The Deep Secret of $E=h\nu$ Uncovered = 0

The value of Planck's constant $h$ is supposed to carry a deep secret of the atomic physics captured in the Schrödinger Equation SE of Quantum Mechanics QM as the foundation of modern physics. A deep secret of a microscopic world which is fundamentally different from the macroscopic world we can fathom by direct experience. A strange world of the modern physics emerging in the beginning of the 20th century, which "nobody understands" including the physical meaning of Planck's constant $h$. 

In the new 2019 SI standard of units, the value of $h$ is specified to be exactly $h=6.62607015\times 10^{−34}$ Joule-seconds, which is a very small number viewed to hide a deep secret, while appearing as an arbitrary unit conversion factor. 

Let us seek to untangle the secret in detail. We recall the message of modern physics of the existence of a smallest quantum of energy $h\nu$ associated to a wave of frequency of $\nu$ showing that the microscopic world is discrete and not continuous like the macroscopic world so well described by continuum mechanics. More precisely, light as a wave phenomenon is viewed to consist of a stream of light particles named photons each one carrying exactly the energy $h\nu$. Mind boggling, suggesting some deep secret.

Let us now trace the connection to SE for the Hydrogen atom taking the form: 

• $ih\frac{\partial\psi}{\partial t} + H\psi =0$                (SE)

where $\psi (x.t)$ is a complex-valued wave function depending on a 3d spatial coordinate $x$ and a time variable $t$ and $H$ is a (Hermitian) operator acting on $\psi$ with a discrete spectrum of real eigenvalues $E$ representing energies of normalised eigenfunctions $\Psi (x)$ satisfying $H\Psi =E\Psi$, which give wave solutions to (SE) of the form 

• $\psi (x,t)=\exp(i\frac{E}{h}t)\Psi (x)=\exp(i\nu t)\Psi (x)$ with
• $\nu =\frac{E}{h}$ or $E=h\nu$.  

We thus see a direct connection between the smallest quantum of energy $h\nu$ and energies $E=h\nu$ of eigenstates/functions of a Hydrogen atom, as a direct reflection of the form of (SE) including a first time derivative: Energy $E$ scales linearly with frequency $\nu$. 

The other way around, one can see (SE) as being formed by Schrödinger to include the connection $E=h\nu$ between energy $E$ and frequency $\nu$ (as a linear dispersion relation), because that fits with observed spectrum of the Hydrogen atom. Mathematical modeling to fit observation.   

More precisely, the spectrum of a Hydrogen atom comes out from differences of eigenvalues/energies $\Delta E$ translated to frequencies by $\Delta E =h\nu$. 

The basic heuristic idea of Einstein in 1905 was that  the energy of the electron of a Hydrogen atom can "jump" from one energy level to another by receiving/delivering exactly one photon of energy $\Delta E =h\nu$ in radiative equilibrium with light of frequency $\nu$: 

• Transition from one energy level to another with an energy jump $\Delta E$ of the electron of a Hydrogen atom involves receiving/delivering exactly the energy $\Delta E=h\nu$ of one photon of frequency $\nu =\frac{E}{h}$. 

This idea is supposed to convince us that the world of a Hydrogen atom is discrete operating with discrete chunks of energy $h\nu$ carried by discrete light particles/photons.

But this is an invented discreteness: SE is a continuum model of classical form in a wave function $\psi$ with $\vert\psi (x,t)\vert^2$ representing charge density, which has a discrete set of eigenvalues just like a vibrating string. The association of energy to frequency by $E=h\nu$ is simply a scaling of between energy and frequency with a scaling factor of $h$ with a value depending on choice of units.

From (SE) it follows that size of a Hydrogen atom scales with $h^2$ which connects to the discreteness of a Hydrogen atom with its only electron, which is described by the continuous model (SE) of classical continuum form. 

We thus find nothing fundamentally different from classical continuum mechanics point of view in the (SE) model of a Hydrogen atom in terms of a charge density. The association of an energy jump $\Delta E =h\nu $ to exactly one photon of frequency $\nu$ lacks real physical meaning and is just a convention which appeared as a heuristic idea in Einstein's mind in 1905. Planck's constant $h$ does not say that the microscopic world is discrete making it fundamentally different from a continuous macroscopic world. Planck's constant has a meaning as setting the physical scale of a Hydrogen atom, but not as a deep secret about the world. Of course atoms have spatial size just as specific macroscopic material objects with specific spatial extension. A Hydrogen atom is a like a continuous string of a violin of certain length and tension. No quantum.

In short, the quantum world of a Hydrogen atom can be understood in terms of classical continuum mechanics. 

The split appears when generalising (SE) to atoms with $N>1$ electrons following the route of Standard QM by Born-Bohr-Heisenberg into a linear wave equation in $3N$ spatial dimensions, with the wave function given a probabilistic unphysical meaning which makes StdQM "not understandable".

RealQM offers a fundamentally different generalisation without split away from classical continuum mechanics, which is understandable.  

Summary: 

1. Planck's constant $h$ serves as a formal conversion factor between energy $\Delta E$ and frequency $\nu$ with $\Delta E=h\nu$ in the setting of a radiating  Hydrogen atom. The size of a Hydrogen atom scales with $h^2$ which gives the specific value of Planck's constant $h$ a physical meaning, which is not some deep secreted of smallest quantum of energy. 
2. The generalisation to any atom by StdQM leaves classical continuum mechanics into a probabilistic quantum world "nobody can understand" where Planck's constant appears as a deep secret.
3. RealQM offers a generalisation staying within the form of classical continuum mechanics which "everybody can understand" where Planck's constant remains the simple conversion factor of 1. = No Secret = 0.
4. RealQM appears as "Quantum Mechanics without Quantum" which opens to unification with electromagnetics-Newtonian gravitation into a Unifies Field Theory as unfinished dream of Einstein. Let's get to work! 

Brief Quantum Story 1900 - 1905 - 1925 - 2025

The first form of the Schrödinger equation presented by Schrödinger in 1926  offered a mathematical model of the Hydrogen atom with one electron in the form of a linear wave equation of classical continuum mechanical form in terms of a (complex valued) wave function $\psi (x,t)$ depending on a 3d space coordinate $x$ and a time coordinate $t$ with $\vert\psi (x,t)\vert^2$ representing charge density at $(x,t)$ with total unit electron charge. The corresponding classical eigenvalue problem with discrete eigenvalues showed to fit exactly with the observed discrete spectrum of Hydrogen. 

The success was immense and Schrödinger rocketed to fame by giving birth to a new form physics of atoms to be named Quantum Mechanics QM, but it was not Schrödinger who coined the concept of quantum, and in fact he disliked it from the bottom of his heart:

• If all this damned quantum jumping were really here to stay, I should be sorry I ever got involved with quantum theory.

Recall from recent posts that that the quantum was the result of desperate actions by first Planck in 1905 introducing a quantum of energy $h\nu$ associated with radiation of frequency $\nu$ with $h$ a very small constant indicating that a quantum of energy is a very small quantity. Einstein followed in 1905 by suggesting that light of frequency $\nu$ could be thought of (heuristically only!) as a stream of light particles or photons each photon carrying exactly one quantum of energy $h\nu$. Vivid fantasy.

Then 20 years passed with the idea of the quantum of energy $h\nu$ kept as a form of easy fix to explain blackbody radiation and photoelectricity believed to be impossible within classical continuum physics. 

Schrödinger gave his revolutionary Hydrogen article the title "Quantisation as Eigenvalue Problem" thus connecting back to the a concept of "quantisation" suggested earlier by Bohr and de Broglie and coming out in Heisenberg's matrix mechanics, which he now reformulated as an eigenvalue problem of the form of classical continuum physics. Schrödinger's goal was to show that the new quantum mechanics of atoms in fact could take the form of classical continuum mechanics. Schrödinger never gave up that goal but could only reach it in the case of the Hydrogen atom with one electron, since already the Helium atom with two electrons appeared to require a new model outside classical continuum mechanics, and so Schrödinger left QM in 1928 disgusted, to let it be formed by Bohr-Heisenberg as a fundamentally new form of physics as QM, which has come to serve as the foundation of modern physics, without Schrödinger the founder of QM 

But back to Schrödinger's equation for the Hydrogen atom, which does not ask for any quantum of energy $h\nu$ carried by a photon. It is a classical continuum physics eigenvalue problem with discrete spectrum of eigenvalues $E_1<E_2<E_3,...$ representing energies of excited states staring from a ground state energy $E_1$. Differences of eigenvalues $E_n-E_m$ with $E_n>E_m$ match with frequencies $\nu$ in the observed spectrum of Hydrogen under scaling with a certain constant $h$. There is here only a superficial connection between a classical continuum physics eigenvalue problem and the new concept of quantum of energy scaling with frequency $\nu$.  Schrödinger managed to turn quantisation into a classical eigenvalue problem. 

Once the Hydrogen atom was secured within classical continuum physics without the real need of any quantum of energy $h\nu$, which he disliked so much, Schrödinger took on the Helium atom with two electrons. And this is where history took a turn with far-reaching consequences into our time. Instead of staying within classical continuum physics, Schrödinger and everyone else took the easy way out by generalising from one electron to many electrons by a purely formal procedure leaving out physics. For some reason, Schrödinger and everyone else missed the possibility demonstrated in Real Quantum Mechanics RealQM of staying within classical continuum physics without need for any quantum of energy. 

The result of taking the easy formal route when generalising Schrödinger's equation from one electron to many and so form StdQM as the textbook version of QM today, is that "nobody understands QM", simply because the easy formal route does not make sense from physical point of view. What does not make sense cannot be understood, and if something cannot be understood, it is because it does not make sense. 

What about giving RealQM a try, if you want to understand QM? RealQM offers an understanding of blackbody radiation and photoelectric effect with a frame of classical continuum physics!

Recall this statement by Lieb and Thirring from this post concerning the easy way out:

• An important historical point is to be noted here. It might have been thought that the correct generalization for N particles is to use N functions of one variable instead of one function of N variables. 
• Such a ‘wrong turn’ did not happen historically, which is, after all, remarkable.

What did not happen was RealQM and so when it now happens 100 years later it may be remarkable.

Shell Structure: StdQM vs RealQM

A fundamental conception of atom physics is that the electrons surrounding an atomic kernel are arranged in a sequence of shells $S_n$ for $n=1,2,3,...$ with $S_n$ containing $2n^2$ electrons when filled, which gives the Periodic Table with periods 2, 8, 8, 18, 18, 32,,,  including repetitions.

A fundamental question in Standard Quantum Mechanics StdQM is if the shell structure of the Periodic Table is carried by solutions of the Schrödinger equation for the atom? Can an answer be given when such solutions are uncomputable because they involve $3N$ spatial dimensions for an atom with $N$ electrons? 

• Does the shell structure of an atom come out from StdQM? 
• Is the Periodic Table well explained by StdQM? 

The view of Eric Scerri as world leading expert on the subject is summarised as follows by chatGPT:

• In short, Scerri agrees that quantum mechanics supplies the essential skeleton of the periodic system, but he rejects the stronger claim that Schrödinger’s equation alone “explains” the periodic table in a purely deductive sense. The full story, in his view, requires a blend of quantum theory, empirical ordering principles, and chemical reasoning.

OK, so the answer is No rather then Yes. 

On the other hand, in RealQM as an alternative to StdQM, the shell structure comes out in a deductive sense as solution to a non-overlapping electron packing problem resulting in the shell structure of the Periodic Table. Details are given in the RealQM book.