The Tragedy of Schrödinger (and QM)

The atomic world was opened to theoretical exploration in 1926 when Erwin Schrödinger formulated a mathematical model of the Hydrogen atom with one electron in terms of a wave function $\Psi (x)$ depending on a 3d spatial coordinate $x$ with $\Psi^2 (x)$ representing electron charge density. So could Schrödinger represent the ground state of the Hydrogen atom by the real-valued wave function $\Psi (x)$ minimising the total energy 

• $E(\psi )=E_{kin}(\psi )+E_{pot}(\psi )$

where 

• $E_{kin}(\psi )=\frac{1}{2}\int\vert\nabla\vert^2dx$  is kinetic (electron "compression") energy 
• $E_{pot}(\psi )=-\int\frac{\psi^2(x)}{\vert x\vert}dx$ is Coulomb potential energy

over all functions $\psi (x)$ defined in 3d Euclidean space with $\int\psi^2(x)dx=1$. Using his solid knowledge of Calculus, Schrödinger was very pleased to find the solution in analytical form:

•  $\Psi (x)=\frac{1}{\sqrt{\pi}}\exp(-\vert x\vert )$. 

In an outburst of creativity in the Alps in the Winter 25-26 together with one of his girlfriends, Schrödinger generalised to a time dependent form capturing the observed spectrum of Hydrogen, and the success was total as a first glimpse into the atomic world to form the new focus of modern physics.

The next step beyond Hydrogen was Helium with two electrons. What could the wave function look like for two electrons?  How to generalise from one to many? Two options presented themselves:

1. Real physics: Add a new non-overlapping charge density for each new electron. 
2. Formal mathematics: Add a new set of 3d spatial coordinates for each new electron. 

Schrödinger contemplated 2. but did not think it was the right way to go because the wave function for Helium would involve 6 spatial dimension and so lack physical representation. For some reason Schrödinger did not pursue 1. either. So right after the big success with Hydrogen Schrödinger hit the wall. 

But other physicists were ready to quickly jump in (1927) following 2. with enthusiasm:

• Born-Oppenheimer: Formal generalisation without Pauli Exclusion Principle PEP. Probabilistic interpretation of wave function.
• Heisenberg-Dirac: Addition of PEP, antisymmetry and Slater determinants.

This forms the basis of Standard QM StdQM also today in its Bohr-Born-Heisenberg text book Copenhagen Interpretation.  

To Schrödinger the take-over of his baby came as a shock:  

• I am not happy with the probability interpretation. In my opinion, it is an ephemeral way to avoid the true problem.
• The whole antisymmetrization seems to me to be a desperate expedient to save the particles’ individuality. Perhaps it is not fundamental but only an approximation.
• The use of Slater determinants in atomic theory seems to obscure the physical picture even more.
• I don’t like it, and I’m sorry I ever had anything to do with it.
• The $\psi$-function as it stands represents not a single system but an ensemble of systems. It does not describe a state of one system in configuration space but rather an ensemble of systems in ordinary space. The $\psi$-function itself, however, does not live in ordinary space, but in the configuration space of the system. And not merely as a mathematical device — it really exists there. That is what is so repugnant about it.

Schrödinger thus quickly became incompatible with StdQM and accordingly was marginalised, along with Einstein sharing similar criticism.

RealQM represents a new initiative to follow 1. as real physics in the spirit of Schrödinger. What would Schrödinger have said about RealQM which lay on the table already in 1927? What would have happened if Schrödinger had been allowed to take care of his own baby and not given it away others?