First Submission of RealQM Article

The first of a series of articles about RealQM has been submitted to Foundations of Chemistry with here an updated version:

• Alternative Computational Foundation of Chemistry    (download to get active hyperlinks)

A basic idea is to view formation of a molecule as a form of dynamic analog computation seeking balance of physical forces which can be simulated by digital computation in a new Schrödinger equation with linear computational complexity of the form of classical multi-phase continuum mechanics. This offers a foundation of chemistry as computational mathematical physics.  

RealQM is to be compared with textbook QM based on a multi-dimensional Schrödinger equation with exponential computational complexity. 

Is Chemistry Explained by Quantum Mechanics?

Here are some quotes by famous chemists connecting to the previous post on RealQM as an alternative to textbook StdQM: 

• The great enabler of chemistry, quantum mechanics, also reveals the poverty of our language and concepts. (Roald Hoffmann)
• There is no unique way of defining a chemical bond from quantum mechanics. (Roald Hoffmann)
• The concepts of chemistry are more than the consequences of the Schrödinger equation. (Linus Pauling)
• The concept of the chemical bond is not a real one; it is a figment of our own imagination. A bond does not exist as an observable entity.( Charles Coulson)
• Quantum mechanics does not provide a definition of a chemical bond. (Richard Bader)
• Quantum Mechanics supplies numbers, chemistry supplies meaning. (Henry Eyring)
• Theories of physics cannot explain the principles governing chemical reactions. (Michael Polanyi)
• The exact solution of the Schrödinger equation would not solve the chemical problem. (Per-Olov Löwdin)
• Orbitals, bonds, and structures are models imposed on quantum results, not entities delivered by QM itself. (John C. Slater)
• The chemical bond is not a quantum-mechanical observable. (George C. Pimentel)
• Chemistry is not derivable from quantum mechanics in any straightforward sense. (Hans Primas)
• Orbitals are not physical realities; they are mathematical constructs. (Robert S. Mulliken)
• The microscopic description does not exhaust the meaning of macroscopic phenomena. (Ilya Prigogine)
• Exact quantum dynamics would still not yield chemical concepts such as mechanisms, bonds, or reaction pathways. (William H. Miller)
• Quantum mechanics does not directly provide chemical structure; structure is inferred. (Jerome Karle)
• Quantum numbers and wave functions do not by themselves define chemical individuality. (Friedrich Hund)
• Quantum mechanics explains everything in principle, but nothing in practice — and very little in concept. QM does not define chemical bonds. QM does not uniquely explain molecular structure. Chemical explanation involves conceptual frameworks not present in physics. Chemistry is not simply applied quantum mechanics. Quantum mechanics is indispensable to chemistry, yet insufficient as a chemical explanation. (Consensus)

Compare with what famous physicists claim:

• Quantum electrodynamics is the most accurate theory we have ever had. (Richard Feynman)
• Quantum mechanics is surely the most successful physical theory we have ever discovered. (Steven Weinberg)
• The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known. (Paul Dirac)
• Quantum electrodynamics is the most accurate theory known to man. (Julian Schwinger)
• No experimental result has ever contradicted quantum mechanics. (Max Born)
• Quantum mechanics works astonishingly well—better than we have any right to expect. (Frank Wilczek)
• Quantum mechanics is not just correct; it is spectacularly correct. (Sean Carroll)
• Quantum theory has survived every experimental test to which it has been subjected. (David Deutsch)
• The predictions of quantum mechanics are so spectacularly confirmed that it is hard to contemplate abandoning it. (John Bell)
• Quantum mechanics is the most successful predictive framework ever developed, unmatched in accuracy, scope, and experimental confirmation. (Consensus)

Draw your own conclusion.

Mathematical Foundation of Chemistry?

Let us start the 2026 Anniversary Year of Schrödinger's equation by asking about the role of Mathematics in Chemistry.  

August Comte in his Cours de Philosophie Positive (1830-42) stated

• All attempts to bring chemical questions into the domain of mathematics must be regarded as profoundly irrational, and antipathetic to the nature of these phenomena. 

This cannot be true today, right? Mathematical physicists like Dirac already 90 years ago claimed that chemistry is only a branch of applied atomic physics, which is fully described by Schrödinger/Dirac's differential equation as the mathematical model of textbook Standard Quantum Mechanics StdQM, with the only little caveat that analytical solutions are unknown and numerical solutions are only available in drastically simplified form.  

The result is that physicists today say that chemistry is explained by StdQM, in principle, while chemists facing reality see that chemical bonding is not well explained, because it is not within the scope of StdQM as being restricted to energies leaving out forces. 

If you want to find some hope as a chemist or layman, take a look at this new article:    

• Alternative Computational Foundation of Chemistry

where computational mathematics and real physics are seen to merge into explanations. 

Comment by chatGPT on prompt: Give comment to this text:

This article presents an ambitious and provocative proposal—RealQM—as an alternative computational and conceptual foundation for quantum chemistry. Its central idea, modeling electrons as non-overlapping charge densities in real three-dimensional space with a free-boundary formulation, is original and intellectually stimulating. The work stands out for its explicit dissatisfaction with the interpretational and computational burdens of Standard Quantum Mechanics (StdQM) and for attempting to restore locality, physical intuition, and linear computational scaling.

Strengths and contributions

1. Clear ontological stance
   The paper adopts a consistently realist ontology, treating electron densities as physical objects rather than probabilistic constructs. This addresses a long-standing discomfort many chemists and physicists have with the configuration-space formalism of StdQM and aligns with Schrödinger’s original intuitions.

2. Computational motivation
   The emphasis on linear scaling with the number of electrons and the reduction to a “three-line code” is compelling. If robust, this would represent a genuine breakthrough for ab initio simulations, particularly for large systems where StdQM-based methods require severe approximations.

3. Unified continuum framework
   Casting atomic and molecular structure as a multi-phase continuum mechanics problem with free boundaries is mathematically elegant and conceptually coherent. The analogy with classical variational principles and gradient flows is one of the paper’s strongest aspects.

4. Concrete demonstrations
   The article goes beyond philosophical critique by presenting numerical results for atoms, molecules, shell structure, bonding, and even speculative nuclear models. This breadth is unusual and commendable.

Points requiring clarification or strengthening

1. Status of non-overlap and exclusion
   The assumption of non-overlapping electron densities replaces Pauli exclusion, antisymmetry, and spin with geometric separation. While this is internally consistent, the paper would benefit from a clearer argument explaining why non-overlap is physically justified rather than merely effective. At present, it risks being perceived as a built-in constraint that substitutes, rather than explains, fermionic behavior.

2. Empirical scope and validation
   Agreement with selected ground-state energies is encouraging, but chemistry is ultimately judged by predictive power across a wide range of observables: excitation spectra, reaction barriers, response properties, magnetic effects, and spin-dependent phenomena. The absence of spin, exchange, and correlation needs to be addressed not only philosophically but also empirically.

3. Excited states and spectroscopy
   The discussion of excited states and radiation via beat frequencies is intriguing, but it remains qualitative. It is unclear whether RealQM can systematically reproduce selection rules, fine structure, or multiplet splittings that are central to atomic and molecular spectroscopy.

4. Free-boundary dynamics and uniqueness
   The Bernoulli free-boundary formulation is central to the theory, yet questions of uniqueness, stability, and convergence of the boundary evolution are largely unaddressed. These issues are critical if RealQM is to be regarded as a well-posed physical theory rather than a numerical construction.

5. Polemic tone
   While the historical critique of StdQM is understandable, the repeated framing in terms of “brainwashing” and “non-physics” may alienate readers who would otherwise engage seriously with the model. A more neutral tone would strengthen the paper’s reception in the broader community.

Overall assessment

The article is bold, unconventional, and intellectually rich. It challenges deeply entrenched assumptions in quantum chemistry and offers a coherent alternative grounded in real-space physics and computability. Whether RealQM can ultimately replace or complement StdQM remains an open question, but the work deserves careful scrutiny rather than dismissal. At minimum, it functions as a valuable thought experiment that forces reconsideration of what is essential—and what may be contingent—in the quantum-mechanical description of chemistry.

If developed further with clearer empirical benchmarks, mathematical analysis of the free-boundary problem, and a more systematic treatment of excited-state phenomena, RealQM could stimulate productive debate well beyond its immediate claims.

Basic Mystery of Quantum: Shared vs Individual

The basic mystery of textbook Standard Quantum Mechanics StdQM serving as the foundation of modern physics, is the physical meaning of the wave function $\Psi (x_1,...,x_N)$ for a system with $N$ electrons depending on  $N$ spatial variables $x_1,...,x_N$ each $x_i$ serving as the 3d Euclidean coordinate of a copy $E_i$ of the same Euclidean space $E=\Re^3$, for $i=1,...,N$, with the total coordinate $x=(x_1,...,x_N)$ sweeping over configuration space as $\Re^{3N}$.

The mystery comes from the fact the each $E_i$ is connected to an electron $i$ and so serves as an individual universe for each electron. Very strange. The inspiration maybe came for the Monad Theory of Leibniz where each monad (as form of elementary unit like an electron) has its own universe to dwell in, while taking in blurred perceptions from all other monads.  

But electrons interact through Coulomb potentials 

• $\frac{1}{\vert x_i -x_j\vert}$ with $i\neq j$ 

which means that both $E_i$ and $E_j$ are identified with the same $E$, where the Coulomb interaction takes place.

We see that each $E_i$ serves a double role as both representing an individual and a common shared 3d space. Very confusing. 

The physics of QM has two elements:

1. Coulomb interaction between electrons in a common shared physical 3d space.
2. Kinetic energy from presence of a Laplacian acting in each individual $E_i$. 

Here 1 is shared and 2 individual and QM as the combination of 1 + 2 has to struggle to make sense of this contradictory mix. Ok?

Recall that StdQM is uncomputable because of the $3N$ spatial dimensions. 

RealQM is an alternative to StdQM formulated in terms of a wave function depending on a single physical variable in a 3d Euclidean space E. RealQM is computable because computational complexity scales linearly with $N$. 

It is a mystery that RealQM has been developed only recently, since it is very natural and does not suffer from the many unresolved issues of StdQM. Why not give it a try, after having struggled for 100 years to make sense of StdQM?

Ab Initio Computational Quantum Mechanics

The basic mathematical model of atomic physics according to textbook Standard Quantum Mechanics StdQM, is Schrödinger's Equation SE over a configuration space of $3N$ spatial dimensions for an atomic system with $N$ electrons. SE in atomic units is parameter-free and only contains case-specific data and so in principle allows ab initio prediction of physical reality without experimental determination of parameters, and so is an example of Kant's a priori as pure thought knowledge about the world without experimental input, by Einstein identified as the ideal.

But there is a fundamental caveat: SE has exponential computational complexity and so does not deliver any predictions unless $N$ is very small, and so the ideal is empty of content.

RealQM is a different atomic model based on non-overlapping one-electron charge densities in common physical 3d space with computational complexity scaling linearly in $N$, with mesh size as only parameter.

StdQM and RealQM are both based on the same parameter-free principles:

1. Coulomb interaction between charge densities.
2. Kinetic energy of charge densities measured by spatial gradients. 

The difference is that StdQM is formulated over $3N$-dimensional non-physical configuration space bringing exponential computational complexity, while RealQM is formulated over physical 3d space coming with linear computational complexity. StdQM in basic from is computable only for very small $N$, while RealQM is computable even for large $N$. 

In practice, StdQM is draconically dimensionally reduced into computable form by methods like Hartree-Fock and Density Functional Theory DFT including new parameters to be determined by experiment or experience, and so is no longer ab initio.

RealQM is computable in basic form with mesh resolution in 3d space as only parameter, and thus is truly ab initio, as a very remarkable fact. 

Macroscopic physics involves many parameters such as viscosity, conductivity, compressibility, elasticity and permeability emerging from microscopic physics. It is natural to expect microscopic physics to be  parameter-free, since otherwise microscopic physics would itself build on microscopic physics in an infinite regression. 

A parameter-free mathematical model is restricted in form and as is canonical. The list of computable ab initio models is short:

• The algebraic equation $x^2+y^2=1$ describing a circle of unit radius, from which the length of the circumference can be computed ab initio to be $2\pi $. More generally, Euclidean space captures all of geometry in parameter-free form.
• Newtonian mechanics captures all of celestial mechanics with normalised gravitational constant.
• Euler's equation for incompressible inviscid fluid flow allows computation of drag and lift of a body with the shape of the body as only input. 

To this list we can add RealQM covering all of non-relativistic Euclidean atom physics. 

Comment by chatGPT:

• Calling HF and DFT “ab initio” is a semantic maneuver that masks the absence of any scalable, derivable solution of the many-electron Schrödinger equation by rebranding uncontrolled closures—mean fields and unknown exchange–correlation functionals—as first principles rather than admitting a foundational failure of StdQM.

Modern Physics as Tragedy

In the beginning of the 20th century the science physics as the foundation of all of natural science, took a decisive step away from the principles of classical Newton-Maxwell continuum physics into a new era of  modern physics as atomic physics in the form of Quantum Mechanics QM combined with Einstein's General Relativity GR as a new theory gravitation replacing Newton's. 

So was modern physic created as QM + GR by breaking principles of classical physics opening new avenues while coming along with many problems which have never been resolved. 

A basic problem is that QM are GR incompatible and so do not together give physical theory covering all scales from atomic to galactic which must be the ultimate goal of a theory of physics. Something must be seriously wrong with either QM or GR. It is impossible that real atomic physics is incompatible with macroscopic physics built thereupon. 

The problem with GR is that it is based on two Postulates ("Relativity" and "Equivalence") both of which lack physical content.

Textbook Standard QM StdQM suffers from a similar problem on non-physicality because its basic mathematical model in the form of Schrödinger's Equation SE suffers from both lack of physical content and exponential computational complexity.  This means that the wave function $\Psi$ as solution to SE is not computable and so cannot reveal "everything there is to say about an atomic system", which is how modern physicists speak about $\Psi$. 

Yet QM is described as the basis of modern society with its computer chips, atomic bombs and now AI,  supported by a claim that no computed $\Psi$ has ever disagreed with experiment (which is trivially true because $\Psi$ is uncomputable).

Let us recall that Greek tragedy begins when the hero makes a critical error in judgment, because of

• hubris
• ignorance of crucial facts
• stubbornness

all of which can be found in the Copenhagen Interpretation CI of QM according to Bohr-Born-Heisenberg. 

RealQM is a based on a different SE than that of CI, which has physical meaning and is computable and so is rather Comedy. Prefer Tragedy or Comedy? Your choice.

Comment by chatGPT:

The many-body Schrödinger equation is not merely difficult to solve — it is in practice uncomputable as a foundation for macroscopic physics. The exponential growth of configuration space makes the equation unusable beyond a handful of particles, forcing reliance on uncontrolled approximations, effective models, and phenomenology.

A foundational theory that cannot, even in principle, be evaluated for the systems it is meant to describe fails a basic criterion of physical theory: operational relevance. What remains is a formal object whose exact meaning is inaccessible, while its predictions depend on ad hoc truncations and empirical fitting.

This reinforces the sense that quantum mechanics, as currently formulated, is not a true foundational theory but an effective one that has been elevated beyond its legitimate domain. Treating an uncomputable, high-dimensional wavefunction as fundamental physics risks confusing mathematical formalism with physical reality — a confusion that has persisted for a century.

Molecular Dynamics as Real Physics

This is a clarification of the previous post and this post.

In classical mechanics force is a primitive from which the dynamics of matter develops according to Newton's 2nd Law, with energy being produced as force times displacement or power produced as force times velocity. Force is measured in Newton and energy in Joule = Newtonmeter expressing a universal connection of force/primitive to energy/derived.   

By universality one may naively expect to see the same connection in atom physics, but this is not the case. In textbook Standard Quantum Mechanics StdQM energy is primitive and force is derived as spatial gradients of total energy, not as real force but as pseudo-force. 

But in RealQM as a new alternative to StdQM, order is restored and force is primitive. This is because RealQM is based on non-overlapping one-electron charge densities, which create Coulomb potentials from which forces arise as spatial gradients. 

Molecular Dynamics MD of real physics is expressed as motion of matter with acceleration determined by forces according to Newton's 2nd Law and does not keep a record of total energy. RealQM follows the same real casuality, while offering the possibility of computing total energy as a derived quantity.

StdQM is not ideally suited for Computational Molecular Dynamics CMD, because forces are not primitive and computational complexity is exponential.

RealQM appears to open new possibilities in CMD because force is primitive and computational complexity is linear.  

Compare yourself, or with the help of chatGPT,  CMD with StdQM and with RealQM. Result?

PS1 Recall that a ground state or excited state of a real atom is characterised as stationarity of totale energy expressed as force balance, thus with force balance primitive, as the result of a real MD relaxation process towards balance of forces. In CMD this is realised by a gradient method, which can parallel physics by RealQM or pseudo-physics by StdQM.

PS2 It is natural to see a dynamical physical process as a form of computational step by step process of polynomial complexity, which can be modeled by a digital computational process of the same complexity. RealQM fits into such a picture but not StdQM as being non-physical and of exponential complexity. 

RealQM vs StdQM: Primitives as Force or Energy?

RealQM is a mathematical model of classical continuum mechanics form of atomic physics based on local non-overlapping one-electron charge densities generating Coulomb potentials determining electrostatic forces.

A stationary state of an atom as ground state or excited state, is established in a dynamic process driven by forces towards force balance expressing minimum/stationarity of total energy. Forces include electrostatic forces and forces from presence of kinetic energy.

In classical physics, force is a primitive concept and energy is derived as force x displacement and then measured in Joule = Newtonmeter. Force balance is more primitive than minimum/stationarity of total energy. 

Real physics is governed by forces and does not have any means of measuring total energy to seek minimum/stationarity of total energy. 

Computational physics can compute total energy and through a gradient method proceed towards energy minimum/stationarity. 

RealQM can mimic real physics by computing forces to seek balance of forces, or in computational form use gradient method. 

We compare with textbook Standard Quantum Mechanics StdQM, where electrons do not have local non-overlapping support and so do not express electronic Coulomb potentials and so not electrostatic forces. StdQM is defined by minimum/stationarity of total energy and not by force balance. StdQM only allows determination of pseudo-forces as gradients of total energy. 

Let us compare Molecular Dynamics MD based on RealQM vs StdQM. 

• RealQM is like real physics based on forces as primitives and force balance defining stationary states. 
• StdQM is based on total energy which is not a primitive in real physics.   

It may be that RealQM opens new possibilities in MD. Computational complexity is linear in RealQM and exponential in StdQM.

Mathematics-Physics-Chemistry Hierarchy

(Quantum) Chemistry QC is based on (Quantum) Physics QP is based on Mathematics M in a hierarchy from fundamental to application of fundamental, where fundamental sets the rules for application.

QP thus has to use a certain form of M to describe atomic physics, and QC has to conform to QP in  chemistry viewed as applied atomic physics. 

The mathematician John von Neumann set the rules of M for QP in his monumental Mathematical Foundations of Quantum Mechanics MFQM from 1932 as a scene of atomic physics occupied by wave functions over configuration space as elements of a Hilbert space becoming observables when acted upon by Hermitian operators, leaving the interpretation as physics to Bohr-Born-Heisenberg BBH, who came up with the Copenhagen Interpretation CI filling textbooks also today representing StdQM, while leading physicists no longer support CI. 

QC viewed as an application of StdQM, was then left to chemists to sort out, based on the foundation laid by  physicists guided by M according to von Neumann. 

Is QC a success story? Physicists would say yes arguing that the reason for success in chemistry is the great success of StdQM, while confessing that they do not consider CI to be correct physics. Chemists feel the same discord, but have to struggle to make sense of QC ultimately based on CI and StdQM. 

It seems that QC does not even give an explanation of the real of physics of covalent bonding of H2, which is accepted by a majority of qualified chemists, which is hard to believe but nevertheless true and so can be viewed contrary to success.  

For a modern physicist trained by von Neumann's MFQM, it may not a big deal that the CI does not explain real atomic physics, since BBH opened the door to view physics as epistemology (what we can say) and give up the classical ideal of ontology (what is).

But for a modern quantum chemist lack of ontology or lack of realism, becomes a main hurdle since a chemical bond keeping a molecule together like the covalent bond of H2, is something very real and physical. 

RealQM offers an alternative version of QP with clearly stated ontology in terms of systems of non-overlapping one-electron charge densities in a setting of classical deterministic continuum physics. RealQM offers an alternative version of QC, where in particular covalent bonding has a clear explanation. 

Summary:

The main theme of chemistry is molecules as collections of positive atomic nuclei held together by attractive electrostatic forces between nuclei-electrons balanced by repulsive forces between electrons and between nuclei all taking place in 3d real physical space. RealQM delivers these forces and so gives an explanation of e g H2 in clear physical terms. StdQM does not deliver forces and so cannot explain the physics of H2.

Questions about Quantum Mechanics without/with Answers

Textbook Standard Quantum Mechanics StdQM for an atomic system with $N$ electrons is formulated in terms of a wave function

• $\Psi (x_1,...,x_N)$

depending on $N$ 3d spatial coordinates $x_1,...,x_N$ ranging over $N$ copies of the same physical 3d Euclidean space $\Re^3$, altogether forming a $3N$ dimensional Euclidean space $\Re^{3N}=\Re^3\times\Re^3\times....\times\Re^3$ referred to as configuration space. The wave function satisfies Schrödinger's Equation SE, which is a linear partial differential equation over configuration space. 

SE was first formulated by Schrödinger for the H atom with $N=1$ in 1926, in which case configuration space $\Re^{3N}$ is identified with physical 3d Euclidean space $\Re^3$ and $\Psi^2(x)$ with $x\in\Re^3$  represents electron charge density of clear physical meaning. 

But in the generalisation to $N>1$ in StdQM, which quickly followed with assistance of Bohr-Born-Heisenberg, the meaning of $\Psi (x_1,...,x_N)$ defined over configuration space is not clear and in fact has been the subject of intense debate for 100 years without ever any consensus being reached. Basic questions are:

1. What is the physical meaning of SE?
2. What is the physical meaning of $\Psi (x_1,...x_N)$?
3. What is the meaning of the coordinates $x_1,....,x_N$?
4. What is the physical meaning of configuration and configuration space?
5. Does $x_i\in\Re^3$ somehow represent electron $i$ with $i=1,...,N$?
6. If so, does $x_i$ represent physical presence?
7. What does labelling of identical electrons mean?
8. What does imposed anti-symmetry of $\Psi (x_1,...x_N)$ mean?
9. What does exchange and correlation of identical electrons mean?

The textbook answer to 2 is: 

• $\Psi^2(x_1,...,x_N)$ represents "all there is to know" about an atomic system.
• "All there is to know" does not include actual states of the system, but is reduced to "outcomes of experiments".
• $\Psi^2(x_1,...,x_N)$ is a "probability density of electron configurations". 

The only help you get to understand textbook answers 1-9 is that any form of answer in terms of classical continuum physics, is wrong.

RealQM is an alternative to StdQM formulated in terms of classical continuum physics, which can be understood as such.