What Is an Electron? RealQM vs StdQM.

Electrons are described by wave functions as solutions to Schrödinger's Equation SE as the basic mathematical model of the quantum mechanics of atoms and molecules which comes in two forms: Standard Quantum Mechanics StdQM and Real Quantum Mechanics RealQM:

• Wave functions of StdQM for a system with $N$ electrons have global supports and depend on $N$ 3d spatial coordinates, altogether $3N$ spatial coordinates. Wave functions are computable only for very small $N$ since computational complexity grows exponentially with $N$.
• Wave functions of RealQM are sums of one-electron wave functions with non-overlapping supports depending on the same 3d spatial coordinate, and meet with continuity at a free boundary between supports. Wave functions are computable for all $N$, since computational complexity grows linearly with $N$. 

We now consider in more detail electrons according to RealQM. Each electron is described as a charge density $\psi (x)^2$ of a real-valued wave function $\psi (x)$ with support (non-zero value) in a certain region $\Omega$ in space with boundary $\Gamma$. The electron charge density does not have to vanish on $\Gamma$. Electrons sharing a common boundary piece meet with continuity as a free boundary condition. This allows the kinetic energy of an electron measured by $\vert\nabla\psi (x)\vert^2$ to be small even if the size of $\Omega$ is very small, which is not possible if $\psi (x)$ is forced to vanish on $\Gamma$. In other words, in the presence of other electrons, an electron can have small support and small kinetic energy, which means that it can circumvent the dictate of Heisenberg's Uncertainty Principle.

RealQM describes protons in the same way as electrons, with only a shift of sign of the charge.  

RealQM allows electrons to appear in two different forms in atoms and atomic nuclei as: 

• Charge densities with large support in atoms around a nucleus of vanishing size, meeting the nucleus freely (RealAtom). 
• Charge densities with very small support in kernels of nuclei surrounded by protons meeting proton charge densities with continuity, making sense since the size of the kernel is not small compared to the nucleus (RealNucleus). 

RealQM thus offers a complete model of an atom + nucleus in terms of: 

• A Schrödinger equation for a collection of positive and negative charge densities with non-overlapping supports interacting by Coulomb potentials, 
• The large difference in mass between proton and electrons allows the electron to serve a double role with presence both outside and inside the nucleus.  

RealQM is non-relativistic with motivation from the fact that there is no charge density motion in neither atom nor nucleus at all, and then certainly not at any relativistic speed. This is fundamentally different in StdQM where inner core electrons of heavy elements like Gold are claimed to move at half the speed of light by a purely formal argument connecting kinetic energy to velocity. Electrons can in giant particle accelerators be accelerated to relativistic speeds with massive input of energy, but its is difficult to fathom that the same thing happens in an atom of Gold...

Summary:

• RealQM describes a collection of $N$ electrons as a charge densities with non-overlapping local supports in 3 space dimensions meeting with continuity and interacting by Coulomb potentials. RealQM is computable for all $N$.
• StdQM the collection as overlapping charge densities with global support in $3N$ space dimensions. StdQM is computable only for very small $N$. 
• RealNucleus describes a nucleus as a collection of non-overlapping electron and proton charge densities meeting with continuity and interacting by Coulomb potentials.
• The Standard Model of StdQM describes a nucleus as a collection of protons and neutrons interacting by a residual of a strong force.   

 

   

The Nucleus Enigma: Proton-Electron Symbiosis

The Standard Model SM offers an explanation of stability/existence of an atomic nucleus of charge $+Z$ consisting of in a basic case $Z$ protons and $Z$ neutrons in terms of new force beyond the Coulomb force of the Schrödinger equation named strong force meditated by force-carrying gluons. SM is an ad hoc model with many parameters invented in the 1960s serving as the main model of nuclear physics still today, as the greatest triumph of theoretical physics of all times.      

RealNucleus as an extension of RealQM for atoms to nuclei offers an explanation of stability/existence of a nucleus consisting of $Z$ electrons and $2Z$ protons (corresponding to $Z$ protons and $Z$ neutrons with formally a neutron = proton + electron), as an extension of the Schrödinger equation to a nucleus including only Coulomb force without the strong force. RealQM thus offers a model of an atom with full quantum mechanical resolution of both atomic electrons and nucleus based on Coulomb potentials/forces. If this model indeed holds up to such a proposition under closer evaluation, it could be viewed as as sensational. 

Let us here do a simple check in a toy model to understand why it is possible for RealNucleus to show stability of a nucleus as a system of protons and electrons interacting by Coulomb potentials. For the real model go to RealNucleus.

We start with the nucleus of 2H consisting of 2 protons surrounding a nucleus kernel of 1 electron. Suppose a linear particle configuration with the protons at coordinates -1 and +1 and the electron at 0. We have the following Coulomb potential energies: 

• proton-electron attraction = -1-1 = -2
• proton-proton repulsion = +0.5 
• total energy as (total potential energy)/2 = -0.75.       

We understand that this is a special case without electron-electron repulsion since self-repulsion is excluded.  

We next consider 4He consisting of 4 protons surrounding a nucleus kernel of 2 electrons, thus a case with non-zero electron-electron repulsion. Suppose a planar quadratic configuration with the electrons at (-0.5, 0) and (0.5, 0) and the protons at (-1.5,0), (1.5,0), (0,1.5) and (0.-1.5) in a 2d coordinate system, which gives the following Coulomb potential energies (with different spatial scale as compared to 2H)   

• proton-electron attraction $ < - 1-1-\frac{1}{2}-\frac{1}{2}-\frac{4}{1.5\sqrt{2}}$
• proton-proton repulsion $  = \frac{1}{3}+\frac{1}{3}+ \frac{4}{1.5\sqrt{2}}$
• electron-electron repulsion $= 1$ 
• total energy as (total potential energy)/2 $< -\frac{2}{3}$.       

We find a total energy which is clearly negative even in the presence of electron-electron repulsion from the kernel. This is made possible by assuming a distance between the electrons in the kernel (=1) to be comparable with the distance to the protons, thus with a nucleus kernel of size comparable to that of the nucleus. This is made possible by the presence in the Schrödinger equation of mass $m$ in the coefficient $\frac{1}{2m}$ of the Laplacian, which sets a spatial scale by the factor $\frac{1}{\sqrt{m}}$. Recalling that the $m$ for the proton Laplacian is much bigger than that of the electron Laplacian (factor 1836) we find a rationale for assuming that kernel is not small compared to the nucleus, thus allowing electron-proton attraction to dominate electron-electron repulsion. 

We thus find that system of $Z$ electrons forming a nucleus kernel surrounded by $2Z$ protons is stable under Coulomb attraction-repulsion, where the small electron mass vs proton mass plays a crucial role to allow domination of electron-electron repulsion by proton-electron attraction. 

We thus find the enigma of the stability of a nucleus can be resolved as marriage between the two components of the system: 

• control of electron-electron kernel repulsion by surrounding protons
• control of proton-proton repulsion by electron kernel
• a kernel of $Z$ electrons binding $2Z$ protons
• $2Z$ protons confining a kernel of $Z$ electrons,  

as an expression a fundamental principle of symbiosis. 

RealNucleus: First Full Quantum Model of a Nucleus with only Coulomb Forces

The article introducing RealNucleus, as an extension of RealQM for atoms as RealAtom, has now been updated. It appears to be the first full quantum mechanical model of an atomic nucleus showing that stability/existence as negative total energy is realised with only Coulomb forces in tests of basic cases, thus without need of the ad hoc strong force of the Standard Model. 

With this extension RealQM appears to offer a computable full quantum mechanical model of an atom-nucleus in a classical setting of Coulomb potentials, thus without need of uncomputable QED/QCD.   

Why No Unified Atom-Nucleus Model?

The previous post presented RealQM = RealAtom + RealNucleus appearing to be the first computable unified model of an atom including nucleus with full quantum mechanical representation in the form of non-overlapping charge densities of both electrons and protons interacting by Coulomb potentials. 

Is this really the first full quantum mechanical model of an atom + nucleus? What have theoretical physicists been doing during the 100 years since the advent of quantum mechanics in 1925? 

Yes, at least according to chatGPT, telling that there is QED for atoms = electrons+point-wise nuclei and QCD for nuclei = quarks and gluons, but QED and QCD represent different "sectors" and cannot be unified:

• There is currently no fully computable, unified quantum mechanical model of an , "atom that includes both the electrons and the atomic nucleus in full quantum detail and that is tractable for general-purpose computation.

How can this be? Is this the concrete meaning of the "crisis of modern physics" proclaimed by leading theoretical physicists? Is it a consequence of all the unresolved contradictions present in the mathematical foundations of quantum mechanics including "wave-particle duality", "complementary principle", "collapse of the wave function", "measurement problem", "statistical interpretation", "uncertainty principle", "exclusion principle", "anti-symmetry", "Born rule", "exchange energy", "indistinguishability of identical particles", "electron orbitals", "superposition", "entanglement", "decoherence", "spin-orbit coupling" + all the wonders of QCD…? 

When I ask if the lack of a unified atom-nucleus model is a sign of failure/crisis of modern physics, chatGPT explains:

• The lack of a unified, computable atomic-nuclear theory is not a failure — it's a reflection of the extraordinary success and specialization of the theories we already have (QED and QCD).  

This is the tragedy of modern physics: Too successful to fail. Like a Big Bank or Great Empire. If you find chatGPT's argument convincing, you have a position at a department of fundamental physics (about to collapse from missing unified theory).

RealQM as RealAtom + RealNucleus

RealQM has now been extended to a full Schrödinger Equation SE for an atom as an electron density of total charge $-Z$ surrounding a nucleus as a proton density of total charge $+2Z$ surrounding a nucleus kernel as an electron density of total charge $-Z$, as RealAtom + RealNucleus. The total energy includes the kinetic energies of both electrons and protons as well as all Coulomb potential energies including both electron-electron repulsion and proton-proton repulsion. The nucleus here appears as in inverted form of the atom with switched roles of electrons and protons, like a Russian doll system with an electron-proton-electron pattern from nucleus kernel -Z to nucleus +2Z to atom -Z (in basic form).

RealAtom computes binding energies of atoms in eV and RealNucleus binding energies of nuclei in MeV with a change of scale of about $5\times 10^5$ reflecting a change of spatial scale from nucleus to atom, while the change of scale from nucleus kernel to nucleus is much smaller around $10$. 

RealQM thus offers a full SE for an atom with nucleus based on a Hamiltonian including all kinetic and Coulomb potential energies. The model is parameter-free modulo the change of scale from nucleus to atom, assuming a mass ratio of 1836 between proton and electron. The computational complexity scales with $Z$.

Note that textbook Standard Quantum Mechanics StdQM including the Standard Model SM does not offer any such complete Schrödinger which is computable. What is offered is (i) SE for an atom with nucleus modeled as a point-wise charge density with zero kinetic and potential energies, and (ii) a shell model of a nucleus consisting of protons and neutrons swimming in a negative potential from a charge density without kinetic and potential energy. StdQM thus does not include the full SE of RealQM. 

In short, RealQM offers the first full SE of an atom including nucleus as a unified model in terms of non-overlapping electron and proton charge densities interacting by Coulomb potentials while adding kinetic energies to potential energies to total energy with computations geared to find minima corresponding to ground states.

Preliminary computations show that RealQM can match observations. RealQM may show a way out of the dead-end of uncomputable StdQM of atom including nucleus. 

Test case 1: 2H

The basic test for RealNucleus is the 2H nucleus in StdQM viewed to consist of 1 proton and 1 neutron, and in RealNucleus viewed to consist of 2 proton charge densities surrounding a nucleus kernel of 1 electron. This a the nucleus analog of an atom consisting of 2 electrons surrounding a nucleus of 1 proton, that is the $H^-$ ion of the H atom with one extra electron, which is known to be stable. If we then assume that the only electron of the nucleus kernel of 2H has zero kinetic energy and no self repulsion, we get the message that 2H should be stable. RealNucleus confirms by giving a binding $E\approx 1$ MeV including the kinetic energy and zero potential energy from no self repulsion of of the nucleus kernel.  

Test case 2: 4He

A more serious test case is the 4He nucleus in StdQM viewed to consist of 2 protons and 2 neutrons, and in RealNucleus viewed to consist of 4 protons surrounding a nucleus kernel of 2 electrons. In this case both kinetic and potential energy of the kernel add to the total energy, and the question is if then the total energy will be negative indicating stability, or not? We use this code realising RealNucleus in a simple implementation with spherical symmetry starting from this input screen with 2 electrons as red spherical nucleus kernel surrounded by 4 protons in a green-blue nucleus: 

Pressing start we get the following output showing electron/proton densities in red and total potential in blue as functions of radius in spherical symmetry:
    

We see that the electron and proton charge densities meet with continuity at the boundary of the nucleus kernel (crest of red curve) with the electron/proton charge density being attracted by the proton/electron charge density into a negative contribution to total energy dominating over kinetic energies, resulting in a total negative binding energy of $E\approx 7$ MeV with a spatial scaling of $4\times 10^5$ between between nucleus kernel and nucleus. We see that the radius of the kernel of the nucleus is about the half of the nucleus. 

Notice that a physicist properly trained by StdQM would say that a nucleus kernel of electrons is impossible because electrons are too big to fit and if fitted the kinetic energy would be in the 100s of MeV. But this is not what RealNucleus tells us as displayed in the output figure above: The radius of the kernel is not so small and the electron kinetic energy can remain small because the electron charge density does not have to vanish on the boundary, only meet the proton charge density with continuity.   

We understand that the balance of $Z$ electrons vs $2Z$ protons is instrumental to overcome the potential energy from electron-electron repulsion in the kernel of the nucleus. A configuration of $2Z$ electrons combined with $2Z$ protons as a form of neutral kernel as an analog to an atom with $2Z$ electrons surrounding a proton nucleus with the same number of charges, is unstable. 

We thus see that RealNucleus explains in particular why a nucleus with an approximately equal number of protons and electrons, can be stable.  

PS1 Use this code to test other nuclei.

PS2 In RealAtom electron densities meet at a free boundary with continuity (and zero normal derivative), and in RealNucleus electron and proton densities meet likewise. This gives an explanation of the fact that electrons and protons do not instantly annihilate under Coulomb attracting, but can coexist by occupying different regions of space meeting a free boundary with continuity of charge density of same or different sign. In StdQM electrons have global overlaying supports which is not compatible with either repulsion or attraction. RealQM resolves this contradiction by assigning charges separate domains in space.