Restart from Schrödinger 1926 into RealQM

This is a follow up the post Anniversary: When Physics Went Wrong 1926

Schrödinger was very happy with his Schrödinger Equation SE for the Hydrogen atom with one electron formulated in early1926 as a partial differential equation of the form of classical continuum mechanics in a Euclidean space $\Re^3$ of 3 dimensions, because he could show by analytical mathematics that the eigenvalues of SE agreed with the already known Rydberg formula for the observed spectrum of the H atom, and so solved an outstanding open problem.  

But Schrödinger was very unhappy with the formal generalisation to atoms with $N>1$ which quickly followed applauded by Bohr-Born-Heisenberg, because it came with an extension of physical space $\Re^3$ to $\Re^{3N}$ referred to as configuration space, which is not physical space for $N>1$.  

Schrödinger wanted a to see a mathematical model with physical meaning as possible to visualize as a model in 3d physical space $\Re^3$.

But the SE was formulated in terms of a wave function $\Psi (x_1,...,x_N )$ depending on $N$ 3d coordinates $x_1,....,x_N$, one for each electron, that is a wave function $\Psi (x)$ depending on $x=(x_1,...,x_N)\in\Re^{3N}$ as configuration space. The water molecule $H_2O$ would then be described by a wave function $\Psi (x)$ depending on $x\in\Re^{30}$ way beyond computational resolution.

A configuration space $\Re^{3N}$ was repugnant to Schrödinger and so he desperately sought a way to compress the wave function over configuration space to $\Re^3$. In a letter to Lorentz on June 6 1926 Schrödinger writes:

• If we now have to deal with $N$ particles, then $\Psi (x_1,...,x_N )$ is a function of $N$ variables $x_1$,...,$x_N$  over $N$ 3d spaces $R_1,...R_N$.  
• Now first let $R_1$ be identified with the real space $\Re^3$ and integrate over $R_2, …,R_N$.
• Second, identify $R_2$ with the real space and integrate over $R_1, R_3,...,R_N$ and so on. 
• The $N$ individual results are to be added after they have been multiplied by certain constants which characterise the particles. 
• I consider the result to be the electric charge density in real space.

Schrödinger thus suggested a compression of $\Psi (x_1,...,x_N )$ into a sum of wave functions $\Psi_i(x)$ with $x\in\Re^3$, where $\Psi_i(x)$ is formed by identifying $R_i$ with $\Re^3$ and averaging over all $x_j$ with $j\neq i$. 

Schrödinger did not follow up this line of thought, because the weight of Bohr-Born-Heisenberg was too big, and so today 100 years later textbook Standard Quantum Mechanics StdQM is formulated in terms of wave functions over configuration space $\Re^{3N}$ and so physics is missing. 

RealQM follows up Schrödinger's suggestion into a SE expressed in a wave function $\Psi (x)$ with $x\in\Re^3$ expressed as a sum 

•  $\Psi (x) =\Psi_1(x)+...+\Psi_N(x)$

where $\Psi_i(x)$ for $x\in \Omega_i$ is the wave function representing an electron charge density with support over $\Omega_i$, where $\Omega_1,...,\Omega_N$ is a subdivision of $\Re^3$ without overlap.

RealQM thus is a realisation of Schrödinger's original idea from June 6 1926, in terms of non-overlapping one-electron charge densities identified by spatial presence. which appears to have been suppressed for 100 years. It seems that Schrödinger requirement of physicality can be met in a very natural way. Why Schrödinger did not invent RealQM,  is a bit of mystery. Why StdQM lacking physics has come to fill textbooks for 100 years, is also a mystery.

What would happen if RealQM indeed shows to be a better description/model than StdQM, following the spirit of Schrödinger, is outlined by chatGPT in this recent post.

The break with classical continuum physics in 3d, hailed as modern physics, happened with the introduction of configuration space $\Re^{3N}$ with $N>1$ in the SE of StdQM. Physicality of actuality in $\Re^3$ was then replaced by probability of possibility in $\Re^{3N}$ and rationality was lost.