Real Quantum Mechanics for Atomic Kernels

Helium atom: 2 electron clouds around a kernel consisting of 2 proton and 2 neutron clouds.

In a moment of heavenly inspiration Schrödinger formulated in 1925 a mathematical model of the Hydrogen atom as a negatively charged electron cloud attracted by a positively charged proton point kernel. The ground state of the atom was given in the form of a real-valued wave function $\Psi (x)$ depending on a space variable $x$ with $x=0$ at the kernel and $\Psi (x)^2$ representing charge density, as the minimizer of the total energy cost functional

• $E(\Psi ) = KE(\Psi ) + PE(\Psi )$,

where

• $KE(\Psi ) = \int\frac{1}{2}\vert\nabla\Psi\vert^2dx$ 

is kinetic energy, and 

• $PE(\Psi ) = -\int\frac{\Psi^2(x)}{\vert x\vert}dx$.

is potential energy expressing Coulomb attraction, under the normalization

• $\int\Psi^2dx =1.$                                  

Qualitatively the minimising charge density representing an eigenstate of minimal energy, concentrates towards the kernel making the negative $PE(\Psi )$ small under the positive gradient cost of $KE(\Psi )$ and takes the analytical form $\Psi (x)=\exp(-\vert x\vert )$ modulo normalisation (with 2d section here), with a charge density dropping off exponential away from the kernel. 

In classical continuum mechanical terms, the kinetic energy can be seen as a form of elastic energy increasing with compression. Real Quantum Mechanics RealQM offers in this spirit a generalisation to atoms with several electrons represented by non-overlapping charge densities. This model has a deterministic physical meaning and is different from the standard Schrödinger equation in configuration space with only probabilistic meaning as stdQM. RealQM thus retains Schrödinger's original conception for the Hydrogen atom of an electron cloud around the kernel with physical meaning, while stdQM already for Hydrogen connects the wave function to the probability of finding an electron particle at a specific location around the kernel (abhorred by Schrödinger).

 

Schrödinger suggested that the atomic kernel similarly possibly could be viewed as a positively charged cloud. For a kernel with several protons to be stable that would require an attractive force named strong force stronger than that of Coulomb repulsion. 

RealQM has a natural extension to a kernel model consisting of non-overlapping positive charge densities with repulsion replaced by attraction from an assumed strong force. In the simplest case with spherical homogenisation this model takes the above form, as a special case of the Cornell potential of Quantum ChromoDynamics QCD. 

RealQM thus can be extended to include also the kernel with the same approach of non-overlapping charge densities interacting by long range Coulomb attraction/repulsion and short range strong force. 

Notice that the contribution to the total energy of positive kinetic energy requires a distributed charge density and thus excludes point particles with infinite kinetic energy, in agreement with Schrödinger's wave mechanics without particles. In the setting of atomic eigenstates, the wave function represents a distributed cloud of charge like the spatial part of a standing wave.

A first challenge would be to model a Helium kernel with 2 protons and 2 neutrons taming the strong force into a stable kernel. Specifically it would give the kernel a small positive radius as required to make RealQM prediction of ground state energy to precisely agree with observation. 

PS Note that the Standard Model of QCD describes interaction of quarks forming hadrons such as protons  mediated by force carrying gluons, both viewed as particles (abhorred by Schrödinger).