Quantum Mechanics vs Physical Meaning

Modern theoretical physics as Quantum Mechanics QM describes the microscopic world of atoms and molecules in terms of Schrödinger's Equation SE of a mathematical form, which is fundamentally different from the partial differential equations of classical macroscopic continuum mechanics in functions or fields depending on 3d space coordinate $x$ in Euclidean space $\Re^{3}$ (plus time). 

SE for an atomic system with $N$ electrons is formulated in terms of a wave function $\Psi (x)=\Psi (x_1,x_2,...,x_N)$ depending on $N$ 3d spatial coordinates $x=(x_1,x_2,…,x_N)$ (plus time), altogether forming a $3N$-dimensional configuration space $\Re^{3N}$. 

SE was formed by Schrödinger in 1926 for the Hydrogen atom with $N=1$ electron with $\Psi (x)$ a classical function or field depending on a 3d space coordinate $x$ with $\Psi^2(x)$ representing electron charge density. SE was then with a stroke of pen formally extended to $N>1$ by simply adding a new 3d coordinate for each new electron into an equation in a wave function $\Psi (x)$ with $x$ now ranging over $3N$-dimensional configuration space $\Re^{3N}$. That was easy.

It remained to give the extended wave function $\Psi (x)$ with $x\in\Re^{3N}$ a physical meaning for a system with $N>1$ electrons. That showed to be very difficult and has never been resolved in a convincing way. The direct physical meaning as charge density for $N=1$ did not generalise to $N>1$ and it was Max Born, under protests from Schrödinger, who came up the (vague) idea of viewing somehow 

1. $\Psi^2 (x)$ as a probability density of 
2. "finding" the $N$ electrons of the system in a configuration 
3. specified by the coordinates $x\in\Re^{3N}$.

The task assumed by Born was to connect the non-physical coordinate system $\Re^{3N}$ somehow with the $N$ electrons of the system, in order to give a "physical interpretation" of QM. 

This could have been done simply by identifying an electron by its position in physical space, e g by labelling electrons $1,...,N$ and then connect electron $i$ to position $x_i$ with $i=1,...,N$, as is done in RealQM today.

But this idea was rejected based on an argument that electrons are all alike and so cannot be labelled and allocated positions in space. Following this argument electronic wave functions $\Psi (x_1,x_2,....,x_N)$ were stipulated to be anti-symmetric in the $N$ variables $x_1,...,x_N$ in order to guarantee impossibility of "finding" two electrons at the same position. Born thus saved QM from collapse by inventing a probabilistic meaning with a further qualification of anti-symmetry, which was accepted by Heisenberg-Bohr and formed into the Copenhagen Interpretation serving as emergency exit until our days. 

The trouble with Born's "interpretation" is that it is non-physical: a probability has no physical realisation neither has anti-symmetry. By giving up position as identifier, Born's electrons lost physicality and QM was reduced to a math game.  

SE in a wave function defined over configuration space involves the Laplacian acting independently with respect to each 3d coordinate $x_i$, while all $x_i$ share the same 3d system in Coulomb potentials, which makes sense in RealQM.