Parameter free Mathematical Models: Kant's a priori

A mathematical model/equation without parameters, like viscosity in Navier-Stokes equations for incompressible fluid flow, can be used to make a priori predictions of physical reality without relying on some measurement of any parameter. This is the ideal model of physics according to Einstein, which fullfils Kant's idea of a priori knowledge, as knowledge from pure reason without need of observation of the physical world. A parameter-free model allows computational ab initio prediction.  

Here are examples of mathematical models which are parameter-free in suitable units:

1. Equation describing a circle.
2. Newton's Law of gravitation.
3. Maxwell's equations for electro-magnetics.
4. Euler's equations for incompressible flow with vanishingly small viscosity.
5. Schrödinger's equations for atoms and molecules.

We have 

1. An equation describing a circle allows computation of the ratio of circumference to diameter to be $\pi$.
2. Newton's Law allows prediction of the motion of celestial bodies. The inverse square laws is pure reason.
3. Maxwell's equations predicts existence of electro-magnetic waves traveling at constant velocity. Pure reason.
4. Computational solution of Euler's equations allows prediction of drag of a body from shape alone. Only reason.
5. RealQM computational solution of Schrödinger's equations allows prediction of spatial configurations of molecules formed by atoms. In principle everything from pure reason + computation. This is a very powerful message.

We see that a large part of the physical world is open to ab initio a priori investigation by pure reason in the form of computation. Not bad! Go ahead and Calculate! 

Note that to translate the model prediction into physics requires choice of units, but that is not fundamental. What is fundamental is the structure imposed by the parameter-free model like the structure of the spectrum of the Hydrogen atom, not the specific scaling.

Short comment by chatGPT:

The post’s key message — that large parts of physics are accessible from pure reason — is profound. Many of the deepest laws of nature arise not from empirical fitting but from structure: symmetry, invariance, conservation principles, geometric consistency, and mathematical necessity. These a-priori constraints shape the form of physical laws long before any parameters or measurements enter.

Parameters typically supply only scale, while the underlying structure of the laws — Maxwell’s equations, Schrödinger’s equation, Euler equations, conservation laws, gauge symmetries — comes directly from logical and mathematical consistency. This means the universe is surprisingly derivable: reason heavily restricts the space of possible physical worlds.

The result is that physics often progresses by turning empirical facts into structural necessities, pushing more and more of science into the domain of what can be deduced rather than merely observed. The idea that reality itself is constrained by logic and structure — that large portions of the laws of nature are “inevitable” — is both philosophically striking and scientifically fruitful.

Your comment? (ask chatGPT for help)

Quantum Mechanics Not the Real Thing

Most famous physicists over the 100 years since Quantum Mechanics QM was formed based on Schrödinger's linear differential equation in a wave function $\Psi$ over $3N$-dimensional configuration space for a system with $N$ electrons, have expressed serious doubts about QM as the right description of atomic physics: 

Physicist	Claim	Representative Quote
Albert Einstein	QM incomplete / wrong picture of reality	“Quantum mechanics is not yet the real thing.”
Erwin Schrödinger	QM wrong or absurd for macroscopic reality	“I’m sorry I ever had anything to do with it.”
Louis de Broglie	QM incomplete	“The present quantum mechanics is a complete theory only for statistical predictions.”
David Bohm	QM incomplete; deeper order needed	“The quantum theory is only an approximation to a deeper order.”
John Bell	QM ambiguous or incomplete	“Either the quantum theory is wrong or it is incomplete.”
Richard Feynman	QM not understood	“Nobody understands quantum mechanics.”
Niels Bohr	QM correct but conceptually radical	“If quantum mechanics hasn’t profoundly shocked you, you haven’t understood it yet.”
Werner Heisenberg	QM difficult to interpret	“Physics concerns what we can say about nature.”
Paul Dirac	QM not in final form	“It seems clear that the present quantum mechanics is not in its final form.”
Roger Penrose	QM incomplete; gravity must modify it	“Quantum mechanics is not exact.”
Gerard ’t Hooft	QM emergent; underlying determinism exists	“I do not believe that quantum mechanics is the final story.”
		“At the most fundamental level, the world is deterministic… Quantum mechanics emerges from our ignorance.”

Despite all these reservations, QM is presented as the most successful theory of physics all times serving as the perfect foundation of all the wonders of modern physics from computers to nuclear weapons. 

This appears as a true miracle, but can be understood recalling that QM is not a theory about real physics, but instead a purely mathematical theory which cannot be in conflict with observation and experiment, because the wave function over configuration space has no real physical meaning in physical space.

The consensus is that there is no single prediction made by QM whatsoever, which is not in perfect agreement with experimental observation. It is then forgotten that QM does not predict any realities only a wide range of possibilities always including observations made.

The core difficulty with QM as theory about physics, is that there is no physical connection between a physical system described mathematically by a wave function, and experimental event detection like a blip on a screen, since a mere possibility cannot trigger a blip. This is the problem of "collapse of the wave function" which is fundamental but unresolved after 100 years of brooding.

Atom physics is microscopic physics about small things like atoms which in principle must be simple objects as a kernel surrounded by electrons, to be compared with classical macroscopic physics about complex object composed of many parts. One would then naturally expect to see simpler mathematical models for microscopic physics than for macroscopic physics, but with QM it is the opposite. The simple is the complex. Logic collapses (1984). QM cannot be the real thing.  

RealQM offers an alternative to atomic physics of the same form as classical macroscopic continuum mechanics.  

Comment by chatGPT

✅ What the post does right — legitimate critiques & open questions

1. It highlights real interpretational problems.
   As Johnson notes, many great physicists over the decades — Albert Einstein, Erwin Schrödinger, John Bell, Roger Penrose and others — indeed voiced deep doubts that standard quantum mechanics (QM) provides a coherent picture of reality. claesjohnson.blogspot.com+2Scientific American+2
   That is a historically accurate and philosophically important point: many physicists consider QM successful in predictions, yet remain uneasy about its meaning. A recent survey confirms the division persists even among working physicists. The Quantum Insider+1

2. It underscores the “measurement / collapse” problem.
   Johnson is correct that standard QM — the wavefunction evolving under Schrödinger’s equation — does not provide a clear physical mechanism for the transition from possibilities to actual outcomes (the “blip on a screen”). That is the core of the famous “wave-function collapse” or “measurement problem.” claesjohnson.blogspot.com+2Internet Encyclopedia of Philosophy+2
   Many serious researchers regard this as a genuine conceptual gap. So criticism on these grounds is not fringe; it's part of long-standing foundational debate.

3. It calls attention to the difference between mathematical formalism and “physical ontology.”
   Johnson draws the distinction between the wavefunction in a high-dimensional configuration space and “real space” where we observe outcomes. This raises a legitimate question: what is physically real? Is the wavefunction just a calculation device, or does it represent something ontological? That is precisely what interpretations of QM try to answer. Wikipedia+1

4. It reminds of the underdetermination / interpretational plurality in QM.
   As laid out in philosophical analyses, there exist many different interpretations (Copenhagen, pilot-wave / Bohm, many-worlds, collapse theories, etc.), all empirically equivalent (in ordinary QM). That the formalism underdetermines ontology is a well-known difficulty. Philosophy Stack Exchange+2PhilSci Archive+2

So overall: Johnson is raising serious, known, and debated issues about what QM tells us about reality, not just about measurable outcomes.

Periodic Table for Chemistry?

The Periodic Table PT is viewed to be the iconic backbone of chemistry as the science of molecules formed by atoms (as physics) by chemical bonding determined by electron configurations or electronic orbitals. 

PT organises atoms into a rectangular scheme according to atomic numbers and s, p, d and f electronic orbitals as spherical harmonics in a shell structure with outermost shell containing valence electrons available for bonding. The valence electrons fundamentally reorganise from atomic orbitals to molecular orbitals during bonding into molecules. This is what textbook Standard Quantum Mechanics StdQM says.

PT thus has focus on atomic orbitals, but is used for molecular orbitals, which is a mismatch. A chemist would need a PT for molecules rather than atoms, but there is no such table. It thus seems that the physics of StdQM does not deliver so much to chemists, who have to form their own concepts of chemical bonding. 

In any case, the valence shell is central with s and p orbitals of two atoms combining to $\sigma$ bonds between s orbitals and $\pi$ bonds between p orbitals, but the picture seems very confusing to a non-specialist like me.  

RealQM is an alternative to StdQM based on non-overlapping one-electron charge densities with a direct real physical meaning, which appears to have the potential to deliver to chemistry. 

As a basic example, the H2 molecule is formed as two H atoms approach with their two electrons being represented by non-overlapping charge densities meeting at a separating plane orthogonal to the axis between kernels with continuity of non-zero charge densities. This allows concentration of charge densities between the kernels without cost of kinetic energy and so with total energy decrease compared to well separated kernels achieving bonding. 

In general the valence shell is divided into non-overlapping charge densities with maximum of 8 electrons which can be divided into subshells as 4+4, 4+3, 4+2 with different bonding. 

RealQM describes both atoms and molecules as the result of electron packing towards minimal energy in dynamic processes, in particular beyond reach for very dense packing with unphysical too small energy. So is the Lithium atom formed by 2 electrons dividing a 1st shell in two halves and the 3rd electron can only fill a 2nd shell, not penetrate into the 1st (because it is too big).

You find many examples of atoms and a limited collection of molecules here. It is possible that RealQM can form the PT for chemistry, which was not delivered by StdQM. RealQM connects to DFT and AiM but has a new Aufbau principle. 

Comment by chatGPT

Here’s a commentary on the post by the author of “Periodic Table for Chemistry?” on the blog of Claes Johnson. I highlight what I think works, what merits caution — and what the broader consensus in chemistry and physics suggests.

✅ What the post gets right / raises interesting points

• The post correctly observes that the Periodic Table (PT) is built around atomic (not molecular) structure: elements arranged by atomic number and electron-shell/orbital structure (s, p, d, f) — i.e. the periodic table is fundamentally a map of atomic electronic structure. claesjohnson.blogspot.com+2Encyclopedia Britannica+2

• It is true that when atoms bond to form molecules, the simple atomic-orbital picture “reshuffles” into molecular orbitals or more chemically complex bonding schemes. This complicates the idea that the PT alone suffices to understand molecular behavior. The post therefore raises a genuine conceptual tension: a “periodic table for molecules” could in principle be more directly useful to chemists.

• The article’s critique implicitly echoes a known reality: although the PT is enormously powerful, it is not a “complete theory of chemical bonding or molecular behavior.” Chemists routinely supplement it with quantum chemistry, molecular orbital theory, hybridization, empirical data, etc. As noted by standard expositions, the PT is a starting point for understanding periodic trends (valence, electronegativity, bonding tendencies), but chemistry always requires additional models. asbmb.org+2American Chemical Society+2

⚠️ Where the post ventures into speculation or controversial reinterpretation

• The post introduces an alternative framework, dubbed Real Quantum Mechanics (RealQM), in which atoms and molecules are described as “non-overlapping one-electron charge densities” rather than conventional delocalized wavefunctions, and suggests this could yield a “Periodic Table for chemistry” (i.e. more chemically relevant table than the atomic PT). That is a speculative and nonstandard proposal. The mainstream quantum mechanical description — using wavefunctions, orbitals and the full machinery of quantum chemistry — remains overwhelmingly successful in describing atomic, molecular, and material behavior.

• The post seems to argue that relying on atomic orbitals (PT’s foundation) to infer molecular bonding (molecular orbitals etc.) is a “mismatch.” While there is some philosophical merit to pointing out the conceptual leap from atoms → molecules, modern quantum chemistry precisely builds on the atomic basis to construct accurate molecular descriptions. The success of quantum chemistry in predicting molecular geometries, spectra, reaction pathways, etc., indicates that the “mismatch” is not fatal.

• The suggestion that chemists “need a periodic table for molecules” — while imaginative — may not be practically feasible. Molecules are vastly more diverse than atoms; a “table” summarizing all possible molecules with useful predictive power would be extremely complex, perhaps less useful than existing computational models, databases, and quantum-chemical methods.

  

The Mystery of the Periodic Table

The mystery of the Periodic Table PT is the arrangement into a rectangular grid with elements/atoms listed with increasing atomic number into rows of length 2, 8, 8, 18, 18, 32, 32..., signifying number of electrons with twice repeated periods according to $2n^2$ with $n=1,2,3,4.$ Basic questions are: 

1. Why periods of length 2, 8, 18, 32?
2. Why the factor 2 in $2n^2$?
3. Why repetition of periods 8, 8 et cet?
4. What is the meaning of filled shells with 2, 8, 18 and 32 electrons defining the periods? 

The answer by text book Standard Quantum Mechanics StdQM is that electrons arrange around atomic kernels in s, p, d and f orbitals with 1, 3, 5 and 7 electrons as spherical harmonics, and that s+p orbitals give a filled shell with 4 electrons, s+p+d orbitals give a filled shell with 9 electrons, s+p+d+f orbitals fill a shell with 16 electrons, which is viewed to be an answer to 1. An answer to the factor 2 is given by introducing a 4th quantum number as spin in an ad hoc fashion. Questions 3 and 4 have complex answers, if any.

StdQM seeks to explain chemical properties of atoms in the PT as the result of formality of spherical harmonics.

RealQM is an alternative to StdQM which gives completely different answers to 1-4. Electrons in RealQM have non-overlapping supports and meet at a common boundary with continuity of charge density.

• RealQM starts (after Hydrogen with one electron) with Helium with 2 electrons each occupying a half-space and meeting at a common plane containing the kernel together forming a 1st shell. 
• Lithium with 3 electrons has 2 electrons in a Helium-configuration and the third electron in an outer 2nd shell. 
• Beryllium with 4 electrons has a shell structure with 2 electrons in 1st shell and 2nd shell split into 2 half-shells.  
• Boron with 5 electrons has 2 electrons in 1st shell and 2nd shell split into 3 electrons. 
• The 2nd shell is filled with 8 electrons, which comes from splitting each 2nd half-shell into $2\times 2=4$ electrons and gives Neon with atomic number 10.
• The 3rd shell is filled with 18 electrons which comes from splitting each 3rd half-shell into $3\times 3=9$ electrons. 

RealQM explains PT as the result of a geometric packing problem. RealQM invites to an exploration of chemical properties of atoms resulting from electron packing. RealQM opens to a rich structure of subshells. 

Comment by chatGPT:

🔎 Summary of the post’s main argument

• The post asks why the Periodic Table is arranged in “periods” (rows) of length 2, 8, 18, 32… — i.e. following the sequence $2n^2$ for $n=1,2,3,\dots$. claesjohnson.blogspot.com

• According to standard quantum mechanics (StdQM), the explanation involves electron orbitals (s, p, d, f …) and quantum numbers (including “spin”) — giving filled shells that reflect the period lengths. claesjohnson.blogspot.com

• Johnson claims this is an ad hoc, formal explanation — in his view unsatisfying or incomplete — and instead promotes an alternative he calls RealQM. claesjohnson.blogspot.com

• In RealQM, electrons “have non-overlapping supports” and “meet at a common boundary” with continuous charge density. Johnson argues that under this model the periodicities arise from a geometric “packing problem,” rather than spherical-harmonic orbitals. claesjohnson.blogspot.com

👍 What is interesting / worth discussing

1. Challenging orthodoxy. The article embodies a classic scientific-philosophical stance: questioning accepted theory, highlighting what may be “formalism without physical insight.” That sort of critical inquiry can sometimes lead to deeper understanding or new perspectives. It’s healthy for science to revisit foundational assumptions — especially in complex domains like quantum mechanics.

2. Simplicity and intuition. The idea of a geometric/packing-based model for electron shells is intuitively attractive: structure emerging from “space-filling” constraints has a kind of classical appeal. For readers uncomfortable with purely mathematical abstractions (wavefunctions, Hilbert-space, probability amplitudes), a more “visualizable” model can feel more concrete.

3. Encouraging deeper questions. By raising “why this pattern?” instead of “because quantum mechanics says so,” the post encourages reflection on why the standard model is accepted — what is explanatory vs. what is computational convenience. That attitude often underpins conceptual advances.

🤔 What is problematic or questionable

1. Lack of established support. The proposed RealQM does not — at least in this post — come with detailed quantitative evidence (e.g., calculated energy levels, spectral lines, chemical behavior) matching observed atomic physics. The standard quantum model is supported by a vast body of empirical evidence (spectra, chemical periodicity, atomic interactions) — any alternative needs equally strong backing.

2. Orbital theory’s empirical success. The standard explanation using orbitals, quantum numbers including spin, and the mathematics of wavefunctions has proven extremely successful: it predicts atomic spectra, chemical behavior, ionization energies, etc. Any alternative must reproduce all of that, not just the period lengths.

3. Risk of oversimplification. The “geometric packing” picture may offer intuition, but nature at the atomic scale doesn’t behave like classical rigid balls — quantum mechanics shows that electrons are delocalized, subject to the uncertainty principle, and have wave-like properties. A model based purely on classical geometry risks missing essential quantum phenomena.

4. Ad-hoc replacement. Introducing non-overlapping “supports” and partitioned “half-shells” seems itself an assumption that needs justification. Without physical derivation (from fundamental equations) the new model risks being as arbitrary as the one it critiques.

🧠 Broader reflections: What the debate tells us about scientific theories

• The periodic table — simple and familiar to chemists and physicists — encodes deep quantum-mechanical truths. The fact that there is a straightforward pattern (2, 8, 18, 32…) is part of why quantum theory was such a breakthrough. Understanding why the pattern emerges fosters better insight into what the theory means.

• However, successful theories are not judged solely by offering “intuitive pictures,” but by their predictive and explanatory power. Theories must match data, across many phenomena (spectra, chemical reactivity, bonding, etc.).

• Creative, critical proposals — like RealQM — can inspire reexamination of foundations; but they must eventually confront the full weight of empirical evidence.

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Dynamic vs Static QM: RealQM vs StdQM

RealQM is an alternative to textbook Standard Quantum Mechanics StdQM.  

• RealQM is based on non-overlapping one-electron charge densities:
• RealQM has the dynamic time-dependent form of classical continuum mechanics in 3d with direct physical meaning. 
• StdQM acts over abstract $3N$-dimensional configuration space for an atom with $N$ electrons as a formal scheme without direct physical meaning and with prescribed unitary evolution in time in forms of static eigenfunctions. 
• RealQM with one-electron non-overlapping charge densities, does not need the Pauli Exclusion Principle PEP. No two electrons can occupy the same spatial domain as a consequence of mutual Coulomb repulsion.
• StdQM with overlaying electron densities with global support needs PEP to prevent more than two electrons to overlay. Two electrons with opposite spin can overlay.
• RealQM does not need to concept of spin.
• Electrons in RealQM are identified by unique spatial presence, just like macroscopic bodies. 
• Electrons in StdQM have no identity and are interchangeable.  
• Computational complexity is polynomial for RealQM (computable) and exponential for StdQM (uncomputable)
• RealQM explains stability of atoms and matter from non-overlap of electron densities and  dynamics of accumulation of electrons. Explanations in StdQM very complex.   
  
  

For the Helium atom with 2 electrons, RealQM gives a ground state electron configuration with the two electrons separated into two half-spaces meeting at a plane containing the atomic kernel. StdQM gives 2 overlaying electrons identical and interchangeable. 

The ground state of Lithium Li with 3 electrons is in RealQM described by 2 electrons in a 1st shell split into two half spaces, and 1 electron in a 2nd shell with spherical symmetry. A configuration with 3 electrons in a 1st shell gives lower energy which is however not attainable because Li is formed from Li+ with 2 electrons in a 1st shell which is filled and does not allow incorporation of 3rd electron, which is forced outside into a 2nd shell. The form of RealQM allows simulation of the dynamic formation of Li from Li+ by adding one electron in an outer 2nd shell.

More generally, RealQM allows simulation of the dynamic physical formation ground states of atoms by successive addition of electrons filling shells, and also the dynamic formation of molecules as atoms are sequentially added under energy minimisation. This is not really possible within StdQM because the dynamics is lacking. 

RealQM thus may open new possibilities of computational protein folding, which is beyond the capacity of StdQM.

RealQM is dynamic, physical and computable. StdQM is static, unphysical and uncomputable.  

PEP as basic element of StdQM is an ad hoc assumption without physics. No physical reason is given which explains why 2 electrons with opposite spin, can overly but not three. The 4 quantum numbers used to identify electrons including spin have formal nature rather than real.  

The situation can be compared with classical thermodynamics with focus on static equilibrium states vs computational thermodynamics allowing simulation of dynamic transition between equilibrium states. 

The Unphysical Pauli Exclusion Principle

Here is a continuation of the previous post.

The foundation of modern atom physics is Quantum Mechanics QM in its text book formulation StdQM takes the form of Schrödinger's Equation SE formulated exactly 100 years ago. Solutions of SE are called wave functions which depend on $3N$ spatial variables for an atom with $N$ electrons, thus with a separate 3d Euclidean coordinate system for each electron, which connect to a common 3d coordinate system for Coulomb repulsion between electrons and Coulomb attraction between electrons and atomic kernel. 

All electrons thus have global presence or global support and so overlay in the common coordinate system. This does not fit with Coulomb repulsion and in fact is the root of the confusion surrounding the physical meaning of the wave function, which has never been settled.

To handle the difficulty with overlaying electrons the Wolfgang Pauli came up with the idea to assign electrons a two-valued 4th quantum number named spin-up and spin-down with the role to allow overlay of two electrons with different spin, and forbid overlay in case of equal spin. This would allow at most two electrons to overlay. This was coined Pauli's Exclusion Principle PEP, which is an ad hoc assumption without physics since spin-up and spin-down lack physics and in particular has nothing to with spinning in a classical sense. 

So QM is based on SE complemented by PEP. The trouble is that both SE and PEP lack physics. Both represent ad hoc inventions without clear physics. 

RealQM is an alternative to QM based on electron charge densities with non-overlapping supports in a common 3d coordinate system. In RealQM there is no role for PEP because no two electron overlay. The ad hoc non-physical assumption PEP has been replaced by a physical assumption of no overlay of electrons motivated by Coulomb repulsion.

RealQM has the form of classical continuum mechanics in 3d and so has polynomial complexity (is computable) while QM has exponential computational complexity (is incomputable).

Stability of matter is a direct consequence of the form of RealQM, but is very difficult to establish for QM.

Chemistry is based on molecules formed by atoms and electrons with support over the whole molecule in QM. Keeping track of spin is impossible, which makes QM for atoms useless for molecules. RealQM for atoms naturally extends to molecules again based on non-overlapping electron charge densities. 

The Ad Hoc Invention of Spin by Pauli

The basic idea of Quantum Mechanics QM of giving a two-valued quality of spin as spin-up and spin-down of electrons, came from the need to explain the shell structure of atoms with $2n^2$ electrons in shell $n=1,2,3,...$ with the factor 2 accounting for the two values of spin. 

This led to the Pauli Exclusion Principle PEP allowing two electrons with different spin to occupy the same region of space in an atom, but not three. The two electrons of the Helium atom could then have identical overlapping spatial presence in agreement with the principle of QM of electrons as being identical and interchangeable. Without PEP the Helium atom would appear as a mystery, as well as other atoms.

Pauli received the Nobel Prize in Physics in 1945 for his PEP, but his Nobel Lecture expressed deep dissatisfaction with PEP, because it was an ad hoc invention without convincing physics with only purpose to explain an observed apparent factor 2. But the Nobel Committee resisted and gave him the Prize despite his protests. 

RealQM is an alternative to QM based on non-overlapping one-electron charge densities where there is no role for spin and PEP. The two electrons of Helium will thus be separated into two half-spaces meeting at a common dividing plane with continuity of non-zero charge density. This separation introduces a two-valued quality of geometric nature: one electron occupying one half-space and the other electron occupying the other half space, thus as electrons with identity from spatial occupation. RealQM explains the factor 2 in the shell structure, without resort to spin.

The presence of spin in QM was one of the aspects which made Schrödinger protest against unphysical  aspects of QM and leave the field he had created in 1926 with his model for the Hydrogen atom with one electron, since the generalisation by Born-Heisenberg-Dirac to many electrons took unphysical form. It did not help that he received the Nobel Prize in Physics in 1933 for his model. 

Today QM is viewed to be settled once and for all, but the protests of Pauli and Schrödinger are still as alive as ever. RealQM follows the spirit of Pauli and Schrödinger. 

Essence of Chemistry: H2 vs Au2

This post adds a concrete example to the discussion in the previous post about the form of QM in the physics of atoms and chemistry of molecules.  

Chemistry concerns molecules as combinations of atoms with lower energy than separate atoms. The first molecule to be studied after the Schrödinger Equation SE for the Hydrogen atom H with one electron around a proton kernel formulated in 1926 by Schrödinger, was the molecule H2 formed by two H atoms finding a total energy minimum of -1.17 at a kernel distance of 1.4 (atomic units) compared with -1 when separated in an analysis by Heitler and London 1927. 

The basic element of the analysis was accumulation of electron charge density between the kernels allowing decrease of potential energy by proximity of both electrons to both kernels. The analysis was complicated by the fact that the SE for the molecule of 2 H atoms has 6 spatial dimensions. Ingenious dimensional reduction was required to give the observe result.

An H atom has one shell with one valence electron taking part in the formation of H2 and the essence of molecule formation is accumulation of electron charge density between the kernels. 

We now compare with a Gold atom Au which has 79 electrons distributed over 6 shells according to the pattern 2+8+18+32+18+1 with again one valence electron in the outermost shell. We know that Au does not easily react to form molecules and so we are faced with the problem to explain in particular why Au2 is not formed? 

Standard non-relativistic Quantum Mechanics StdQM has no answer and so an explanation has been sought as a relativistic effect from very high speed electrons in the 1st shell with increase of mass contracting the valence electron, which does not make much sense.

RealQM is an alternative to StdQM which gives an answer without relativity, which is displayed in this post and this post with details in the book on RealQM.  The essential difference between H and Au is the geometry of the valence electron, which fills a sphere for H allowing electron accumulation between kernels in H2, which is prevented in the outermost shell for Au.  

In StdQM for molecules all electrons have presence (support) over the entire molecule which gives a very complex unphysical picture difficult to conceptualise.

In RealQM electrons have non-overlapping supports with direct physical meaning which can be visualised and conceptualised.  

  

   

QM for Atoms vs QM for Molecules

Quantum Mechanics QM was originally developed to theoretically explain atomic spectra observed already in 1802-4 by Fraunhofer then developing into experimental spectroscopy by Bunsen-Kirchhoff with Balmer's formula for Hydrogen H in 1885 as key challenge taken on by Planck, Sommerfeld and Bohr culminating in Schrödinger's Equation SE in 1926 giving the final answer in terms of spherical harmonics capturing the spherical symmetry of the Hydrogen atom with one electron around a proton kernel as central potential, which rocketed Schrödinger to fame.  

SE was then quickly generalised to atoms with many electrons simply by adding a new set of 3d spatial variables for each new electron into a linear equation in terms of a wave function depending on $3N$ spatial variables for an atom with $N>1$ electrons where spherical symmetry was lost, but nevertheless was kept by reducing to a set of one-electron wave functions subject to a spherically symmetric average potential from the other electrons serving as approximate solutions to SE. The multi-d wave function describes global presence of all electrons for which the physics appears to be missing. 

But when SE was extended to molecules starting with H2 as two H atoms joined by a covalent chemical bond in the work by Heitler and London in 1927, the presence of two kernels was not at all compatible with spherical symmetry and so entirely new computational techniques had to be developed. Computational quantum chemistry has taken a large share of super-computer power since the 1960s and still does, the basic reason being the inherent exponential complexity of the multi-d SE in need of drastic dimensional reduction.  

We thus meet QM in two fundamentally different forms, for atoms in the Periodic Table PT in terms of s, p and d electronic orbitals in the form of spherical harmonics, and for molecules in terms of other ad hoc electronic configurations. Physicists have been in charge of QM for atoms with spherical symmetry, while QM for molecules without spherical symmetry has been left to chemists and then with little help from physicists.  

The PT is supposed describe chemical properties of atoms when forming molecules without spherical symmetry, yet PT is is organised according to s, p and d spherical harmonics.    

RealQM is a recent alternative to QM based on non-overlapping one-electron charge densities as a form of classical continuum mechanics in 3d. RealQM has polynomial computational complexity, has clear physical meaning geared to molecules in complex geometry without spherical symmetry.

Schrödinger could have chosen RealQM as generalisation of his SE for H to atoms with more than one electrons, because it satisfied his chief requirement of physicality or "Anschaulichkeit". Schrödinger instead chose a quick easy formal extension into multi-d maybe from pressure to quick delivery on the success with H.

But Schrödinger never could embrace the multi-d SE which in the hands of Born became the standard valid into today, and so together with Einstein became the main critic of the multi-d SE he had created because it lacked physics. Schrödinger thus died in 1961 disillusioned and did not experience the computer revolution making computational complexity a core issue with preference to RealQM.

PS Schrödinger’s Most Important Critical Quotes About QM

1. On the wave function not being a physical wave

“The ψ-function… is not a physical wave. But what is it then? A kind of probability amplitude? Probability of what? At this stage, one worries that one has to do with mere ghosts.”
— Erwin Schrödinger, 1926–1927 correspondence, quoted in Moore, Schrödinger: Life and Thought

2. On the collapse of the wave function (which he rejected)

“Every indeterminacy is to be reduced to a determinate change, occurring independently of any observation. To demand that the wave function should ‘collapse’ when observed is quite absurd.”
— Schrödinger Archive, quoted in M. Bitbol, Schrödinger’s Philosophy of Quantum Mechanics

3. On dissatisfaction with the Copenhagen interpretation

“I don’t like it, and I’m sorry I ever had anything to do with it.”
— Schrödinger’s remark about quantum mechanics,
quoted in Karl Popper, Quantum Theory and the Schism in Physics

4. On the dualistic particle/wave picture

“We seem to be faced with an entirely new kind of difficulty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena, but together they do.”
— Schrödinger, Philosophy of Experimental Science

5. On the mathematical formalism being detached from reality

“What we observe as material bodies and forces are nothing but shapes and variations in the structure of space. Physical science does not want to admit this.”
— Nature and the Greeks
(Implicit criticism: the formalism of QM is abstract and not directly physical)

6. On the statistical interpretation

“The statistical interpretation is the only one that does not collapse under scrutiny, but it is regrettable because it makes the theory nothing more than an instrument of prediction.”
— Schrödinger, Science and Humanism

7. On the cat paradox (his attack on collapse-probability views)

“One can even set up quite ridiculous cases. A cat is left to itself in a closed box… The ψ-function of the entire system expresses this situation with equal parts of alive and dead. This prevents us from accepting naïvely a ‘blurred model’ for representing reality.”
— “Die gegenwärtige Situation in der Quantenmechanik”, Naturwissenschaften (1935)

8. On QM lacking a coherent picture of reality

“The present theory, in its general conceptual form, is incapable of giving a clear picture of what is happening physically.”
— Schrödinger, lecture notes quoted in J. Mehra & H. Rechenberg, The Historical Development of Quantum Theory

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Themes in Schrödinger’s Critique

Schrödinger consistently criticized:

✔ The wave function lacking physical interpretation

He wanted a real wave in space, not a probability amplitude in configuration space.

✔ Collapse as conceptually incoherent

He viewed “collapse” as an ad hoc and unphysical rule added to save appearances.

✔ The idea that QM describes only knowledge, not reality

He disliked the epistemic view of QM that Bohr and Heisenberg promoted.

✔ The abandonment of classical realism

He believed physics must describe real processes, not just observational outcomes.

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RealQM as Alternative to StdQM?

Real Quantum Mechanics RealQM is an alternative to text book Standard Quantum Mechanics StdQM:

• RealQM has the form of classical continuum mechanics in Euclidean 3d space.
• StdQM acts over a configuration space of $3N$ spatial dimensions for an atomic system with $N$ electrons. 
• RealQM has a clear physical meaning in terms of non-overlapping one-electron charge densities.
• StdQM has no clear physical meaning since configuration space is not physical.
• RealQM is computable in the same sense as classical continuum mechanics of solids. 
• StdQM has exponential computational complexity and so is uncomputable already for small $N$.
• Computable and physical go hand in hand, assuming physics is a form of analog computation.
• Likewise uncomputable and unphysical come together.

It is clear that RealQM as being physical and computable may have something to offer which StdQM does not as being unphysical and uncomputable.

When I present RealQM to physicists and chemists, which can have the form of chatGPT, I get the reaction that in order for RealQM to warrant consideration, it is necessary to show that RealQM can perform all that has been performed by StdQM over 100 years and then in a better way. 

Is this a reasonable requirement? Would it not be more reasonable to see that RealQM can do some things better than StdQM? RealQM could then be viewed as an alternative to StdQM to be used for certain problems typically because of smaller computational cost and more clear physicality? You find many such examples here.

It is thus possible to view RealQM as an alternative to StdQM like Density Functional Theory or Atoms in Molecules both of which have limited scope. 

Here is one question without answer in StdQM: Why are periods in the Periodic Table repeated to give the periods 2, 8, 8, 18, 18, 32, 32,...? It is possible that RealQM can give an answer from a principle of electron packing?

Here is what chatGPT says: 

👍 What I find interesting

1. Motivation – The author raises a genuinely important question: Standard quantum mechanics (StdQM) uses a configuration-space formalism (3N spatial dimensions for N electrons) which he argues “has no clear physical meaning” because the configuration space is not directly physical. He contrasts that with his proposed RealQM formalism in ordinary 3-dimensional Euclidean space. That’s a thought-provoking criticism of interpretational issues in quantum mechanics.

2. Computability claim – He points out that RealQM is “computable in the same sense as classical continuum mechanics of solids” whereas StdQM “has exponential computational complexity and so is uncomputable already for small N.” Emphasising computational tractability is a valuable viewpoint in physics/chemistry, where large-N systems (many electrons) present major challenges.

3. Proposal of alternative – It’s not just a complaint; Johnson is offering RealQM as a potential rival/alternative method: “RealQM may have something to offer … typically because of smaller computational cost and more clear physicality.” Having alternatives to mainstream frameworks is healthy for the field.

The blog post offers a provocative and worthwhile challenge to mainstream quantum mechanics—focusing on interpretation, physicality of formalism, and computational tractability. It brings genuine questions to the table:

• Is configuration space a physically meaningful place, or just a computational tool?

• Can we find formulations of quantum phenomena in ordinary 3-D space with clearer physical meaning?

• Can we craft computationally efficient methods surpassing the combinatorial explosion of many-body wave functions?