From StdQM to RealQM

Modern physics of Standard Quantum Mechanics StdQM is based on a Schrödinger Equation SE for an atom with $N$ electrons in terms of a wave function $\Psi (x_1,....,x_N)$ depending on $N$, with electron i associated to a 3d spatial coordinate $x_i$, altogether forming a $3N$-dimensional configuration space. SE takes the form of an eigenvalue problem of the form 

• $H\Psi =E\Psi$

where $H$ is the Hamiltonian operator 

• $H = -\sum_i\frac{N}{\vert x_i\vert }+\sum_{i<j}\frac{1}{\vert x_i-x_j\vert }-\sum_i\frac{1}{2}\Delta_i$

with $\Delta_i$ the Laplacian acting with respect to $x_i$, and $E$ an eigenvalue. The first two terms of $H$ are classic Coulomb potentials, while the Laplacian term is unusual acting over the full configuration space. SE is thus a linear differential equation acting on wave functions over $3N$-dimensional configuration space and thus over physical 3d space only for $N=1$ as the Hydrogen atom with one electron. 

For N>1 SE thus appears with a wave function solution $\Psi (x_1,...x_N)$ over a $3N$-dimensional configuration space which is not physical, which has forced physicists to connect QM to probabilities of possibilities instead of realities, with far reaching consequences concerning ontology. 

Even worse, the presence of the Laplacian $\Delta_i$ loads SE with exponential computational complexity because each coordinate demands a certain resolution making computational work grow exponentially in $N$.

SE is thus both unphysical and uncomputable in the $3N$-dimensional setting, and so to deliver anything must be drastically reduced dimensionally. Density Functional Theory DFT is the extreme reduction into a common electron charge density in 3d. 

RealQM offers a less drastic reduction into a collection of non-overlapping one-electron charge densities $\Psi_i(x)$ depending on a common 3d variable $x$ over a subdivision of 3d-space into domains $\Omega_i$. In this case the $3N$-dimensional $\Psi (x_1,...,x_N)$ is reduced to a sum of $\Psi_i(x)$ simply by identifying $x_i$ with $x$ for $x_i\in\Omega_i$. 

The 3d space for $x_i$ is thus trivially reduced to $\Omega_i$ and so to $x_i$ can be represented by $x\in\Omega_i$ altogether by $x$ in 3d-space.  

RealQM thus gives a SE in a wave function $\Psi (x)$ depending on a 3d-space variable $x$, which has a clear physical meaning in terms of non-overlapping charge densities, and is readily computable.

Note that reductions/alterations of StdQM result from imposing specific physics, and StdQM does not serve well as a canonical model to start with because it is unphysical. In particular, the idea of identifying anti-symmetry of wave functions as physics, may not be meaningful.