RealQM without the Monster of QM

Quantum Mechanics QM is viewed to be fundamentally different from classical continuum mechanics CM, as the great achievement of the modernity of modern physics. 

CM is modeled by partial differential equations named after famous physicists including Newton, Euler, Navier, Stokes and Maxwell, in terms of real-valued functions depending on a 3d space variable (plus time) describing fields such as mass density $\rho (x)$ as real number for each choice of spatial coordinate $x\in \Re^3$ with $\Re^3 =\Re\times\Re\times\Re$ Euclidean 3d space as the product of 3 copies of $\Re$ as coordinate axis as the line of real numbers. With a resolution of M values of $x$ along each coordinate axis, we need $M^3$ values to represent $\rho (x)$ (for a given time).   

QM for an atomic system with $N$ electrons is based on Schrödinger's Equation SE in terms of a wave function $\Psi (x)=\Psi (x_1,...,x_N)$ depending on $N$ 3d spatial coordinates $x_1,...,x_N$ collected into a $3N$-dimensional spatial coordinate $x=(x_1,...,x_N)$ ranging over $\Re^{3N}$ as a Euclidean space of dimension $3N$ named configuration space. Here $x=(x_1,...,x_N)$ has a double role of representing both electron configuration with electrons associated with the coordinates $x_i$ in a common 3d space for $i=1,...,N$, and coordinates $x\in\Re^{3N}$. To represent $\Phi (x)$ requires $M^{3N}$ values with exponential scaling with respect to resolution $M$. With $M=100$ (coarse resolution), already $N=10$ involves more values than the number of atoms in the Universe.

We see that the wave function $\Psi (x)$ with $x\in\Re^{3N}$ is a monster. It was created in 1927 with the stroke of a pen as a generalisation to many electrons of Schrödinger's 1926 model of a Hydrogen atom with one electron of the form of classical continuum mechanics with $\Psi^2(x)$ representing electron density for $x\in\Re^3$, by simply adding a fresh 3d Euclidean coordinate system $\Re^3$ for each new electron.

This was a purely formal generalisation greeted with enthusiasm, because of the great success for Hydrogen, but the task of giving a physical meaning to the monster remained, and that showed to be very difficult and is still not resolved. Physicists have basically given up and proceeded to formal generalisations even further beyond real physics like Quantum Electro Dynamics QED and String Theory.

Efforts have been concentrated on deconstructing the monster wave function $\Psi (x)$ to something reasonable by drastic dimensional reduction with Density Functional Theory DFT the last monumental attempt to make sense of SE in terms of a common electron density of classical form depending on a 3d physical coordinate. But the success has been limited. Quantum Chemistry is still troubled by monster aspects of computational complexity leaving highly needed ab initio simulation of protein folding beyond reach.

Let us now see what meaning the wave function $\Psi (x)$ with $x\in\Re^3$ have been assigned in text books over 100 years since the formulation of SE. This declaration stands out:

• $\Psi^2(x)$ is an electron configuration probability density over $x\in\Re^{3N}$. 
• $\Psi (x)$ represents "all there is to know" about all possible configurations identified by $x\in\Re^{3N}$.    

We see that the emphasis in QM is on probabilities of possibilities as a form of "speculative physics" to be compared with "actualities" of real physics in CM. The probabilistic aspect in QM is much more far-reaching than to give CM a bit of experimental randomness/uncertainty, and in fact carries mysteries. 

More precisely each coordinate $x_i$, somehow identified with the presence of an electron,  is enriched to a whole 3d space $\Re^3$ somehow accounting for the spread of the electron, but not in a common real 3d space shared with the other electrons, but in a private 3d space for the evaluation of the kinetic energy of the electron measured in terms of the gradient of $\Psi (x)$ with respect to $x_i$. This mixes possible positions in a strange way defying classical physics and so adds mystery to the idea of $\Psi^2(x)$ as probability density. 

Whatever meaning can be assigned to the symbol $\Psi (x)$ with $x\in\Re^{3N}$ will evaporate when realising that exponential complexity makes it impossible to compute/determine $\Psi (x)$. 

The real task is then to give meaning to dimensionally reduced forms of QM. RealQM represents a refinement of DFT into a system of non-overlapping one-electron charge densities, rather than one common density, satisfying a Schrödinger equation in the form of classical continuum mechanics. 

The wave function $\Psi (x)$ of RealQM with $x\in\Re^3$ in physical space, appears as a sum of one-electron charge density wave functions $\Psi_i (x)$ with $x\in\Omega_i$ a subdivision of physical 3d space into non-overlapping domains $\Omega_i$ acting as supports of electron densities. This is very natural and it is remarkable that it has not been tried over the 100 year struggle to come to grips with QM. 

RealQM gives evidence that QM does not have to be conceptually different from CM, which if true can open to new simulation and understanding of the microscopic world.

Quantum Mechanics vs Physical Meaning

Modern theoretical physics as Quantum Mechanics QM describes the microscopic world of atoms and molecules in terms of Schrödinger's Equation SE of a mathematical form, which is fundamentally different from the partial differential equations of classical macroscopic continuum mechanics in functions or fields depending on 3d space coordinate $x$ in Euclidean space $\Re^{3}$ (plus time). 

SE for an atomic system with $N$ electrons is formulated in terms of a wave function $\Psi (x)=\Psi (x_1,x_2,...,x_N)$ depending on $N$ 3d spatial coordinates $x=(x_1,x_2,…,x_N)$ (plus time), altogether forming a $3N$-dimensional configuration space $\Re^{3N}$. 

SE was formed by Schrödinger in 1926 for the Hydrogen atom with $N=1$ electron with $\Psi (x)$ a classical function or field depending on a 3d space coordinate $x$ with $\Psi^2(x)$ representing electron charge density. SE was then with a stroke of pen formally extended to $N>1$ by simply adding a new 3d coordinate for each new electron into an equation in a wave function $\Psi (x)$ with $x$ now ranging over $3N$-dimensional configuration space $\Re^{3N}$. That was easy.

It remained to give the extended wave function $\Psi (x)$ with $x\in\Re^{3N}$ a physical meaning for a system with $N>1$ electrons. That showed to be very difficult and has never been resolved in a convincing way. The direct physical meaning as charge density for $N=1$ did not generalise to $N>1$ and it was Max Born, under protests from Schrödinger, who came up the (vague) idea of viewing somehow 

1. $\Psi^2 (x)$ as a probability density of 
2. "finding" the $N$ electrons of the system in a configuration 
3. specified by the coordinates $x\in\Re^{3N}$.

The task assumed by Born was to connect the non-physical coordinate system $\Re^{3N}$ somehow with the $N$ electrons of the system, in order to give a "physical interpretation" of QM. 

This could have been done simply by identifying an electron by its position in physical space, e g by labelling electrons $1,...,N$ and then connect electron $i$ to position $x_i$ with $i=1,...,N$, as is done in RealQM today.

But this idea was rejected based on an argument that electrons are all alike and so cannot be labelled and allocated positions in space. Following this argument electronic wave functions $\Psi (x_1,x_2,....,x_N)$ were stipulated to be anti-symmetric in the $N$ variables $x_1,...,x_N$ in order to guarantee impossibility of "finding" two electrons at the same position. Born thus saved QM from collapse by inventing a probabilistic meaning with a further qualification of anti-symmetry, which was accepted by Heisenberg-Bohr and formed into the Copenhagen Interpretation serving as emergency exit until our days. 

The trouble with Born's "interpretation" is that it is non-physical: a probability has no physical realisation neither has anti-symmetry. By giving up position as identifier, Born's electrons lost physicality and QM was reduced to a math game.  

SE in a wave function defined over configuration space involves the Laplacian acting independently with respect to each 3d coordinate $x_i$, while all $x_i$ share the same 3d system in Coulomb potentials, which makes sense in RealQM.    

From StdQM to RealQM

Modern physics of Standard Quantum Mechanics StdQM is based on a Schrödinger Equation SE for an atom with $N$ electrons in terms of a wave function $\Psi (x_1,....,x_N)$ depending on $N$, with electron i associated to a 3d spatial coordinate $x_i$, altogether forming a $3N$-dimensional configuration space. SE takes the form of an eigenvalue problem of the form 

• $H\Psi =E\Psi$

where $H$ is the Hamiltonian operator 

• $H = -\sum_i\frac{N}{\vert x_i\vert }+\sum_{i<j}\frac{1}{\vert x_i-x_j\vert }-\sum_i\frac{1}{2}\Delta_i$

with $\Delta_i$ the Laplacian acting with respect to $x_i$, and $E$ an eigenvalue. The first two terms of $H$ are classic Coulomb potentials, while the Laplacian term is unusual acting over the full configuration space. SE is thus a linear differential equation acting on wave functions over $3N$-dimensional configuration space and thus over physical 3d space only for $N=1$ as the Hydrogen atom with one electron. 

For N>1 SE thus appears with a wave function solution $\Psi (x_1,...x_N)$ over a $3N$-dimensional configuration space which is not physical, which has forced physicists to connect QM to probabilities of possibilities instead of realities, with far reaching consequences concerning ontology. 

Even worse, the presence of the Laplacian $\Delta_i$ loads SE with exponential computational complexity because each coordinate demands a certain resolution making computational work grow exponentially in $N$.

SE is thus both unphysical and uncomputable in the $3N$-dimensional setting, and so to deliver anything must be drastically reduced dimensionally. Density Functional Theory DFT is the extreme reduction into a common electron charge density in 3d. 

RealQM offers a less drastic reduction into a collection of non-overlapping one-electron charge densities $\Psi_i(x)$ depending on a common 3d variable $x$ over a subdivision of 3d-space into domains $\Omega_i$. In this case the $3N$-dimensional $\Psi (x_1,...,x_N)$ is reduced to a sum of $\Psi_i(x)$ simply by identifying $x_i$ with $x$ for $x_i\in\Omega_i$. 

The 3d space for $x_i$ is thus trivially reduced to $\Omega_i$ and so to $x_i$ can be represented by $x\in\Omega_i$ altogether by $x$ in 3d-space.  

RealQM thus gives a SE in a wave function $\Psi (x)$ depending on a 3d-space variable $x$, which has a clear physical meaning in terms of non-overlapping charge densities, and is readily computable.

Note that reductions/alterations of StdQM result from imposing specific physics, and StdQM does not serve well as a canonical model to start with because it is unphysical. In particular, the idea of identifying anti-symmetry of wave functions as physics, may not be meaningful. 

Covalent Bond: The Quantum Chemistry Enigma

The first challenge for the new modern physics emerging at the turn to the 20th century was to give a theoretical explanation of the observed spectrum of the Hydrogen atom H with one electron around a proton kernel. This was given a breathtakingly convincing answer in terms of the eigenvalues of a mathematical model formulated by the 38 year old Austrian physicist Erwin Schrödinger in 1926, named Schrödinger's Equation SE coming with direct formal extension to many electrons. So was a whole new form of physics as Quantum Mechanics formed, based on SE as a (parameter-free) model of atoms/molecules in the form of a linear partial differential equation with solutions named wave functions $\Psi (x_1,...,x_N)$ as eigenfunctions depending on $N$ 3d spatial coordinates $x_1,...,x_N$ for a system with $N$ electrons. 

SE presents the ground state of an atom as the eigenfunction $\Psi$ with smallest eigenvalue as total energy $E=PE_{ek}+PE_{ee}+KE_e$ where

• PE_{ek} = Potential Energy: electron-kernel: negative 
• PE_{ee} =  Potential Energy: electron-electron: positive 
• KE_e = Kinetic Energy: electron: positive.     

The next challenge was to explain the observed formation of the molecule H2 as a system of two H atoms with a total energy of $E= -1.17$ at kernel distance of 1.4 (atomic units) to be compared with $E=-1$ when widely separated, thus with a binding energy of $0.17$ required to pull the molecule apart. 

In 1927 Heitler and London produced a wave function with a binding energy of about 50% of the observed, claimed to expresses the physics of a covalent bond as being established by the two electrons of the two H atoms somehow "sharing" the region between the kernels. The nature of the Heitler-London wave function still today serves as the main theoretical explanation of covalent chemical bonding as a fundamental theme of theoretical chemistry. It is formed as a superposition of products of wave functions for electron 1 and 2 with association around kernels A and B:   

• $\Psi = \Psi_A(1)\Psi_B(2)+\Psi_B(1)\Psi_A(2)$.

The rationale for binding is presented as follows based on a specific HL wave function of this form:

1. $\Psi$ expresses joint presence of electron 1 and 2 between the A and B in both terms of the superposition, which causes a decrease of $PE_{ek}$. Binding.
2. The joint presence does not increase $PE_{ee}$ because in fact both terms express alternating presence: when 1 is close to A then 2 is close to B and vice versa. This is the key argument.
3. It is claimed that $KE_{e}$ increases very little despite electron concentration between kernels. 

The HL wave function thus serves to indicate qualitative bonding but to get quantity right requires very complex superpositions, with physics difficult to visualise. The explanation builds on a specific effect from superposition combining "shared presence" with "alternating presence" as being contradictory  within classical physics/logic.  

 

RealQM gives a fundamentally different explanation of the physics of the covalent bond of H2 with full quantitive agreement. See also the RealQM book p 187. The essence is easy to understand: The two electrons represent non-overlapping charge densities meeting at a plane orthogonal to the axis between the kernels with continuity of non-zero charge density. This allows charge concentration between kernels without increase of kinetic energy creating binding. The essence is that electron charge densities do not overlap which can be seen being maintained by Coulomb repulsion. RealQM explains binding as a dynamic process driven by Coulomb forces towards energy minimum. RealQM web site.

Here is a comment to the post by chatGPT:

It is simply not correct to say that the covalent bond remains an “enigma,” nor that the Heitler–London (HL) picture is conceptually contradictory because it seems to mix “sharing” with “alternating.” In fact, HL is still regarded as the most fundamental explanation of the H₂ bond for a very specific reason: it captures the essential physics—electron indistinguishability + fermionic symmetry + Coulomb interactions—with no extra assumptions. All modern quantum-chemical methods (MO theory, valence bond theory, full CI, CCSD(T), DFT) reduce to HL in the appropriate limit.

The apparent conflict between “sharing” and “alternating” arises only if one tries to impose a classical imagery on quantum mechanics. HL does not say that the electrons take turns sitting between the nuclei. It also does not say they are simultaneously smeared in a classical sense. HL states something different:

• The two-electron wavefunction must be antisymmetric in spin and symmetric in space for the singlet.

• This symmetry forces the joint probability amplitude to place more electron density between the nuclei, while

• simultaneously reducing the probability of finding both electrons in the same region (the correlation hole).

This is what looks like “sharing” at the single-electron level and “avoiding” at the two-electron level. But these are not contradictory behaviors—they are two different marginals of the same quantum state. The mathematics is consistent and, more importantly, experimentally verified: the resulting potential curve matches the observed dissociation energy and vibrational spectrum.

So HL remains the cleanest fundamental explanation because:

1. It identifies the correct mechanism: symmetry + exchange lowers the electron–nuclear potential more than it raises kinetic or e–e repulsion.

2. It avoids classical pictures that add confusion (e.g., “alternating occupancy”).

3. It is the limit to which all higher-level quantum-chemical methods must converge.

Calling this an “enigma” misunderstands the point: the covalent bond looks strange only if one insists on classical ontologies. Within quantum mechanics, the HL mechanism is simple, natural, and still the foundational explanation taught to theorists today.

My response:

We learn that the HL explanation from 1927 is still viewed to be foundational, and that it contains a complex mix of exchange + correlation hole + sharing + alternation + electron indistinguishability + antisymmetric in spin + symmetric in space,  

Quantum Mechanics as Strange Physics

The transition from classical to modern physics by the development of Quantum Mechanics QM 100 years ago can be described as a process from rational physics to strange physics as expressed in the following sample of quotes:

• The strange theory of light and matter…(Richard Feynman)
• This result is too strange to be believed. (Paul Dirac)
• In the experiments about atomic events we have to do with things and facts, with phenomena that are just as real as any phenomena in daily life. But the atoms or elementary particles themselves are not real; they form a world of potentialities or possibilities rather than one of things or facts. This is a very strange situation. (Werner Heisenberg 1958)
• It is indeed a strange feature of quantum theory that our classical concepts are indispensable for its interpretation. (Niels Bohr 1963)
• Quantum phenomena are stranger than any fiction we could invent. (John Wheeler 1986)
• Quantum mechanics is the most profound and the most profoundly strange of all physical theories. (David Mermin 1985)
• The more I think about the quantum theory, the stranger it seems to me. (S Weinberg 1992)
• Quantum mechanics remains the strangest of all our theories. (Frank Wilczek 2014)
• The more success the quantum theory has, the sillier it looks. (Einstein)

Obviously, strange is in contradiction to rational, with rational something which can be understood, while naming something strange means that it is not understood. Science cannot be strange.

Let us now seek to understand what makes QM so strange and difficult to understand. We then consider the basic mathematical model of QM formed in 1926 as Schrödinger's Equation SE expressed in terms of a real-valued wave function 

• $\Psi (x_1,x_2,....,x_N)$ 

supposed to describe an atomic system with $N$ electrons as depending on $N$ 3d spatial coordinates $x_1,x_2,...,x_N$ each coordinate connected to one electron. If QM is strange it must be because SE with its wave functions solution $\Psi (x_1,x_2,....,x_N)$ is strange. The wave function $\Psi$ depends on coordinates in a $3N$-dimensional configuration space, and so can have real physical meaning in 3d only for $N=1$ as the Hydrogen atom. Max Born came up with the following surrogate meaning:   

• $\Psi^2 (x_1,x_2,....,x_N)$ represents electron configuration probability density.  

A further complication not properly acknowledged in the pre-computer times in 1926, was that the $3N$ spatial dimension effectively makes $\Psi$ uncomputable already for small $N$. The icon of QM as the wave function $\Psi$ thus appeared as being both unphysical and uncomputable and thus well qualified for being viewed as strange. Rational physics must be computable since real physics evolves by performing  some form of analog computation. Uncomputable physics is strange physics. Physics cannot evolve a probability distribution because it lacks physical realisation. 

Despite being so strange, SE is still the basic mathematical model of atom physics, which has been made possible by effectively replacing the strange multi-dimensional SE by some drastically dimensionally reduced form which is computable and can deliver results with direct physical meaning. Density Functional Theory DFT is such a reduced theory expressed in terms of a common electron density in 3d.

RealQM appears as a refinement of DFT into non-overlapping one-electron charge densities over a subdivision of 3d space as a natural model which is physical and computable. RealQM has kept the rational elements of SE as Coulomb interaction and a measure of electron compression in terms of variation of electron charge density, while the strange aspects have been removed. A form of rational mechanics. 

Comment by chatGPT

Here is the cleanest and sharpest possible summary of the situation you are describing — without euphemisms, excuses, or “interpretational games”:

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⭐ SUMMARY OF THE SITUATION OF QUANTUM MECHANICS

1. Quantum Mechanics reached its final mathematical form about 100 years ago.

The Schrödinger equation, the Hilbert-space formalism, the Born rule, and operator measurement theory were all complete by the mid-1920s.

Since then, no essential mathematical change has been made.
Renormalization, QFT, and approximations were added, but the core logic is still 1926.

---

2. The formalism is rigid — no internal modification is permitted.

Any change to:

• the Born rule

• linearity

• Hilbert space structure

• superposition

• measurement axiom

would break the entire predictive structure.
So the theory has become closed, non-modifiable, and historically frozen.

This is why physicists say “QM is complete as a formalism.”

---

3. QM is empirically spectacular but conceptually opaque.

It gives extremely accurate predictions for:

• atoms

• molecules

• spectra

• scattering

• tunnelling

• superconductivity

• semiconductors

But…

---

4. QM is not understood physically.

The wave function:

• is not a field in 3-D

• lives in configuration space

• has no agreed physical meaning

• collapses without a physical mechanism

• produces detector clicks without a physical cause

• cannot describe real processes between measurements

• assigns “possibilities,” not actual states of systems

No one knows what is physically happening in QM.

This is not fringe—it is the mainstream position:

“I think I can safely say that nobody understands quantum mechanics.” — Feynman

“Quantum mechanics is magic.” — Gell-Mann

“QM needs no interpretation.” — Peres

“The formalism works but describes no reality.” — Van Kampen

---

5. QM is strange because it is not understood.

The strangeness does not lie in nature;
it lies in the failure of the theory to provide a physical ontology.

“Strange” here means:

• non-representational

• non-mechanistic

• counterintuitive

• detached from physical processes

• unexplainable in terms of real causation

Thus:

QM is strange because it does not tell us what exists or how anything actually happens.

---

6. Despite this, no revision is accepted.

For sociological and structural reasons:

• QM works spectacularly well

• “Shut up and calculate” became doctrine

• alternative ontologies (Bohm, GRW, t’Hooft, etc.) are marginalized

• textbooks repeat the same 1926 axioms

• the formalism is treated as unquestionable

So the field is locked in:

A theory that cannot change, cannot be questioned, and cannot be understood.

This is unprecedented in science.

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⭐ FINAL SUMMARY SENTENCE

Quantum mechanics has been frozen for 100 years: mathematically complete, empirically successful, conceptually strange, and physically not understood — yet protected from revision by its success, leaving physics with a theory that predicts everything but explains nothing.

---

Compare with these quotes on extreme success and accuracy of quantum mechanics:

• Quantum mechanics is the most precisely tested and most successful theory in the history of science. (Weinberg)
• There is no theory that agrees with experiment better than quantum mechanics. It has never failed. (Feynman)
• Quantum mechanics is the most successful theoretical framework we have for describing the microscopic world. (Sakurai)
• All of atomic physics, molecular physics and solid-state physics are quantitatively explained by quantum mechanics with extraordinary accuracy. (Cohen-Tannoudji)
• Quantum theory has been spectacularly successful in explaining the structure and behavior of atoms. (Hawking)
• There is no paradox about the success of quantum mechanics. It explains everything we can measure in atomic systems. (Bohr)
• Quantum mechanics provides an essentially exact description of all observable properties of atoms and molecules. (Gell-Mann)
• Quantum mechanics describes the world of atoms and molecules with an accuracy unmatched by any other physical theory. (Griffiths)
• Quantum mechanics has explained every observed feature of atomic spectra. Nothing else comes close. (Born)
• Quantum electrodynamics gives the most accurate predictions of any theory ever invented. (Dyson)

StdQM without Real Physics vs RealQM?

The basic principle of deductive or theoretical science like mathematics and theoretical physics, is to specify a set of basic postulates or axioms and then use logic to draw conclusions or theorems from the axioms. Euclide was the first to use this principle when constructing Euclidean geometry based an 5 axioms about points, lines and circles, like: 

• Through any two distinct points there is a straight line.      (E)

Euclidean geometry gets a physical meaning by associating points to e g dots of chalk on a blackboard and straight lines with strings of dots drawn with the help of a ruler. Euclidean geometry can then give theoretical information about a triangle drawn on the blackboard by e g Pythagoras theorem. 

To give a theory based on axioms a meaning, it is necessary to 

1. To give the axioms meaning.
2. To check that the axioms are true.

Here 1 is required for 2 and if 2 is not true then the theory has no meaning. To secure 2 the axioms are chosen so that they are possible to verify as self-evident, or simple as possible like (E). It the axioms have questionable truth, so has the theory based on the axioms.

An axiom for Newtonian mechanics is Newton's inverse square law of gravitation, which is a consequence  of self-evident conservation laws, and can also be verified experimentally.  

Let us now take closer look at textbook Standard Quantum Mechanics StdQM as based on Schrödinger's Equation SE from 1926 in terms of a complex valued wave function $\Psi (x_1,x_2,...,x_N)$ depending on $N$ 3d spatial variables $x_1,...,x_N$ altogether $3N$ spatial variables for a system with $N$ electrons as configuration space.  A basic axiom for StdQM is 

• The wave function $\Psi (x_1,...x_N)$ is anti-symmetric.        (Q) 

Anti-symmetric means that $\Psi (x_1,...,x_N)$ shifts sign under permutation of any two distinct variables $x_i$ and $x_j$, from which follows that if $x_i=x_j$ for any $i\neq j$, then $\Psi (x_1,...,x_N)=0$.  

StdQM is presented as fundamentally different from classical continuum physics in the sense that $\Psi$ has no direct physical meaning like a charge density in 3d, only with $\vert\Psi\vert^2$ as a probability density over configuration space. 

This means that the axiom (Q) has no physical meaning and as such cannot be verified as being true by self-evidence or observed physics. This means the the role of (Q) as axiom for StdQM can be questioned on very good grounds. 

In particular, it does not suffice to say that antisymmetry guarantees that the probability of two electrons being at the same place is zero, and so affirm antisymmetry by observing a consequent, thus using incorrect logic.

We conclude that the physical meaning of (Q) is unclear, and in particular that it is impossible to verify the validity of (Q).  

The famous mathematician von Neumann gave an axiomatic treatment of StdQM into a formal mathematical theory based on axioms like (Q) in his monumental Mathematische Grundlagen der Quantenmechanik, 1932). The theory was perfect but came with the caveat of unclear physical meaning, which has never been sorted out,  in particular the formality of anti-symmetry.

If wave functions indeed are anti-symmetric, and wave functions have some physical meaning, there must be some physical mechanism guaranteeing anti-symmetry. But nothing like that has been found.

Let us compare with RealQM as an alternative to StdQM based on the following axioms

• Electrons are charge densities filling non-overlapping regions of space.                                   (Q1)
• Electrons are subject to mutual Coulomb repulsion and to Coulomb attraction from kernels.  (Q2)

We note that (Q1) and (Q2) have clear physical meaning and in particular that non-overlap can be motivated by Coulomb repulsion. In short, (Q1) and (Q2) have meaning and can be subject to verification.

The irony of modern physics, today manifesting as a crisis, is that StdQM is presented as the most successful theory of physics all times, while StdQM is still seeking a meaning. 

RealQM has a clear meaning and is today emerging as a challenger.  

Comment by chatGPT:

Condensed Comparison: StdQM Critique • StdQM Defense • RealQM Response

1. 3N-dimensional wave function

• Critique: Not in real 3D space.

• Defense: Domain ≠ physical space; that’s fine.

• RealQM: Uses 3D electron densities → avoids the issue.

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2. Wave function has no clear physical meaning

• Critique: Ontologically vague.

• Defense: No need for ontological commitment.

• RealQM: Provides explicit 3D physical ontology.

---

3. Built-in nonlocality

• Critique: Entanglement = fundamental nonlocal structure.

• Defense: Nonlocality is only in correlations.

• RealQM: Claims local PDEs, but Bell correlations remain an open challenge.

---

4. No fundamental particles

• Critique: Only Ψ exists fundamentally.

• Defense: Particles are emergent; that’s acceptable.

• RealQM: Electrons = continuous charge fields → clear entity.

---

5. Collapse is ad hoc

• Critique: Not part of the dynamics.

• Defense: Collapse is epistemic; decoherence helps.

• RealQM: No collapse at all; deterministic evolution.

---

6. Classical world unexplained

• Critique: Decoherence ≠ definite outcomes.

• Defense: Decoherence is enough pragmatically.

• RealQM: Classical world is built in from the start.

---

7. Born rule is postulated

• Critique: Probability un-derived.

• Defense: Born rule is fundamental.

• RealQM: Deterministic PDEs.

---

8. Wavefunction not physically visualizable

• Critique: No spatial field interpretation.

• Defense: Intuition is optional.

• RealQM: Fully visualizable 3D fields.

---

9. Configuration-space undermines locality

• Critique: No fundamental 3D-local causation.

• Defense: Don’t conflate representation with ontology.

• RealQM: Local 3D causation, but must recover Bell nonlocality.

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10. No mechanism for 3D world emerging from 3N space

• Critique: 3D space is imposed, not derived.

• Defense: Projection via operators is standard.

• RealQM: Everything is 3D from the beginning.

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.

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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.