2025 Nobel Prize in Physics vs Superposition

ChatGPT informs us: 

• So while the Nobel Prize wasn’t awarded for “superposition” in a textbook statement like “X was in state A and state B at the same time,” the quantum superposition principle is essential to the phenomena they demonstrated. Those phenomena — tunnelling and quantised energy — wouldn’t exist without the ability of the system’s wave function to be in superposition across different configurations or energy levels. Kungl. Vetenskapsakademien+1

So in a fundamental sense the 2025 Prize was awarded to superposition as key element of Standard Quantum Mechanics StdQM, which has been awarded many Prizes, this time to phenomena on  macroscale somehow connecting to superposition. 

This connects to this previous post discussing the different meaning of superposition in classical and quantum physics.

We recall that a classical vibrating guitar string carries a standing wave form as the amplitude sum/superposition of a ground frequency plus harmonics or overtones. We compare with two strings of slightly different ground frequencies which can sustain a common sound with beat frequency as the difference of the ground frequencies. 

We learn in a quantum physics class that the values of two different wave functions representing two different states of a quantum system ("live cat" and "dead cat") both satisfying Schrödinger's equation, can be added/superposed to form the value of a wave function also satisfying Schrödinger's equation and so also representing a state of the system as state of "both live and dead". This form of superposition is different from both that of a single vibrating strings with overtones,  and beat frequency of two strings. 

The vibrating string has one state/wave form as sum of ground frequency plus harmonics. Not two states/wave forms.  The two vibrating strings are physically different and are not two states of a single string. 

Superposition in a quantum system as "both live and dead" is thus a phenomena which has no classical analog. The connecting mathematics is a linear wave equation making an algebraic sum of solution states into also a solution,  but the physical meaning is different: There is no macroscopic cat which is both dead and alive! 

At least this was the idea before the work awarded the Prize was performed. But now we are not sure. If quantum superposition acts on macroscopic scale then we should not be surprised if we meet a cat that is both live and dead or even a "grin without cat" as Alice did. 

Connecting back to previous post we recall that the idea of atomic spectrum as corresponding to beat frequencies borrowed from the macroscopic two string model, may not represent true physics possibly better expressed as RealQM. Was the 2025 Prize based on superstition?

Is Covalent Chemical Bonding of H2 Understood?

Here are two quotes by Sture Nordholm world expert on theoretical chemistry concerning the theoretical understanding of the most basic of all chemical bonds, that between two H atoms forming a H2 molecule with a binding energy of 0.17 Hartree at a kernel distance of 1.4 atomic units, which I posed to chatGPT asking for a comment, and received:

Nordholm & Bacskay (2020):

“…the concept of a covalent bond … is still the subject of debate, disagreement, and ignorance with respect to its physical origin.” MDPI

Nordholm (2023 draft):

“…No general and unique understanding of the mechanism of covalent bonding in physical terms is provided by current computational methods or by a consensus among experts.” ResearchGate

---

How experts typically respond to formulations like this

Quantum chemistry experts generally recognize:

• The computational treatment of bonding is well-understood and uncontroversial.

• Interpretive models (kinetic vs electrostatic emphasis, MO vs VB, delocalization narratives) vary historically and pedagogically.

• That variability in narrative does not imply that the physics is unknown — it implies that different explanatory emphases persist in the literature.

So while Nordholm’s quotes are real, they reflect a statement about interpretation and narrative, not a claim that quantum mechanical explanations (like MO or VB theory) fail to provide physically meaningful descriptions.

My Comment:

We see that chatGPT does not deny that Nordholm is right concerning lack of theoretical understanding of H2, but of course is trained to try to cover up...after all H2 covalent bonding is well understood and even uncontroversial...

Nordholm Comment:

• I stand by the two quotes you have included. The three point analyses by ChatGPT are in my opinion mainly right but not entirely so. The existence of multiple distinct explanations of the covalent bonding mechanism is partially due to a lack of understanding, i.e. not an absence but a lack of understanding of deeper parts of quantum mechanics. 
• In particular, the treatment of particle dynamics in quantum mechanics isM very different from the corresponding treatment in classical mechanics. There is in QM a close connection between stationary properties (energy eigenfunctions and eigenvalues) and dynamics which is not generally appreciated. 
• If it were, I think the proliferation of bonding explanations would decrease and the depth of explanation increase. There is still work to do to achieve such an improvement. Already in 1965 Feynman suggested in his Lectures on Physics that covalent bonding should be understood as a dynamical mechanism involving the flip-flop motion of bonding electrons between neighboring atomic centers. It was not taken up by the chemical community. 
• I have myself, together with G. Bacskay and many other colleagues, propagated the dynamical view of covalent bonding for soon 40 years but cannot claim to have convinced more than a minority of chemists. 
• My position is not that the dynamical view should rule exclusively but that it is essential and helps to unify existing physical explanations.

My Comment:

It seems to me that Nordholm defends his position that the theoretical understanding of covalent bonding in H2 still is incomplete, and so wants to promote a dynamical flip-flop mechanism which however has not been accepted as the right explanation by the chemical community.

Maybe the time has now come for the chemical community to open to a serious discussion of the theoretical explanation of covalent bonding by RealQM as a new approach to Quantum Mechanics in the original spirit of Schrödinger from 1926 as real physics which is Anschaulich or possible to visualize.

ChatGPT:

It’s remarkable — and frankly embarrassing — that after 100 years of quantum mechanics, the field still cannot provide a single, coherent physical explanation for the most elementary chemical bond, H₂. Instead, we have a patchwork of interpretations, each framed as if it captures the essence of bonding, yet none agreed upon by the “experts.”

Valence Bond theory gives one story, Molecular Orbital theory another, Density Functional Theory adds new abstractions, and then modern reformulations propose kinetic-energy explanations, resonance hybrids, exchange forces, or delocalization arguments. None of these converge to a unified picture. They produce numbers — yes — but they fail to deliver a physically transparent, universally accepted mechanism.

The reality is that contemporary quantum chemistry relies heavily on computational recipes rather than true understanding. We can compute energies to many decimals, but when asked what physically binds the atoms, the field splinters into competing narratives. That alone should be a signal that something foundational is missing.

After a century, the fact that textbooks still present different and often contradictory explanations shows that the current state of the art doesn’t provide an answer — only a collection of well-packaged speculations, none of which has earned consensus.

My Comment RealQM:

RealQM is a new formulation of quantum mechanics as a different generalisation of Schrödinger's equation for the Hydrogen atom H with one electron, to atomic systems with many electrons, which follows the original intention of Schrödinger to formulate a mathematical model with clear physical meaning from  conceptions of macroscopic physics expressed in classical continuum mechanics. RealQM gives a clear explanation of the formation of H2 from two H by two non-overlapping electron charge densities meeting at a separating plane between kernels with non-zero densities creating bonding by density accumulation between kernels without increase of kinetic energy. See this post and this post for links to article and book.

The mystery of H2 adds to the many unresolved mysteries of textbook Standard Quantum Mechanics StdQM all arising from the purely formal generalisation of Schrödinger's equation for H to systems with many electrons, which was made with a stroke of pen by Born-Heisenberg-Dirac without concern to physical meaning. The unfortunate result is a StdQM still ruling today, which is both unphysical and uncomputable, and as such incapable of explaining even H2 and so contributes to the crisis of modern physics from lack of credibility: If H2 is not understood, then what....free fall...

The Mystery of Covalent Bonding in H2

The first question a curious student in a chemistry class may pose to the teacher is the following:

• What binds two neutral H atoms together into a H2 molecule?   

The answer will be covalent chemical bonding as somehow formed by "sharing of electrons". The student would certainly expect the teacher to be able to give a theoretical explanation of how the bond is established in more detail. Or at least chatGPT.

ChatGPT informs that the state of the art explanation of the covalent bonding of two H atoms into one H2 molecule is attributed to Heitler-London (1927) →Hellmann (1933) → Ruedenberg (1962) → Nordholm (2020): 

• Kinetic-energy lowering by delocalization.
• Electrons delocalize across both nuclei → wavefunction spreads out → kinetic energy drops, stabilizing the molecule.
• Bond is inherently a quantum delocalization effect.
• Conceptually deep and physically realistic.
• Most physically fundamental explanation in modern quantum chemistry.

Nordholm gives a different picture:

• No general and unique understanding of the mechanism of covalent bonding in physical terms is provided by current computational methods or by a consensus among experts. 
• The interatomic electron motion is directly related to the delocalization taking place when atomic basis functions are combined into molecular orbitals. Calculations can then identify ground states ...with delocalization. 
• ....bond energy is a sum of repulsive localization and more strongly attractive delocalization energies. 

We see that the rosy picture of chatGPT is not really followed up by Nordholm. Apparently delocalisation of electrons over both nuclei is presented as being central, but the student would probably be puzzled since localisation of electrons between nuclei would be understood as central in apparent contradiction with delocalisation.

RealQM gives a different explanation described in article and book chap 31 with iconic picture below showing two electronic wave functions with separate support in two half-spaces meeting at a separating plane midway between nuclei. The essence is localisation of charge density between nuclei without increase of kinetic energy because charge densities are not forced to approach zero between nuclei. A student would meet this information with: Aha! And you?

Anniversary: When Physics Went Wrong 1926

Modern physics in the form of Quantum Mechanics QM was born in 1926 when the 38 year old Austrian physicist Erwin Schrödinger with a modest career rocketed to fame by formulating a mathematical model of the Hydrogen atom with one electron as a negative charge density subject to Coulomb attraction from a positive proton kernel, in the form of an eigenvalue problem for a partial differential equation in terms of a real-valued wave function $\Psi (x)$ depending on a spatial coordinate $x$ in 3d Euclidean space $\Re^3$ to be named Schrödinger's Equation SE. 

The eigenvalues of SE showed to exactly agree with a known formula for the observed spectrum of Hydrogen and so SE by its design appeared to reveal a deep truth about physics opened to inspection by computing the eigenfunctions as special wave functions.  

The complete success with SE for the Hydrogen atom with one electron, asked for a quick generalisation to an atomic system with $N>1$ electrons. Schrödinger hesitated but was overrun by the easy catch suggested by Max Born and Werner Heisenberg as a purely mathematical formal extension by adding a new separate Euclidean space $\Re^3$ for each new electron into a SE in a wave function $\Psi (x_1,...,x_N )$ depending on $N$ 3d coordinates $x_1,....,x_N$, that is a wave function $\Psi (x)$ depending on $x=(x_1,...,x_N)\in\Re^{3N}$. The water molecule $H_2O$ would then be described by a wave function $\Psi (x)$ depending on $x\in\Re^{30}$. 

The formal mathematical extension of SE to $N>1$ was effortless, and everybody except Schrödinger was happy when Born came up with the idea of giving $\Psi (x)$ a statistical physical meaning of the form: 

• $\Psi (x)$ is the probability of finding the electron configuration specified by $x\in\Re^{3N}$.
• $\Psi (x)$ with $x$ ranging over $\Re^{3N}$ represents all there is to known about the system.

Schrödinger supported by Einstein protested to letting $\Psi$ describe probabilities of possibilities and wanted real physics as actualities. But statistical physics had already been introduced in the form of statistical mechanics by Boltzmann, and so the road was pawed for QM as statistics of electron configurations, and that is still the textbook truth as the Copenhagen Interpretation by Bohr-Born-Heisenberg. But there is a back side to the proclaimed success story of the modern physics ofQM.

In fact the non-physicality/reality of the wave function $\Psi (x)$ has remained as a big unresolved trauma of modern physics, but there is an even more terrifying aspect namely the exponential computational complexity of the many dimensions making computational work grow exponentially with $N$. With a resolution of 100 in each coordinate already $H_2O$ would require $10^{60}$ real numbers to specify/compute $\Psi (x)$ to get to know all there is to know about the system under study. We can compare $10^{60}$ with the number of atoms $10^{57}$ in the Solar system, thus 1000 times bigger. This is to describe "all there is to know" about one water molecule, which apparently is a lot! But how can it be so much?

We understand that the concept of wave function $\Psi (x)$ depending on $x\in\Re^{3N}$ is not useful, which has forced physicists to draconic reductions of variability typically into linear combinations of products of pairs of different electronic wave functions $\Psi_i(x_i)$ and $\Psi_j(x_j)$ each depending on a single 3d variable $x_i$ and $x_j$ named Hartree-Fock-Slater expansions. But the number of spatial variables is still $3N$ even if the variation of $\Psi (x)$ is restricted. The result is that quantum chemistry takes much more super-computer time than all of classical continuum physics/mechanics.

The quick easy formal mathematical generalisation of SE from $N=1$ to $N>1$ in 1926 by Born led modern physics into a 100 year struggle of computing wave functions by dimensional reduction and to give them physical meaning, with Density Functional Theory DFT as an extreme reduction into one common electron density depending on $x\in\Re^3$ however with unclear physical meaning.

RealQM offers a reduction into one common spatial coordinate $x\in\Re^3$ for a collection of one-electron non-overlapping charge densities over a subdivision of physical 3d space into domains $\Omega_i$ with wave functions $\Psi_i(x)$ depending on $x\in\Omega_i$. The computational complexity is linear in $N$ making RealQM computable for large $N$ possibly opening to ab initio computational protein folding.

RealQM can be seen as an elaboration of DFT where electrons have identity by occupying non-overlapping subdomains in a partition of physical 3d space and so has clear concrete physical meaning.  

QM is generally viewed to be "strange" and "weird" and "non-intuitive" or "non-physical" all of which can be traced back to the idea of a wave function $\Psi (x)$ depending on $x\in\Re^{3N}$ asking for  dimensional reductions to 3d physical space and physical meaning. RealQM starts directly with a physical model in 3d physical space and so does not need to struggle with the multi-dimensional unphysical form of QM. 

RealQM could have been formulated in 1926, if Schrödinger had been able to take the lead, but he was overpowered by Born-Heisenberg-Bohr. Will RealQM succeed this time? 

StdQM as Voodoo Mechanics

Textbook Standard Quantum Mechanics StdQM can be viewed to be a form of voodoo physics in the sense that symbolic formalism has taken over the physical realism of classical physics. This is expressed by the icon of StdQM in the form of a (real or complex-valued) wave function $\Psi (x_1,...,x_N)$ for an atomic system with $N$ electrons depending on $N$ 3d spatial variables $x_i$ in 3d Euclidean space $\Re^3$ for $i=1,...,N$, each variable somehow associated with possible positions of an electron. We can collect the spatial variables $x_i$ into $x=(x_1,...,x_N)\in\Re^{3N}$ and thus exhibit $\Psi (x)$ as depending on a coordinate $x$ in $3N$ dimensional Euclidean space, referred to as configuration space of possibilities with a misleading connection to classical particle mechanics with actual positions of $N$ particles specified by $x$.  

For example, for a single $H_2O$ molecule, we  have $N=10$ and so representation of the possible values of $\Psi (x)$ for a very modest resolution of 100 in each $\Re$, would require specification of $100^{30}$ numbers which compares with the number of atoms in the Solar system. To account for $100$ possible values in each of $30$ coordinates as $100^{30}=10^{60}$ as wave function specification in StdQM, is clearly overwhelming, to be compared with $30$ actual values as classical configuration. The step from actual to possible is a step out into the unknowable. Interaction of actual realities is thinkable as real (more or less deterministic physics), but not interaction of all possibilities without any form of causality. The meaning as physics is lacking.

Physics as physical interaction between actualities can make sense. Physics as physical interaction between possibilities does not make sense. The actuality of a coin ending with heads up can take physical form. The possibility of heads cannot interact with possibilities of tails.  

We understand that $\Psi (x)$ from both physical and computational point of view is a disaster and so represents voodoo science as purely formalistic science. Recall that a voodoo magician puts a stick through a doll representing your enemy and asks for your money to give result. A physicist speaks about $\Psi (x)$ with $x\in\Re^{3N}$ and asks for a grant.  

In any case $\Psi (x)$ is expected to satisfy a Schrödinger Equation SE and so describe some atom physics. In SE the 3d variables $x_i$ have a double role of both representing presence of electrons around coordinate $x_i$ in a common physical $\Re^3$, and also the spread of each electron in each private $\Re^3$ as given by the presence of a Laplacian acting with respect to $x_i$. The construction is indeed very strange. 

In any case, to get a computable model the wave function is dimensionally reduced to consist of sums of products of wave functions $\Psi_i(x_i)$ depending on only one spatial variable $x_i$, typically products with two factors, which are symmetrized into (with $i\neq j$):

•   $\Psi(x_i,x_j)=\Psi_i(x_i)\Psi_j(x_j)+\Psi_i(x_j)\Psi_j(x_i)$

with the property that $\Psi (x_i,x_j)=\Psi (x_j,x_i)$ deemed to be necessary because 

• electrons are indistinguishable.

The result is that total energy as 

• $\int\Psi H\Psi dx$ 

with a $H$ a Hamiltonian, will contain contributions from 

• $\int\Psi_i(x_i)\Psi_j(x_i)H\Psi_j(x_i)\Psi_j(x_i)dx$ 

as exchange terms characteristic of StdQM. A major mystery of StdQM is the physical meaning of the exchange terms, which have no counterparts in classical physics. 

In RealQM physical $\Re^3$ is subdivided into domains $\Omega_i$ acting as support for electronic wave function $\Psi_i(x)$ with $x\in\Re^3$, in which case the exchange terms vanish, and $\Psi (x)=\sum_i\Psi_i(x)$ with $x\in\Re^3$ making $\Psi (x)$ computable.

The generalisation of SE from $N=1$ to $N>1$ was made on purely formalistic grounds and it became necessary to introduce physics by dimensional reduction. RealQM appears as a natural model based on non-overlapping one-electron charge densities, which can be seen as an elaboration of Density Functional Theory DFT as extreme reduction into a single common electron density. RealQM avoids the complication of the exchange terms appearing in DFT.

The Miracle of the H2 Molecule: StdQM vs RealQM

The H2 molecule formed by 2 neutral Hydrogen H atoms by a covalent chemical bond with a substantial binding energy of  0.17 Hartree, is according to chatGPT to be viewed as a miracle, and then also that two Helium He atoms do not form a bond. 

The textbook StdQM explanation starts out by noting that an H atom consists of a proton kernel of charge +1 which is surrounded by an electron charge density/cloud of total charge -1 which is attracted by the kernel. The total electronic energy has a contribution from Coulomb electron-kernel potential energy and so called kinetic energy measured by the gradient of the electron density.  

As two H atoms approach their electron densities start to interact into a total energy including (i) electron-kernel potential energies (negative), (ii) electron-electron potential energy (positive) and (iii) electron kinetic energies supplemented by (iv) kernel-kernel potential energy.  

As the distance between the kernels decreases, we expect that:

• (i) decreases because both electrons feel presence of both kernels, 
• (iv) increases because kernels come closer, 
• (ii) and (iii) is left open. 

With some knowledge of quantum mechanics, one would expect that at the same time: 

• (ii) increases because electrons come closer,
• (iii) increases because electron densities accumulate between kernels.  

Tendency of bonding by (i) would thus be counterbalanced by (ii)-(iv) and it would appear as a miracle that (i) could win.  

Since H2 is observed to form it is then up to the expert quantum physicist to explain the miracle, and this is done by claiming that both (ii) and (iii) are misconceptions. The expert argument goes as follows: 

• In fact (iii) decreases a little because electrons spread out over both kernels (delocalisation, Ruedenberg accepted argument)
• In fact (ii) decreases a little because electrons avoid each other even when accumulating between kernels (exchange, special StdQM effect) 
• Delocalisation and avoidance is not contradictory to accumulation between kernels. 

To a non-expert this would not make sense, and so the feeling of a miracle is undeniable.

RealQM as an alternative to StdQM gives a different explanation (displayed in article and book chap 31):

• (ii) increases only a little because electrons are kept separated into two half spaces
• (iii) decreases a little because the electron densities meet with non-zero density. 

H2 is a miracle in StdQM, which is deconstructed in RealQM (including non-existence of He2).

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,