91)\times 10^{-15}\, eVs$,

with Millikan's value from 1916 within $0.5\%$.

Determining $h$ this way makes Einstein's law of photoelectricity into an energy conversion standard attributing $h\nu$ electronvolts to the frequency $\nu$, without any implication concerning the microscopic nature of the photoelectric effect.

The award motivation "discovery of the law of the photoelectric effect" reflected that Einstein's derivation did not convince the committee as expressed by member Gullstrand: 

• When it was formulated it was only a tentatively poorly developed hunch, based on qualitative and partially correct observations. It would look peculiar if a prize was awarded to this particular work. 

To give perspective let us as an analog of the law of the photoelectric effect consider a water dam with sluice gates which automatically open when the level of water is $\nu_0$.  The sluice gates will then remain locked as long as the water level $\nu <\nu_0$.  Lock the sluice gates and let the dam fill to some water level $\nu >\nu_0$ and then unlock the sluices. The sluices will then open and water will flow through under transformation of potential energy into kinetic energy. Assuming the work to open the sluices corresponds to a level loss of $\nu_0$, a net level of $\nu -\nu_0$ potential energy will then be transformed into kinetic energy by the water flow through the sluices. 

The dam can be seen as an illustration of the photoelectric effect with the water level corresponding to frequency $\nu$ and the gravitational constant corresponding to $h$ and the width of the dam corresponding to the amplitude of incoming light. If $\nu <\nu_0$ then nothing will happen, if $\nu >\nu_0$ then the kinetic energy will scale with $h\nu$ and the total flow will scale with the width of the dam.

Notice that noting in this model requires the water to flow in discrete lumps or quanta. The only discrete effect is the threshold $\nu_0$ for opening the sluices.

Universal Quantum of Action: Standard Without Universality

In recent posts on we have seen that Plank's constant $h$ in physics text books being presented as a universal quantum of action as a smallest "packet of action" as a fundamental constant of fundamental significance in the "quantized" world we happen to be part of, in fact is nothing but a conversion standard between two measures of energy, in terms of frequency $\nu$ in periods per second and electronvolt (eV), determined by Einstein's law of photoelectricity

• $h(\nu - \nu_0) = e\, U$,

where $\nu_0$ is the smallest frequency releasing electrons from a metallic surface upon exposure of light, $U$ in Volts $V$ is the stopping potential bringing the current to zero for $\nu >\nu_0$ and $e$ is the charge of an electron. Observing $U$ for different $\nu$ shows a linear relationship between $\nu -\nu_0$ and $U$ with $h$ as the scale factor measured in $eVs$ $electronvolts\times second$ as $energy \times time$ as action. The reference value obtained this way is 

• $h = 4.135667516(91)\times 10^{-15}\, eVs$,

with Millikan's value from 1916 within $0.5\%$. Determining $h$ this way makes Einstein's law of photoelectricity simply into a conversion standard (that is, a definition) of energy attributing $h\nu$ electronvolts to the frequency $\nu$. Another way of finding the conversion from frequency to electronvolt is using a Josephson junction.

We now turn to Schrödinger's equation

• $i\bar h\frac{\partial\psi}{\partial t}+H\psi=0$,

where $\bar h=\frac{h}{2\pi}$ is Planck's reduced constant as conversion from periods $\nu$ per second to angular velocity $\omega$ with $h\nu =\bar h\omega$, and $H$ is a Hamiltonian of space dependence. An eigenvalue $E$ of the Hamiltonian represents energy with $\psi_E$ a corresponding space dependent eigenfunction satisfying $H\psi_E =E\psi_E$ and $\exp(i\omega t)\psi_E$ a corresponding solution of Schrödinger's equation with 

• $h\nu = \bar h\omega =  E$, 

expressing energy in terms of frequency. We see that the appearance of $\bar h$ with the time derivative in Schrödinger's equation accounts for the energy conversion and is completely normal and without mystery. 

Next, we consider the space dependent Hamiltonian in the basic case of the Hydrogen atom:

• $H\psi =  \frac{\bar h^2}{2m}\Delta\psi +  \frac{e^2}{r}\psi$  

where $\psi =\psi (x)$ with $x$ a space coordinate, $r =\vert x\vert$, and $m$ is the mass of the electron. Normalising by changing scale in space $x=a\bar x$ and time $t=b\bar t$, we obtain the Hamiltonian in normalized atomic units in the form

• $\bar H = \bar\Delta + \frac{2}{\bar r}$ with smallest eigenvalue $1$, 
• $a=\frac{\bar h^2}{me^2}$ as $Bohr\, radius$,
• $b=\frac{\bar h2a}{e^2}$ as $Bohr\, time$ with $\omega =\frac{1}{b}$ angular velocity
• $E =\frac{e^2}{2a}$ as $Rydberg\, energy$.

We now observe that

• $E\, b = \bar h$,

• $E = \bar h\, \omega$, 

which shows that the also the space dependent part of Schrödinger's equation is calibrated to the energy conversion standard. 

Finally, Planck's constant also appears in Planck's radiation law and then in the high-frequency cut-off factor

• $\frac{\alpha}{\exp(\alpha )-1}$
• $\alpha = \frac{h\nu}{kT}$,

where $k$ is Boltzmann's constant and $T$ temperature. We see that again $h\nu$ appears as an atomic energy measure with a value that is not very precisely determined in its role in the cut-off factor.

The value of $h$ from photoelectricity can then serve also in Planck's law.

We conclude that Planck's constant $h$ is a conversion standard between two energy measures and as such has no meaning as a universal quantum of action or as integral multiples $nh\nu$ with $n=1,2,3,..$ of special significance other than by connection to eigenfunctions and eigenvalues.   

Ultimately, what is measured are atomic emission spectra in terms of frequencies and wave lengths which through Planck's constant can be translated to energies expressed in electronvolts (or Joule). Nothing of the internal atomic structure (in terms of $e$ and $m$) enters into this discussion.

Planck introduced $h$ in a statistical argument in 1900 long before atoms were known, Einstein picked up $h\nu$ in his 1905 article on photoelectricity, before atoms were known, and Schrödinger put $h$ into his equation in 1926 to describe atoms. This line of events supports the idea that Planck's constant $h$ is a convention without any universal significance.

Understanding the real role of Planck's constant may help to give Schrödinger's equation a physical interpretation which is free from mysteries of "quantization" and statistics. Versions of Schrödinger's equation based on an idea of smoothed particle mechanics then naturally present themselves, with $h$ acting as a smoothing parameter.

PS Notice that the fine structure constant $\alpha = \frac{e^2}{\bar hc}=\frac{1}{137}$ can be expressed as $\alpha =\frac{2}{c}\frac{a}{b}$ which shows that $\alpha$ relates $Bohr\, speed\, =\frac{a}{b}$ to the speed of light $c$. This relation is viewed to be fundamental, but why is hidden in mystery.