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.