Why Was RealQM Not Found in 1926?

Schrödinger's equation SE in its linear multi-dimensional form with wave function solution $\Psi (x)$  depending on a $3N$-dimensional spatial variable $x$ for a system with $N$ electrons is viewed to be the foundation of quantum mechanics of atoms and molecules as Standard Quantum Mechanics StdQM. 

Without SE physics would be thrown back to 1924 with only Bohr's model of the atom in a form of classical physics. There are no physicists of today that advocate that this makes any sense. 

But SE has been subject to deep dispute since its formulation by Schrödinger in 1926 followed by Born's suggestion of giving the wave function $\Psi (x)$  a meaning by saying that $\vert\Psi (x)\vert^2$ represents the probability of an electron configuration described by $x$. This means that $\Psi (x)$ is given an epistemic meaning as what "a physicist can say", and not an ontic meaning as physics independent of what a physicist may have to say, as expressed by Bohr. 

But Bohr's view was not satisfactory to many physicists who wanted to find an ontic meaning of $\Psi$ independent of what people may have to say, but nothing really convincing was ever found. The $3N$-dimensionality of $\Psi (x)$ defied real physical meaning and then the only option was an epistemic statistical meaning. 

This means that the basic foundational problem of SE has never been resolved despite intense debate over 100 years with no consensus in reach, except an agreement that after all the physical meaning of $\Psi (x)$ does not matter much, since it has shown to always deliver predictions in full agreement with observations, but then predictions with unclear physical meaning of course. 

But there is an alternative to StdQM which could have been formulated in 1926, but for some reason was missed, see this post. This is RealQM as a different form of Schrödinger's equation as a non-linear system of one-electron wave functions $\psi_i(x)$ for $i=1,...N,$ with non-overlapping supports depending on a common 3d space variable $x$ with $\vert\psi_i(x)\vert^2$ as physical charge density with direct ontic meaning. 

Connecting to the previous post, RealQM can be seen as a Platonic/mathematical generalisation from $N=1$ to $N>1$ where the physicality for the Hydrogen atom with $N=1$ is kept. On the other hand, StdQM can be seen as an Aristotelian/pseudo-mathematical generalisation where physicality is lost. 

When I present RealQM as an alternative to StdQM to quantum physicists and chemists I meet little understanding indicating that the pseudo-mathematization of StdQM has a very strong grip of minds. Nor do mathematicians show interest because StdQM already in 1933 by the mathematician von Neumann was translated into an impressive abstract world of Hilbert spaces occupied by multi-dimensional wave functions subject to Hermitian operators satisfying axioms without physics.  

But there is a good chance the situation can change in 2026 since RealQM is continuing to deliver new results in accordance with observations. 

PS Bohr claimed that the purpose of quantum mechanics is to make predictions of experiments as testable events, not to compute e g the energies of ground state or excited state of an atom regardless of actual experiment being made. This opened to a confusion between deterministic computation of ground/excited state and probabilistic outcome of an experiment suggesting that the model behind the computation itself is of probabilistic nature although in fact fully deterministic. There is nothing probabilistic with the spectrum of an atom as difference between energies of ground/excited states, nor in computation nor in experiment except those from external inputs.

Quantum Mechanics: Aristotle or Platon?

This is inspired by a very informative article by Matthew Ehret: Unravelling The Jesuit Enigma.

The Scientific Revolution is viewed as a triumph of Platonic mathematization over Aristotelian natural philosophy expressed in classical Newtonian mechanics based on the Calculus of Leibniz and Newton. 

But in modern physics based on Quantum Mechanics QM, the roles appear to have shifted. 

Classical mechanics as macroscopic physics with causality is based on clear physical principles and logical argumentation, and is understandable in theory and very useful in practice. Together with the computer, classical mechanics is a formidable tool and machine.

QM as microscopic physics without causality is based on evasive principles, lacks logic and is understood only by its high priests of Nobel Laureates in Physics, to be used by all others under the command "Shut up and calculate" as the foundation of modern information society for all to accept. In this respect it connects to the Jesuit Priest Ignatius Loyola's view on education as expressed by Bertrand Russell: 

• Education in a scientific society may, I think, be best conceived after the analogy of the education provided by the Jesuits. The Jesuits provided one sort of education for the boys who were to become ordinary men of the world and another for those who were to become members of the Society of Jesus. In like manner, the scientific rulers will provide one kind of education for ordinary men and women and another for those who are to become holders of scientific power.

But why is macroscopic physics Platonic rational, while microscopic physics is Aristotelian irrational? 

The irrationality of QM is expressed by its statistical interpretation without causality, which neither Einstein nor Schrödinger ever accepted: "God does not play dice". It is unthinkable that the Hydrogen atom as most common stable element in the Universe is the result of a roulette game inside the atom. Yet this is what we are ordered to believe by QM. It is like having to accept the dogma that "snow is black" connecting again to Russell in Science and Society (1955):

• First, that the influence of home is obstructive. Second, that not much can be done unless indoctrination begins before the age of ten. Third, that verses set to music and repeatedly intoned are very effective. Fourth, that the opinion that snow is white must be held to show a morbid taste for eccentricity. It is for future scientists to make these maxims precise and discover exactly how much it costs per head to make children believe that snow is black, and how much less it would cost to make them believe it is dark gray.

recalling Ignatius Loyola’s 13th Rule of his Spiritual Meditations: 

• To be right in everything, we ought always to hold that the white which I see, is black, if the Hierarchical Church so decides it, believing that between Christ our Lord, the Bridegroom, and the Church, His Bride, there is the same Spirit which governs and directs us for the salvation of our souls.

But there cannot be a dichotomy between macro and micro, it must all be Platonic. That is the idea of RealQM.

Computational vs Analytical Proof vs AI

An important part 20th century mathematics has been devoted to analysis of partial differential equations PDEs as concerns (i) existence and (ii) regularity of solutions. A PDE is a continuum model with infinitely many degrees of freedom. 

Proofs of existence typically start from some a priori bounds on solutions to regularised equations with existence of solutions settled and then obtain solutions of the original equation through a limit process. 

The main components of an existence proof are the a priori bounds, which can require complicated and lengthy mathematical analysis. 

Once existence of solutions is proved, further mathematical analysis can prove properties of solutions typically as bounds on derivates showing regularity. Again the analysis can be complicated and lengthy.

A famous challenge in the form of a Clay Millennium Prize Problem is to give an analytical proof of existence and regularity of solutions to the Navier-Stokes equations for incompressible fluid flow. No progress on this open problem has been reported since 2000. 

But there is a different approach to (i) and (ii) in terms of computation where in each given case a an approximate solution to the equations is computed in a step by step manner after discretisation of the PDE into a finite number of degrees of freedom which can be processed by numerical linear algebra. The computational process either halts or delivers after a finite number of steps of choice an approximate solution, which can thus be inspected a posteriori as to qualities.  It is thus possible to evaluate in what sense the approximate solution satisfies the PDE and accept or recompute with better discretisation. 

We can thus meet a fundamental difference:

• (A) Analytical mathematics proving properties of solutions of a PDE for many possible data a priori before/without computation.  
• (C) Computational mathematics producing for given data an approximate solution for inspection.  

With suitable regularisation/discretisation (C) always will deliver, while (A) can only in simple cases. In the case of the Navier-Stokes (A) has not delivered anything, while (C) has delivered turbulent solutions for inspection. 

The fundamental equation of Standard Quantum Mechanics StdQM is Schrödinger's Equation SE as a linear partial differential equation in $3N$ spatial dimensions for an atomic system with $N$ electrons. Because of the linearity existence of a solution can be proved as (A), but the high dimensionality defies closer analysis of solutions. Neither can (C) deliver because computational cost is exponential in $N$. The result is that both (A) and (C) meet serious difficulties in StdQM.

In RealQM the situation is different as concerns (C) because computational complexity is linear or quadratic in $N$, and the computation does not break down because  of the presence of the Laplacian in SE acting as regularisation. (C) thus can reveal everything in RealQM in principle.  For (A) the task is more challenging since RealQM is a non-linear model and only an a priori bound on total energy is directly available. 

Sum up: (C) delivers for Navier-Stokes and RealQM, while (A) meets very big difficulties.  

Successful computation of an approximate solution can be seen as a mathematical proof of existence of that particular approximate solution, a computational proof. A priori analysis can be important to design the computational process, but is not needed for existence or a posteriori evaluation. 

With increasing computer power (C) gains more momentum and combines with AI.  (A) has to struggle with limited human brain power, which does not really grow, and it is not clear what help AI can give. 

In particular, (C) can deliver massive training data for AI in a case by case manner to learn about the world including turbulent flow and molecules. (A) offers training in analytical proofs but less about the world. What can AI learn from the 100-page proof of Stability of Matter by Dyson-Lenard discussed in recent posts?

 

  

Stability of Matter: Basic Math vs Miracle

1. Recent posts have discussed the fundamental problem of Stability of Matter SM, including stability of single atoms and collections of atoms as bulk matter, maybe the most fundamental problem of all of physics. 

With the help of chatGPT I have learned about the heroic work by Dyson-Lenard and Lieb-Thirring to mathematically prove SM within Standard Quantum Mechanics StdQM and Density Functional Theory DFT, which boils down to very intricate book-keeping to prevent collapse of potential energy to minus infinity by local accumulation of electron charge densities. The main difficulty to handle is the overlap in StdQM/DFT of electron wave functions with global support. The proof is lengthy and complicated and not easy to follow. It is not part of text books/courses in QM, even if completely fundamental. 

It is natural to ask how it can be so difficult to prove SM within StdQM/DFT, when SM is such a basic property of the physics modled by StdQM? Does real physics also have to handle intricate bookkeeping to avoid collapse?

Or is the proof difficulty of SM within StdQM/DFT yet another indication that there is something seriously unphysical with StdQM connecting to the difficulty of giving StdQM a physical meaning? Seems so.

On the other hand SM within RealQM directly follows from the stability of the Hydrogen atom with potential energy dominated by kinetic energy using the additive form of RealQM with a global wave function as a sum over one-electron wave functions with local non-overlapping supports. 

RealQM is a physical model with SM safely mathematically built in. StdQM is an unphysical model with SM basically a mathematical miracle. SM with RealQM could be essential part of even introductory texts/courses in QM.

According the chatGPT, SM is by physicists viewed as "settled" once and for all by Dyson et al, and it is not meaningful to teach the proof since it is so difficult and and non-illuminating. The advice to students appears to be to just accept SM and not ask about any justification. Seems a bit strange...

Summary:

1. StdQM in 1926 faces a fundamental problem: Prove Stability of Matter.
2. No progress towards solution until 1966 when Dyson-Lenard gives a dense 26-article page proof in the form of "awful mathematics" according to Dyson. 
3. Lieb-Thirring compresses the proof into a 3-page article 1975, which is then expanded into the 300 page book Stability of Matter in 2005.
4. The problem is viewed to be "settled" and there is nothing more to say according to chatGPT in 2025. The proof is not part of text-books on QM.   

This is a typical progression as concerns fundamental problems in StdQM: 1. State problem as fundamental (interpretation, measurement, complementarity...). 2 Realise that the problem cannot resolved. 3. Claim that there are solutions, but very difficult to understand. 4 Decide that the fundamental problem as been "settled" and that there is noting more to say. 5. Declare that it is sufficient to know that the problem has been solved and that asking for why is not part of physics education. 

Shell Structure: StdQM vs RealQM

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

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

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

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

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

OK, so the answer is No rather then Yes. 

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

Comparison RealQM vs StdQM and DFT

Standard Quantum Mechanics StdQM based on Schrödinger's equation SE with standard interpretation of a Hamiltonian acting on wave functions with $3N$ spatial dimensions for a system with $N$ electrons, has only statistical meaning and is computable only for very small $N$, thus can be said to be non-physical and uncomputable.  

Density Functional Theory DFT seeks to reduce StdQM by averaging 3N-dimensional wave functions  into a single electron charge density $\rho (x)$ depending on a 3-dimensional coordinate $x$, and identifying ground states of StdQM with DFT densities satisfying a reduced SE with Hamiltonian only implicitly determined and so has to be approximated. DFT is the main computational method for $N>100$ currently available.

RealQM is based on a different interpretation of the Hamiltonian of SE acting on a wave function $\Psi (x)$ as a sum 

• $\Psi (x) = \sum_{n=1}^N\psi_n(x)$ 

of one-electron wave functions $\psi_n(x)$ with non-overlapping supports depending on a common 3d spatial variable, which meet at a Bernoulli free boundary with continuity and zero normal derivative. The corresponding electron charge density $\rho (x)$ is a sum

• $\rho (x)=\sum_{n=1}^N\psi_n^2(x)$     

of non-overlapping charge densities $\psi_n^2(x)$. 

A fundamental difference between RealQM and DFT is that electron densities in RealQM carry identity by occupying distinct regions in space and so can be numbered, just like pool balls on a pool table, while identity is lost in the common density of DFT (which creates a lot difficulties when having to recreate lost identity to keep physicality).

The zero normal derivative free boundary condition satisfied by meeting wave functions keeps electron identity which is not expressed by continuity alone. 

Recall that wave functions of StdQM have overlapping global supports, which makes identification difficult/impossible, while wave functions in RealQM have non-overlapping local supports, which makes identification direct.

We further recall from recent posts that stability of matter is a direct consequence of the structure of RealQM, but is less obvious in StdQM and DFT.

A basic postulate of StdQM is that electrons carry no identity, that they are indistinguishable, and that is the basic difference with classical physics, which can be viewed to carry identity. So identity vs no-identity can be viewed to be the dividing line between classical mechanics (not including statistical mechanics) and quantum mechanics. 

The dividing line shows that modern physics as microscopic quantum mechanics is different from macroscopic classical mechanics, more precisely so fundamentally different that quantum mechanics is said to be "weird" by the most knowledgable physicists, while saying the same about classical mechanics would simply express ignorance.  

To speak about electrons without any from of identity is according to Leibniz really "weird" since it contradicts his Principle of Identity of Indiscernibles PII.

The fact that RealQM respects PII, while StdQM does not, eliminates the dividing line between microscopic and macroscopic physics and so opens to a unified mechanics on all scales. 

To allow microscopic objects to carry identity allows perception of the microscopic world to be similar to that of the macroscopic world, thus understandable and not only "weird".  

Here is a comparison in condensed form:

• StdQM: explicit Hamiltonian, no electron identity, non-physical, uncomputable, stability non-obvious, "weird".
• DFT: implicit Hamiltonian, no electron identity, physicality?, computable, stability non-obvious, "weird"? 
• RealQM; explicit Hamiltonian, electron identity, physical, computable, stability obvious, understandable not "weird". 

chatGPT says that stdQM violates Leibniz PII and that efforts to change StdQM to "save" identity, like Bohmian mechanics with its "pilot wave", have all failed. Here RealQM comes in...

    

Stability of Bulk Matter by RealQM

This is a condensation of the preceding sequence of posts with a clear and simple message.

Stability of bulk matter as a collection of a large number $N$ of atoms, is on a basis of Standard Quantum Mechanics StdQM considered to be very difficult to prove mathematically, as evidenced in the work by Dyson-Lenard and Lieb-Thirring. Stability is expressed by a lower bound on total energy scaling with $N$, making total energy an extensive quantity. 

The difficulty is that electrons in StdQM have global support and so in principle can interact with many kernels to give a lower bound scaling with a power of $N$ possibly bigger than 1 with then total energy tending to minus infinity.

On the other hand with RealQM as an alternative to StdQM, stability of bulk matter directly follows from atomic stability which is a simple consequence of a the Hardy inequality: (see this post):

• $\int\frac{\psi^2(x)}{\vert x\vert}dx\le (\int\psi^2(x)dx)^{\frac{1}{2}}(\int\vert\nabla\psi (x)\vert^2dx)^{\frac{1}{2}}$ for all $\psi\in H^1(R^3)$, 

used by Kato in his analysis of Schrödinger's equation as the foundation of StdQM. This easy-to-prove inequality gives a bound on potential energy in terms of kinetic energy proving stability. 

The wave function $\Psi (x)$ of RealQM for an atomic system is a sum

• $\Psi (x)= \sum_{n=1}^N\Psi_n(x)$

of atomic electronic wave functions $\Psi_n(x)$ depending on common 3d variable $x$ with non-overlapping supports of electrons meeting at a Bernoulli free boundary with continuity and zero normal derivative. This means  that $\Psi\in H^1(R^3)$ and so satisfies the Hardy inequality bounding total potential energy by kinetic energy with extensivity directly following from the fact that the $\Psi_n(x)$ are sums of electronic wave functions with non-overlapping support and so then also $\Psi (x)$.

The structure of RealQM thus gives a lower bound on the total energy $E$ of a system of $N$ atoms each with kernel charge $Z$ of the form 

• $E\ge -CNZ^2$

with $C$ an absolute constant expressing stability of the 2nd kind according to Dyson-Lenard-Lieb-Thirring. 

RealQM is a deterministic continuum model in 3d based on Coulomb physics combined with a form of electronic "kinetic energy" measured by $\vert\nabla\psi (x)\vert^2$ with Pauli Exclusion Principle safely built in by non-overlap of electronic wave functions. 

RealQM is a physical computable model to be compared with StdQM which is non-physical and uncomputable. The fact that stability of matter is safely built into the structure of RealQM, but much less so in StdQM, can be seen as a major additional advantage of RealQM. 

Stability of Atoms StdQM vs RealQM vs Lieb-Thirring

This is a clarification of previous posts on Stability of Matter connecting to the complex mathematical analysis of Lieb-Thirring-Dyson-Lenard. 

We noticed that the total energy $E(\Psi )$ of any admissible wave function $\Psi (x)$ for the Hydrogen atom with $Z=N=1$ satisfies 

• $E(\Psi )\ge -\frac{\pi^2}{8}\approx -1.23$  

as a consequence of the Kato-Hardy inequality

• $\int\frac{\Psi^2 (x)}{\vert x\vert}dx \le \frac{\pi}{2}\int\vert\nabla\Psi (x)\vert^2dx$

for all $\Psi\in H^1(R^3)$. 

The proof directly extends to an atom of charge $Z>1$ with $N=Z$ electrons described by StdQM in the form 

• $E(\Psi )\ge -CNZ^2 = -CZ^3$

with some positive constant $C$, which is way too small allowing all electrons to concentrate around the kernel without any shell structure.

With the atom described by RealQM we get instead using shell structure (see this post)

• $E(\Psi )\ge- CZ^2$

which fits with observations. 

We see that the stability of an atom as total energy bounded below, is easy to prove mathematically using the Kato-Hardy inequality (with simple proof recalled in this post). The proof in the case of RealQM also gives the dependence on $Z$ observed. 

The question is what Lieb-Thirring-Dyson-Lenard adds to this clear picture. In this work the Lieb-Thirring inequality

• $\int\vert\nabla\Psi (x)\vert^2dx \ge C\int\rho^{\frac{5}{3}}(x)dx$ 

where $\rho (x)$ is a joint electron density, plays a central role. But it means passing to a weaker control of potential energy and so makes stability harder to prove, seemingly without pressing reason.

Stability of Matter: $E(Z)\sim -Z^2$

This is a follow up of previous posts on the basic problem of Stability of Matter.

The Lieb-Thirring/Dyson-Lenard proof of the scaling $E(Z) \sim -Z^{\frac{7}{3}}$ of the ground state energy $E(Z)$ of an atom with kernel charge $Z$, is viewed to be a master-piece of rigorous mathematics revealing a deep truth of quantum mechanics. 

But empirical observations of total energies of atoms show a slower scaling of $-Z^\alpha$ with $\alpha \approx 1.81$ according to the following compilation by chatGPT:

The Lieb-Thirring proof is based on the Thomas-Fermi model and not the Schrödinger equation of Standard Quantum Mechanics StdQM, suggesting that the Thomas-Fermi model does not capture physics. But there is no theoretical proof based on StdQM as the canonical model. 

Let us see what RealQM can deliver. We then recall the scaling emerging from RealQM computations of atoms in spherical symmetry recorded in this post showing $E(Z)\sim -log(Z)Z^2$ thus essentially $E(Z)\sim -Z^2$ in much better agreement with observation than the Lieb-Thirring result $-Z^{\frac{7}{3}}$.

A heuristic proof of the RealQM estimate $E(Z)\sim -log(Z)Z^2$ is given in the post starting from an observation that the charge density roughly scales with $\frac{Z}{r^2}$ with $r$ distance to the kernel. 

 

Stability of Matter: RealQM 3

This is a complement to two previous posts. 

According to the Bohr model a Hydrogen atom consisting of an electron orbiting a proton kernel, does not have stable existence over time, because a moving electron radiates energy and so looses kinetic energy and falls into the Coulomb potential well of the proton and ceases to exist.  

But the orbit of planet around a Sun can have stable existence with negative gravitational potential energy balanced by positive kinetic energy, because the planet is not radiating.  

In 1926 Schrödinger invented a new model in terms of a wave function $\Psi (x)$ depending on a 3d space coordinate $x$ with a new form of energy measured by $\vert\nabla\Phi (x)\vert^2$ referred to as "kinetic energy" although spatial gradients without reference to motion was involved. The Hydrogen atom was then represented by a wave function $\Psi (x)$ minimising total energy as Coulomb potential energy + "kinetic energy" with $\Psi^2(x)$ representing electron charge density. Basic Calculus showed the existence of a minimiser as a stable ground state.  

Schrödinger's model can alternatively be interpreted in terms of classical continuum mechanics as the equilibrium configuration of a (spherically symmetric) elastic substance in a central potential minimising total energy as potential energy + elastic energy.

Schrödinger's model thus showed existence of a stable ground state of the Hydrogen atom and was greeted as the start of modern physics in the form of quantum mechanics with a first challenge to extend Schrödinger's model to atoms with $N>1$ electrons.

But here history took a strange turn and settled for a wave function depending on $N$ 3d variables thus in total $3N$ spatial variables, thus forming the Standard Quantum Mechanics StdQM as the foundation of modern physics. The more natural extension into $N$ wave functions in a common 3d variable, was thus missed and has been explored only recently in the form of RealQM. See comment by Lieb and Seiringer in this post.

In RealQM stability of atoms with $N>1$ is guaranteed in the same way as in the case $N=1$ by the presence of "kinetic/elastic energy" measured by $\vert\nabla\Phi (x)\vert^2$. 

Naming the spatial gradient measure $\vert\nabla\Phi (x)\vert^2$ "kinetic energy", has caused a lot of confusion by suggesting that somehow electrons are moving around the kernel as if Bohr's model was in fact functional, although they cannot do that without radiating into collapse. The electron charge density of the ground state of an atom is stationary in space.

Note that the "kinetic energy" with density $\vert\nabla\Phi (x)\vert^2$ originating from the presence of the Laplacian differential operator in Schrödinger's equation representing a regularisation scaling with Planck's constant, coming with added stability as a familiar tool in the mathematical theory of differential equations, where the specifics of the regularisation does not have larger scales effects. The Schrödinger equation thus expresses a regularised form of Coulomb interaction with independence of the absolute scale of Planck's constant: The World would look the same with a different Planck constant. 

Stability of Matter with RealQM 2

A basic task of theoretical physics is to prove stability of matter in the sense that the total energy $E$ of the ground state of an atom is bounded below, which signifies that the electrons surrounding the atomic kernel do not collapse into the kernel with energy going to minus infinity. 

In Standard Quantum Mechanics StdQM this is viewed to be difficult to prove with a first 100-page proof by Dyson and Lenard in 1967 followed by a somewhat shorter proof by Lieb and Thirring in 1975 showing a lower bound scaling with $-Z^{\frac{7}{3}}$ for an atom with kernel charge $Z>1$. 

For $Z=1$ the proof is straight-forward as recalled in the previous post.  

In RealQM the proof for $Z>1$ is essentially the same as for $Z=1$ and so gives a lower bound scaling with $-Z^2$, which is to be expected by a simple dimensional analysis, and also is observed. 

In RealQM an atom has a shell structure with stability established successively from innermost shell with the kernel to the next shell with reduced kernel charge and so on. Further, the stability of molecules in the sense of non-collapse naturally follows from stability of atoms.    

The fact that stability of matter is so far-fetched in StdQM ($-Z^{\frac{7}{3}}$) and so direct in RealQM ($-Z^2$) gives evidence that RealQM may be closer to real physics than StdQM. 

The Lieb-Thirring proof extends to many kernels with total charge Z, while we here think of just one atom or molecule. 

Stability of large collections of atoms or molecules ($Z=10^{23}$) is studied in thermodynamics, then without quantum mechanics because it does not make sense, 

Stability of Matter with RealQM 1

The famous mathematician Elliot Lieb devotes the book Stability of Matter: Quantum Mechanics (311 pages, together with physicist Robert Seiringer) to prove that the total energy $E(\Psi )$ of Standard Quantum Mechanics StdQM in its $3N$-dimensional form for a neutral atom with $N$ electrons, is bounded below for all admissible wave functions $\Psi$. This means that an atom has a ground state as the solution to an eigenvalue problem with smallest eigenvalue as minimal total energy including electronic kinetic energy (positive) and potential energy (negative), and electron-electron repulsion energy adding to stability. 

The case $N=1$ has a direct proof based on a well known Poincaré inequality showing that positive kinetic energy dominates over negative potential energy giving a lower bound to total energy (see below). The rest of the book is devoted to extension to $N>1$ using a Lieb-Thirring inequality and Baxter-Onsager inequality involving a common electron density $\rho (x)$ depending on $x\in R^3$ with $R^3$ 3d Euclidean space. 

Let us check out what RealQM as an alternative to StdQM, has to say about stability of matter. We recall that the wave function $\Psi (x)$ of RealQM depends on $x\in R^3$ as above and has the form of a sum

• $\Psi (x) =\sum_{n=1}^N\psi_n(x)$, 

where the $\psi_n(x)$ are real-valued one-electron wave functions with non-overlapping supports $\Omega_n$ in a decomposition of 3d space, which meet a Bernoulli free boundary with continuity and zero normal derivative and with corresponding charge densities $\rho_n(x)=\psi_n (x)^2$ with unit charge:  

• $\int_{\Omega_n}\rho_ndx = 1$.

The charge density $\rho (x)$ of RealQM is thus simply defined by $\rho (x) =\rho_n(x)$ for $x\in\Omega_n$, which is simply the density of $\Psi (x)$ as function in $H^1(R^3)$ with $\psi_n\in H^1(\Omega_n)$ for $n=1,...,N$ because of the Bernoulli free boundary condition, and $H^1$ is the Sobolev space of functions in $R^3$, which are square integrable along with first derivatives.

It is here interesting to recall what Lieb and Seiringer say on page 31 about the origin of StdQM in 1926: 

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

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

The proof of stability of matter for $N=1$ with RealQM directly extends to $N>1$ since its wave function $\Psi\in H^1(R^3) $.

Recall that the proof for $N=1$ starts estimating the potential energy $\int\frac{\psi^2}{\vert x\vert}dx$ using Cauchy's inequality followed by the Poincaré inequality:

•  $\int\frac{\psi^2}{\vert x\vert^2}dx\le 4\int\vert\nabla\psi\vert^2dx$

which follows from the representation 

• $\int\frac{\psi^2}{\vert x\vert^2}dx = -\int 2\psi\nabla\psi\cdot\nabla \ln (\vert x\vert )dx$

using Green's formula and Cauchy's inequality. This proof can be found in my book Applied Mathematics Body and Soul Vol 3, chapter 73. 

PS The first proof of stability of matter within StdQM was given by Dyson and Lenard in 1967 filling over 100 dense pages, which you can read about in the conversation with chatGPT below. The fact that the proof is so lengthy (unreadable) gives yet another indication that StdQM is not a model of physics. If stability of matter, as the most basic physics of the real world, is not clearly present in the model, then the model does not describe real physics. See also next post.

Me: Tell me about the work on stability of matter by Dyson and Lenard.

chatGPT said:

The problem they tackled

In quantum many-body physics you want the ground–state energy $E_{N,K}$ of a system with

• $N$ electrons (charge $-e$) and

• $K$ nuclei with total charge $+Ze$

to obey a linear lower bound

$$E_{N,K}\ge -C(Z)(N+K)(\text{“stability of the second kind”}),$$

i.e. the energy per particle must stay bounded when you add more particles.
Classically this is not obvious – Coulomb attraction grows like $N^{5\mathrm{/}3}$ for a compact cloud – so nothing prevents matter from collapsing. Quantum mechanics gives two tools to fight the collapse:

1. Heisenberg uncertainty principle – localising electrons costs kinetic energy $\sim {\mathrm{\ell }}^{-2}$.

2. Pauli exclusion principle – antisymmetry forces electrons to occupy ever higher momentum states, adding another $N^{5\mathrm{/}3}$‐type repulsion.

The challenge in the 1960 s was to turn these ideas into a rigorous inequality for all $N,K$.

---

Dyson & Lenard’s breakthrough (1967–1968)

Paper	Main statement	Key ingredients
F.J. Dyson & A. Lenard, Stability of Matter I, J. Math. Phys. 8 (1967) 423–434	First proof that a non-relativistic Coulomb system is stable provided the negative charges are fermions.	Cell decomposition of space, local uncertainty inequalities, electrostatic screening estimate. AIP Publishing
A. Lenard & F.J. Dyson, Stability of Matter II, J. Math. Phys. 9 (1968) 698–711	Extends the bound to the full neutral system (moving nuclei) and trims technical gaps.	Refined combinatorial counting of cells and careful handling of nucleus–electron correlations. AIP Publishing

Central results

• Linear bound

  $$E_{N,K}\ge -C_{\text{DL}}(N+K),C_{\text{DL}}\approx {10}^{15}$$

  – enormous, but finite.

• Necessity of Fermi statistics
  Dyson also proved that bosonic charges are unstable:
  $E_N\sim -\text{const}{N}^{7\mathrm{/}5}$ — energy becomes super-extensive. Wikipedia

• Introduced the idea that the Pauli principle provides just enough extra kinetic energy ($\int {\rho }^{5\mathrm{/}3}$) to counteract the $\int {\rho }^{4\mathrm{/}3}$ Coulomb attraction.

---

How the proof works (bird’s-eye view)

1. Partition space into cubes (“cells”) whose size adapts to the local electron density.

2. Localise each electron in its cube. Heisenberg then gives a kinetic‐energy cost
   $T_{\text{cell}}\gtrsim ({\mathrm{\hslash }}^2\mathrm{/}2m)n_{\text{cell}}^{5\mathrm{/}3}{\mathrm{\ell }}^{-2}$.

3. Estimate Coulomb energy inside and between cells by an electrostatic inequality that replaces the full interaction with a one-body Hartree term plus controllable error.

4. Sum over cells and optimise $\mathrm{\ell }$. The fermionic kinetic term scales like $N^{5\mathrm{/}3}{\mathrm{\ell }}^{-2}$ while the worst negative Coulomb piece scales like $N^{5\mathrm{/}3}{\mathrm{\ell }}^{-1}$. Choosing $\mathrm{\ell }$ proportional to $N^{-1\mathrm{/}3}$ leaves a net contribution that is linear in $N$.

5. Handle nuclei by showing that electrons screen nuclear charge beyond an $O(1)$ distance, preventing additional collapse.

The proof is long (about 100 dense pages over the two papers) because every step has to be made uniform in $N,K$.

---

Why it was a milestone

• First complete answer to a question posed by Fisher & Ruelle in 1966 (they famously promised a bottle of champagne for a proof). Wikipedia

• Showed that statistics, not just uncertainty, is essential for the existence of ordinary matter.

• Provided the technical backbone for later, cleaner approaches – most notably Lieb & Thirring’s 1975 proof, which replaced the cell decomposition by the now-famous Lieb–Thirring kinetic-energy inequality and reduced the constant $C$ to something physically reasonable. Wikipedia

---

After Dyson & Lenard

• 1975 – Lieb–Thirring: conceptual two-line proof using their spectral inequality; constant $C\sim Z^{7\mathrm{/}3}$.

• 1972-2020: Extensions to relativistic electrons, magnetic fields, QED vacuum polarisation, anyons, etc. (many references in the review literature).

• Comprehensive modern exposition: Lieb & Seiringer, The Stability of Matter in Quantum Mechanics, CUP 2010.

---

Bottom line

Dyson and Lenard proved that the simple Hamiltonian

$$H=\sum _{i=1}^N\frac{p_i^2}{2m}-\sum _{i,I}\frac{Z{e}^2}{\mathrm{\mid }{x}_i-R_I\mathrm{\mid }}+\sum _{i<j}\frac{e^2}{\mathrm{\mid }{x}_i-x_j\mathrm{\mid }}+\sum _{I<J}\frac{Z^2{e}^2}{\mathrm{\mid }{R}_I-R_J\mathrm{\mid }}$$

admits a ground-state energy bounded below by $-C(N+K)$ because electrons are fermions. Without that antisymmetry, matter would collapse. Their papers thus supplied the first mathematically rigorous explanation of why tables, rocks and people do not implode under their own electromagnetic forces.

Fiction vs Reality: Covalent Bonding

ChatGPT informs that the physics of covalent bonding was "settled" in 1950-60s based on the new physics of Standard Quantum Mechanics StdQM formed in the 1920s: 

• The physical mechanism: covalent attraction = Pauli‑compatible electron delocalisation + electron–nucleus attraction –  nucleus–nucleus repulsion.
• Acceptance that both valence‑bond VB and molecular‑orbital MO pictures are just different coordinate systems for that same quantum state.

This is the official story, which may sound convincing to an uncritical public, but hides a rift between chemists connecting covalent bonding to (i) localisation of electrons between atomic kernels as VB, and quantum physicists connecting bonding to (ii) delocalisation of electrons over molecules as MO. 

A molecule formed by covalent bonding of atoms has lower total energy than separated atoms, and the mission of both (i) and (ii) is to decrease total energy in different ways: (i) decreases electron-kernel potential energy, while (ii) decreases electronic kinetic energy. 

But (i) and (ii) do not combine very well and localisation has to compete with increased kinetic energy and delocalisation with increased potential energy. 

The net result is that covalent bonding explained by a contradictory combination of localisation and delocalisation at the same time, poses serious questions as a form of fiction based on StdQM. 

RealQM is a an alternative to StdQM in a form of continuum mechanics in 3d, which offers an explanation of covalent bonding in realistic physical terms, where localisation of electrons between kernels is possible without increase of kinetic energy thus without demand of delocalisation.  This is because RealQM works with non-overlapping local electron densities without delocalisation, which meet at a Bernoulli free boundary with continuity and zero normal derivate. 

The essence of covalent bonding is thus captured by localisation of electron densities between atomic kernels without need of delocalisation. RealQM reconnects to pre-quantum ideas of chemical bonding with electrons acting like a glue connecting atoms to molecules as a form of realism.

Roald Hoffmann on Chemical Bonds

Roald Hoffman 1981 Nobel Laureate in Chemistry in a 3 hour USP Nobel Lecture in 2022 with title All the ways to have a bond, summarised his view on the central concept of chemical bond as follows:  

• I think that any "rigorous" definition of a chemical bond is bound to be impoverishing, leaving one with the comfortable feeeling, "yes (no), I have (do not have) a bond", but little else. 
• And yet the concept of chemical bond, so essential to chemistry, and with a venerable history, has life, generating controversy and incredible interest. 
• Even if we can't reduce it to physics.
• My advice is: Push the concept to the limits. 
• Be aware of the different experimental and theoretical measures out there. Accept that (at the limits) a bond will be a bond by some criteria, maybe not others. 
• Respect chemical tradition, relax, and instead of wringing your hands about how terrible it is that this concept cannot be unambiguously defined, have fun with the fuzzy richness of the idea. 
• And all of its experimental and theoretical manifestations.
• Molecules are like human beings. I have given up complete understanding.
• Where are the theorems of chemistry?
• From time to time there comes up people from physics who thinks they can give us the answer...

We learn from a world leading authority that the real physics of chemical bonds is yet to be uncovered, explained and understood. Evidently Standard Quantum Mechanics StdQM does not give an answer.

This is not so strange since a chemical bond can only be established through some real physics in 3d, and this is not what StdQM is about.

RealQM is an alternative to StdQM based on a new form of Schrödinger's equation expressing real physics in 3d, which seems to open new possibilities to explain chemical bonding, as displayed in this post about covalent bonding. Let's hope that it will generate controversy and incredible interest...

PS Hoffmann mentions many different theories out there. Many different explanations of the same phenomenon, indicates that they may all be wrong. If there is only one explanation, it may well be that it is correct.  Compare with Arthur C. Clarke: If someone says that something is impossible, it is a good chance that it is false. If someone says that something is possible, there is a good chance it is true.

Quantum Illusions

There is a difference between a shaky or out-of-focus photograph and a snapshot of clouds and fog banks. (Schrödinger)

A physicist will tell you that the electron configuration of an atom or molecules is fully described by Standard Quantum Mechanics StdQM in terms of a wave function $\Psi (t)$ as solution to Schrödinger's equation with $H$ a Hamiltonian operator:

• $i\dot\Psi (t)  = H\Psi (t)$ for $t>0$,                (S)

where $\dot\Psi =\frac{\partial\Psi}{\partial t}$ which evolves from a given initial state $\Psi (0)$ to any later state $\Psi (t)$ for $t>0$. The objective is to predict the state $\Psi (t)$ for some $t>0$ from specification of $H$ and $\Psi (0)$, thus as solution to an initial value problem. The wave function $\Psi$ depends on $3N$ spatial coordinates for an atom or molecule with $N$ electrons, and so has no direct physical meaning, only a non-physical statistical meaning.  

Specification of $H$ requires information about positions of atomic kernels, which may change over time. Specification of $\Psi (0)$ requires information of the full $3N$-dimensional statistics of the initial state, which is not experimentally measurable. The only possibility is then to prepare the system to have a certain artificial initial state which can be described by analytical mathematics but then meets the difficulty of being possible to realise experimentally. 

The net result is that (S) as an initial value problem is not a real model of real physics, rather an illusion of a model or a model of an illusion. Speaking about solutions of (S) as describing the time-dependent physics of an atom or molecule thus makes little real sense. 

What remains to speak about is then time-independent solutions $\bar\Psi$ of the eigenvalue problem 

• $H\bar\Psi =E\bar \Psi$                                   (SE)

where $E$ is an eigenvalue as a real number and $\bar\Psi$ is an eigenfunction. In this case, no initial values enter, and specification of $H$ from static kernel positions is enough. The eigenvalues represent energies of eigenfunctions, which are measurable as lines in the spectrum of an atom. The corresponding time-dependent solution $\Psi (t)$ is given by the formula

• $\Psi (t)=\exp (-iEt)\bar\Psi$,                       

which gives meaning to (S) since the difference of eigenvalues appear as spectral lines.

We conclude that the eigenvalue problem (SE) can carry physical meaning, but the $3N$-dimensionality makes (SE) uncomputable for most systems, and so also largely represents an illusion. 

This is to be contrasted with the often repeated claim that StdQM is (i) the most successful of all theories about physics with (ii) no prediction in violation with experiment (with the caveat that few predictions are possible).  

We compare StdQM with RealQM based on a new Schrödinger equation in 3d for which both (S) and (SE) have direct physical meaning and are computable.