Weak Reactivity of Gold by RealQM

 Here is a new article to the series about RealQM as an alternative to StdQM:

• Real Quantum Mechanics: Weak Reactivity of Gold

RealQM explains why Au with electron shell configuration 2+8+18+32+18+1 with 1 valence electron in a last shell outside a sphere of radius R containing all other shells, does not form a molecule Au2 by covalent bonding as a geometric effect of large R preventing electron accumulation between kernels.

RealQM explains covalent bonding of H into H2 and non-bonding of He2 in this earlier article in the series:

• Real Quantum Mechanics: Covalent Bonding. 

Download files and open in Acrobat to get active hyperlinks.

Origin of CMBR?

Here is an interesting video about the possible origin of the Cosmic Microwave Background Radiation CMBR other than an "afterglow" of a Big Bang proving the Big Bang hypothesis to be correct, as the currently most popular cosmological theory:  

The video recalls early theories about a static Universe filled with some form of interstellar dust radiating a Planck spectrum of around 2-5 Kelvin. Such a thing was detected in 1967 at 2.7 K and was then  connected to Big Bang deleting the old theories from the map. In particular the Big Bang theory claimed to resolve the mystery of the observed red-shift of galaxies increasing with distance suggesting an accelerated expansion of the Universe. 

One of the old theories was the "tired light" hypothesis presented by Zwicki in 1929 suggesting that the red-shift could be the result of a loss of energy of light passing through interstellar dust over long distance with energy scaling with frequency.  

The idea of an active interstellar dust actively radiating at 2.7 K seems more natural than that of a Big Bang "afterglow" still hanging on after 13.8 billion years. 

Zwicki's "tired light" hypothesis can be connected to some form of dark matter as source of gravitation which shows up as CMBR resulting from sucking up energy from passing light.  I have discussed dark matter in posts on Neo-Newtonian Gravitation.   

Computational vs Theoretical Mathematics in Physics

The mathematical models of physics take the form of partial differential equations like Euler's Equations for incompressible inviscid fluid flow EE, corresponding Navier-Stokes equation for viscous flow NSE and Schrödinger's equations for atoms and molecules SE. 

The task of a theoretical mathematician has been to prove by symbolic analytical techniques (i) existence, (ii) uniqueness and (ii) regularity of solutions to a given equation with data given in some large class of possible data with data including initial data, forcing and parameters like viscosity in NSE. 

The task of a computational mathematician has been to compute solutions for specific choices of data which in each specific case can answer (i)-(iii) by inspection of the computed solution. 

It has been argued that computation is not enough, even if for each specific choice of data (i)-(iii) can be answered, because only a limited number of specific choices can be inspected. The possibly very large class of data can thus never be exhausted by computation, which gives analytical symbolic mathematics a role to play by covering a large class of data.

It is natural to ask if there are examples of equations for which the class of relevant data is so small that it can be exhausted by computation. This means first that the equation cannot contain any parameter like viscosity. Are there any models of interest which are parameter free? Inspection of EE and SE shows that they are both parameter free, and so meet the requirement of Einstein of an ideal mathematical model opening to say something about the world without measuring anything. This is like learning the area of a circular disc by computation with unit radius as only input.

Solving EE computationally thus delivers the drag of a body moving through a slightly viscous fluid such as air and water at a subsonic speed with the only data being the shape of the body and not any viscosity as parameter. This limits the class of data to shapes of bodies with a limited range of shapes of interest to be covered by computation. This is all described here.

The case of SE is in its traditional form of Standard Quantum Mechanics StdQM troubled by the fact SE by its multi-dimension nature is uncomputable and so needs dimensional compression which introduces parameters. 

RealQM is different realisation of the same parameter-free Hamiltonian as StdQM into computable form without introduction of any parameter. RealQM thus expresses SE in parameter-free computable form and so opens the possibility of saying something about the atomic world without experimental input. RealQM thus computes the ground state of an atom with the only input being the number of electrons and so can exhaust the Periodic Table.      

An analytical estimate of ground state energy as the result of a longer or shorter sequence of successive bounds, can be seen as a form of symbolic computation, while a numerical computation can be seen as very long arithmetic proof.

Computation with a parameter-free mathematical model can produce a rich set of outputs from very limited structural input, which can serve as data for AI in need of rich data. Computation is then used both to produce data and to learn from data. Symbolic mathematics has an important role to set up computation.

The Clay Institute Millennium Problem on (i)-(iii) for NSE is still open in the form of symbolic mathematics with no progress reported over 25 years. Can computation get the million dollar Prize?

Covalent Bonding by RealQM

Here is the next article in a sequence of articles exhibiting the capabilities of RealQM as an alternative to StdQM as the canon of modern physics: 

• RealQM: Covalent Bonding

with earlier articles in recent posts. Download article and open in Acrobat to get hyperlinks to essential codes to run, inspect and modify.

Shut up and Calculate vs Compute, Learn and Speak

Here is a reflection connecting to the previous post on RealQM as a computable model of atoms and molecules and Mearmin's Shut up and Calculate desperate reaction to the difficulty of making sense of the theory of Standard Quantum Mechanics StdQM. 

Let us recall the following views on the divide between StdQM as the theoretical foundation of chemistry according to physicists and the theoretical chemistry actually used by chemists. 

Eberhardt 2012:

• Chemistry is a discipline of two faces, one applied and the other theoretical. The applied face focuses on the design and synthesis of molecules and solids, while the theoretical face looks for explanations of a molecule or solid’s properties.

Bader 2011:

• ....the divide that exists in chemistry between those who seek their understanding within a universe wherein the laws of physics apply and those who prefer alternative universes wherein the laws are suspended or bent to suit preconceived ideas.

Dirac 1929:

• The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.

Dirac identifies the root cause of the divide as the impossibility of solving Schrödinger's equation of StdQM for the complex atomic systems of chemistry. If computational solution was possible there would be no divide and the whole of chemistry would be like an open book to read by computation. 

RealQM is an alternative to StdQM which is ab initio computable for the complex atomic systems of chemistry and is also understandable in the same sense as classical continuum mechanics. If RealQM indeed shows to models physics, then there is no longer any reason for a divide between theory and practice. The Shut up and Calculate can then be replaced by Compute, Learn and Speak. 

 

RealQM Article: Stability of Atoms

Here is another article in a series of articles about RealQM to be submitted:

• Real Quantum Mechanics: Stability of Atoms 

Recall the first article in the series:

• Real Quantum Mechanics: Introduction

Download file and open in Acrobat to get active hyperlinks.

RealQM Article to Submit

I am now now preparing to submit a sequence of articles about Real Quantum Mechanics to relevant journals and here is a first test to check out reaction:

• Real Quantum Mechanics

Take a look and give a comment. 

Political Role of Quantum Mechanics

The shift from the old quantum mechanics of Niels Bohr happened with Werner Heisenberg's July 1925  "Reinterpretation" article introducing a new form of matrix mechanics without Bohr's electron trajectories to form the new quantum mechanics.  

Heisenberg was a student of Max Born at the University of Göttingen with the mathematician Hilbert as world authority of mathematical physic acting as Born's mentor. Göttingen mathematical physics was an important part of the "Weimar Renaissance" during the recovery of Germany after the defeat in WWI getting momentum in 1925. 

It is thus possible to give Heisenberg's new quantum mechanics political dimensions stretching into WWII with Heisenberg as leading scientist in Germany's quest for an atomic bomb in the "Uranium Club".  

Planck had a similar role at the height of the German Empire when he in 1900 took on the responsibility to resolve the outstanding open physics problem of black-body radiation, when to avoid failing he resorted to statistics of quanta which resurfaced in Born's interpretation of Heisenbergs new quantum mechanics.

Heisenberg worked as Bohr's assistant in 1925 and Bohr invited Born to Copenhagen in 1926, and so they came to form the Bohr-Born-Heisenberg Copenhagen school setting the agenda for quantum mechanics from its beginning into our time. 

Schrödinger entered in 1926 with his equation showing to be equivalent to matrix-mechanics, but Schrödinger did not accept the Copenhagen Interpretation and so left the field to come back only in 1954 to discover that BBH still controlled the scene. 

After WWII the US took over quantum mechanics still in the spirit of BBH.

All physicists of today say that following the idea of the mathematician von Neumann that the wave function of quantum mechanics lives in a Hilbert space of infinite dimension and most physicists will confess to the Copenhagen Interpretation even if its meaning is unclear. Schrödinger's request of physicality is met with "shut up and calculate". 

What do the Chinese say? Is it time for Schrödinger to come back in the form of RealQM in the "China Renaissance" that is now reshaping the world? It is not impossible since the Chinese are very clever, very organised and result oriented towards a clear plan.

ChatGPT: After completing his doctoral work in Budapest and Zurich, von Neumann spent the academic year 1926–27 at Göttingen. There he

• Studied under David Hilbert, attending Hilbert’s lectures on the mathematical foundations of quantum mechanics,

• Sat in on Max Born’s seminar on the new quantum theory, and

• Published his first quantum‐mechanical notes out of Göttingen later that year.

Issues with Standard Quantum Mechanics 1926-2026

There are basic unresolved foundational "issues" with Standard Quantum Mechanics StdQM based on Schrödinger's Equation SE, basically issues with SE: 

1. Unitary deterministic evolution of the wave function.
2. Collapse of the wave function upon observation.
3. Statistics of collapsed wave function by Born's Rule. 

There are many more issues (ontology, correlation, non-locality...) but the all connect to 1-3 in one way or the other. 

Major efforts have been made since the 1926 when Schrödinger formulated SE, to resolve the issues but there is still no resolution in sight accepted by most physicists. There are several very different proposals (Copenhagen, Many-Worlds, Bohmian Pilot Wave...) with the great variety suggesting that they are all wrong. 

The previous post exhibited the "weirdness" of SE in the sense of electrons having both separated existence in different worlds and existence in a common shared world. This is expressed in the multi-dimensional nature of the Hamiltonian $H_{weird}$ underlying SE. 

But there is a different interpretation of $H_{weird}$ named RealQM, which is not weird because the electrons in RealQM have only shared existence with the corresponding Schrödinger equation taking the form of classical continuum mechanics.  

By restricting electrons to share the same 3d world/coordinate system, all the issues troubling StdQM evaporate. RealQM comes out as a form of classical continuum mechanics without issues beyond those of classical physics. RealQM thus offers a unified continuum model including both micro and macro-scopic physics. 

It is a mystery why RealQM was not tried in 1926, since it is the most natural way of generalising Schrödinger's equation from one electron to many electrons, staying within classical continuum mechanics without all the issues of StdQM, which was Schrödinger's approach.

It seems that it was Heisenberg and Born who made history turn in the direction of StdQM and not RealQM. 

The following quotes of Heisenberg shows his ideas rooted in his matrix-mechanics as a new form of physics, which express essence of StdQM:

• What we observe is not nature itself, but nature exposed to our method of questioning.
• Not only is the Universe stranger than we think, it is stranger than we can think.
• The reality we can put into words is never reality itself.
• The atoms or elementary particles themselves are not real; they form a world of potentialities or possibilities rather than one of things or facts.
• The existing scientific concepts cover always only a very limited part of reality, and the other part that has not yet been understood is infinite.
• The ontology of materialism rested upon the illusion that the kind of existence, the direct "actuality" of the world around us, can be extrapolated into the atomic range. This extrapolation is impossible, however. 
• The conception of objective reality ... has thus evaporated ... into the transparent clarity of mathematics that represents no longer the behavior of particles but rather our knowledge of this behavior.

Born came to the help of Heisenberg with the statistical interpretation of the wave function of StdQM:

• If God has made the world a perfect mechanism, He has at least conceded so much to our imperfect intellect that in order to predict little parts of it, we need not solve innumerable differential equations, but can use dice with fair success.
• I am now convinced that theoretical physics is actually philosophy.
• The universe is not a puzzle to be solved, but a mystery to be embraced.

Bohr from his failed attempt to give physics to an atom, also jumped in to help Heisenberg in support of StdQM:

• It is wrong to think that the task of physics is to find out how Nature is. Physics concerns what we say about Nature. 
• Nothing exists until it is measured.
• Opposites are not contradictory but complementary.
• When we measure something we are forcing an undetermined, undefined world to assume an experimental value. We are not measuring the world, we are creating it.
• When it comes to atoms, language can be used only as in poetry.
• There is no quantum world. There is only an abstract quantum physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about Nature.

Together Bohr-Born-Heisenberg formed a very strong team setting the agenda of modern physics to be  StdQM against the will of Schrödinger who wanted something like RealQM. 

It may be that all the unresolved issues with StdQM having taken toll for 100 years finally will open for a come back of Schrödinger and RealQM.

Why Quantum Mechanics is "Weird"

All leading modern physicists agree that quantum mechanics is so "weird" or "absurd" that it cannot be understood: 

• Quantum mechanics describes nature as absurd from the point of view of common sense. And yet it fully agrees with experiment. So I hope you can accept nature as She is - absurd. (Feynman)
• I think I can safely say that nobody understands Quantum Mechanics. (Feynman)
• If quantum mechanics has not profoundly shocked you, you have not understood it yet. (Bohr)

The "weirdness" of quantum mechanics comes from its foundational principle in the form of Schrödinger's equation based on a Hamiltonian $H_{weird}$ of the following form for an atom with kernel of positive charge $Z$ at the origin of a 3d Euclidean coordinate system $R^3$ surrounded by $N=Z$ electrons labeled $i=1,2,...,N$:

• $H_{weird}= \sum_{i}(-\frac{1}{2}\Delta_i -\frac{Z}{\vert x_i\vert}) +\sum_{j<i}\frac{1}{\vert x_i-x_j\vert}$                                        

where each $x_i$ is a 3d coordinate for a separate copy of $R^3$ and $\Delta_i$ the Laplacian differential operator with respect to $x_i$. The Hamiltonian $H_{weird}$ acts on wave functions $\psi (x_1,x_2,...x_N)$ depending on $N$ 3d spatial variables $x_i$, each $x_i$ serving to represent an electron with presence over the whole of its own copy of $R^3$, thus based on electronic wave functions having global supports.

The weirdness comes from the many dimensions where each electron $i=1,...,N$ is equipped with a separate copy of $R^3$, where it is acted upon by a Laplacian differential operator $\Delta_i$ giving it kinetic energy, yet the electrons interact in a common $R^3$ by the presence of $\vert x_i-x_j\vert$ in the term representing Coulomb electronic repulsion. The electrons thus have both separated individual existence and shared existence. That is weird from classical continuum mechanics point of where a shared single physical 3d space is the only one available. With a travesty of Bohr, one could maybe say

• If from a  knowledge of classical continuum mechanics, you are not shocked by $H_{weird}$, then you have not understood classical continuum mechanics.

Quantum mechanics is thus weird unphysical because it is based on a Hamiltonian $H_{weird}$, which is weird unphysical. Of course, if you are dealing with something which is partly weird, then you have to get rid of the weirdness and keep whatever is not weird and maybe useful.  Efforts to this end e g in the form of Density Functional Theory have been made without however completely getting rid on the weirdness. RealQM is a based on $H_{weird}$ but with a new meaning which is physical and not at all weird, see this post for an intro. 

 

Physics and Mathematics of Schrödinger's Equation

The Schrödinger equation describes the ground state of the Hydrogen atom by the wave function $\Psi (x)$ with $x$ a 3d spatial variable, which minimises the total energy

• $E =E_{kin} + E_{pot}$

as the sum of 

• $E_{kin} =\frac{1}{2}\int\vert\nabla\Psi (x)\vert^2dx$     (electronic kinetic energy)
• $E_{pot} =  -\int\frac{\Psi^2(x)}{\vert x\vert}dx$              (electronic Coulomb potential energy)

under the side condition

•  $\int\Psi^2(x)dx =1.$

The model contains the following three components as functions of $x$: 

1. Distributed charge density: $\Psi^2(x)$ with unit total charge.
2. Distributed kinetic energy: $\vert\nabla\Psi (x)\vert^2$.
3. Distributed potential energy: $-\frac{\Psi^2(x)}{\vert x\vert}$.  

The solution can be computed analytically to be $\Psi (x)=\frac{1}{\sqrt{\pi}}\exp(-\vert x\vert )$. The total energy represents the ground state energy of a Hydrogen atom with kernel at $x=0$. The Coulomb potential is classical physics, while the kinetic energy is a new form of energy measured by the gradient $\nabla\Psi (x)$ as an analog to classical elastic energy. The model has a clear physical meaning and the ground state is characterised by a charge density which concentrates around the kernel paying a kinetic energy cost. 

The Schrödinger equation for the Hydrogen atom in charge density is an example of an Eulerian continuum model of the same form as the Navier-Stokes equations for fluid flow in terms of velocity and pressure as distributed functions of a 3d space coordinate, where individual particle trajectories are not followed. A major advantage of a continuum model is that it allows very efficient computation under discretisation of different spatial resolution.

Schrödinger's equation does not involve point positions of electrons, just distributed charge density, and thus has nothing to say about point positions of electron. At least this is what a mathematician would say understanding that a mathematical model does not contain more than what is put in. Even an emergent phenomenon is a consequence of input. It is not meaningful to ask about point positions of the planets in the Solar system at some given time from a model that only contains time-less orbits of the planets. 

A mathematician would add that neither can exact point position of the electron of a Hydrogen atom be determined experimentally. Wittgenstein would agree that asking about electron position does not make sense in the Schrödinger charge density model and so should not no be spoken of.  

But physicists would not hesitate to say that it is meaningful to ask about the position of the Hydrogen electron, even if it is not contained in the model and cannot be experimentally determined. A physicist would insists that Schrödinger's equation is to be viewed to be formulated in terms of a probability density of electron point position, and not charge density. 

But a probability density is not by itself any physical quantity, and is instead by physicists described as a catalog of possible electron point positions. But physics does not keep such a catalog. Changing from charge density to probability density thus turns Schrödinger's equation from being a model of physics into a model of non-physics. 

So why did physicists take this step in 1926 when seeking to give a meaning to Schrödinger's equation for the Hydrogen atom, which had shown to accurately capture the spectrum of the Hydrogen atom from electronic energies of excited states of the ground state. Schrödinger certainly viewed his equation for Hydrogen in terms of charge density and not probability density.

The switch to probability density came with an ad hoc generalisation of Schrödinger's equation for atoms with more than one electron in terms of a multi-dimensional wave function depending on $3N$ spatial variables for an atom with $N>1$ electrons. With the help of Max Born this model was given an interpretation in terms of probability density to form Standard Quantum Mechanics StdQM as the foundation of modern physics. Here Born's rule states that experimental observation of a prepared quantum state as a given linear combination of eigen-states, choses one of the eigen-states with probability measured by its coefficient in the linear combination. 

This is called collapse of the wave function and has remained a true mystery from physical point of view. The eigen-states are deterministically determined by Schrödinger's equation and so the spectrum. Statistics thus enters only in experimental observation of prepared states while the spectrum is always the same. 

This connects to the view of Bohr that the objective of StdQM is to predict outcomes of prepared experiments, not to model reality. This is mind-boggling and does not seem to make any sense. 

 

Main efforts have been made over the 100 years since 1926 to give StdQM back some physical meaning but no consensus about interpretation in terms of charge density has been reached. 

RealQM presents a new different generalisation to $N>1$ of the Schrödinger equation for $N=1$ in terms of a system of electronic non-overlapping charge densities, which keeps the physicality of the Hydrogen atom. RealQM is a continuum model in 3d with computational cost scaling linearly with $N$, compared to StdQM with exponential scaling. 

The linearity of the Schrödinger equation of StdQM invited to a mathematical analysis using the machinery of the new field of functional analysis developed by Hilbert in terms of Hilbert spaces, which von Neumann exploited in a form of axiomatic formal mathematics (in uncomputable form) with axioms without clear physical meaning.   

Mathematics can thus serve to keep physicality of quantum mechanics as in RealQM, but also by von Neumann abstraction leave physicality as in StdQM. Physics education has been locked on StdQM with all its complications from non-physicality as expressed by Nobel Laureate Murray Gell-Mann 50 years ago:

• Niels Bohr brainwashed a whole generation of theorists into thinking that the job of interpreting quantum theory was done 50 years ago.

The brainwash has continued since text-book physics still today is StdQM. Attempts have been made to give StdQM physical meaning like Many-Worlds and Bohmian Pilot Wave but are not viewed to be successful. RealQM opens a new way of thinking, which apparently has been brainwashed away for 100 years... 

Bohr quotes:

• Physics is not about how the world is, it is about what we can say about the world.
• Those who are not shocked when they first come across quantum theory cannot possibly have understood it.

Yes, it is shocking to learn that StdQM it is not about how the world is, that StdQM is non-physical. Schrödinger would have welcomed RealQM as a theory about physics, while such a thing would have shocked Bohr again...

RealQM is based on the same physics as the Hydrogen atom: Coulomb potential energy and kinetic energy. RealQM appears to be about how the microscopic world is...just like Newton's mechanics based on Newton's laws of motion appears to describe how the macroscopic world is...and RealQM connects seamlessly to Newton's mechanics in a unified continuum model.  

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

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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.

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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.