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.