Schrödinger's Equation in Physical Space as Physics of Chemistry

This connects to this recent post asking if chemistry is well explained by quantum mechanics.

The novelty of modern physics of the 20th century vs classical physics of the 19th century was a  mechanics for atoms and molecules in its canonical form Standard Quantum Mechanics StdQM based on Schrödinger's equation SE (1926), which was fundamentally different from classical continuum mechanics of macroscopic objects based on Newton's equations.  

Modern physicists are educated to claim that the physics of an atom (kernel + electrons) is described by StdQM and so will naturally argue that chemistry is simply physics of molecules and that so also chemistry can in principle be described by StdQM, as famously stated by the famous physicist Dirac in 1933. Modern chemists will say that chemical bonding as the essence of chemistry is not well described by StdQM and so chemists still have a role to play, then referring to Dirac's follow up that Schrödinger's equation is uncomputable and so in practice chemistry cannot be reduced to physics. 

Newton's equation's for a collection of macroscopic objects are computable since computational complexity grows linearly or quadratically with the number of objects. But the computational complexity of Schrödinger's equation grows exponentially with number of atoms/electrons making it uncomputable even for small molecules. The reason Schrödinger's equation is uncomputable is that it has new multi-dimensional form with a separate full 3d coordinate for each electron demanding computation in $3N$-dimensional space for an atom with $N$ electrons with exponential growth in $N$.

Over the 100 years since 1926, many attempts have been made to compress the multi-dimensionality of SE to computable form, most drastically in Density Functional Theory with a reduction to a single electronic charge density depending on a single physical 3d coordinate. I have come up with a less drastic reduction in the form of Real Quantum Mechanics RealQM based on non-overlapping electron densities. My hope is that RealQM can bring Dirac's idea of chemistry as physics into more practice. 

StdQM and RealQM both start from SE for the Hydrogen atom with one electron based on a Coulomb Hamiltonian of the form:

• $H=-\frac{1}{2}\Delta-\frac{1}{\vert x\vert}$     

where $\Delta$ is acting on an electronic wave function $\psi (x)$ depending on a 3d coordinate $x$. The corresponding total energy $E$ is given by 

• $\frac{1}{2}\int\vert\nabla\psi (x)\vert^2dx -\int\frac{\psi (x)^2}{\vert x\vert}dx$

as a sum of kinetic energy measured by $\vert\nabla\psi (x)\vert^2$ and potential energy measured by $\psi (x)^2$ weighted with the kernel potential $-\frac{1}{\vert x\vert}$ for a positive unit kernel/proton at the origin. We understand that the kinetic energy has nothing to do with motion in space but measures spatial variation of $\psi (x)$, which is a source of great confusion.

But StdQM and RealQM generalise to an atom with more than one electron in different ways. 

The Coulomb Hamiltonian $H_{std}$ for an atom with kernel of positive charge $Z$ at the origin of a 3d Euclidean coordinate system $R^3$ surrounded by $N=Z$ electrons, takes the form 

• $H_{std}= \sum_{i}(-\frac{1}{2}\Delta_i -\frac{Z}{\vert x_i\vert}) +\sum_{j<i}\frac{1}{\vert x_i-x_j\vert}$ for $i=1,2,...,N$,                                           

where each $x_i$ is a 3d coordinate for a copy of $R^3$ and $\Delta_i$ the Laplacian differential operator with respect to $x_i$. The Hamiltonian $H_{std}$ acts on a wave function $\psi (x_1,x_2,...x_N)$ depending on $3N$ spatial variables. Compared to classical mechanics in physical 3d space, this is a new strange construction with $N$ versions of $R^3$ so to speak stacked upon each other into a product space

$R^{3N}$ of $N$ versions of $R^3$ which are separated but yet share the same $R^3$ in the electronic repulsion potential $\frac{1}{\vert x_i-x_j\vert}$. The result is that $H_{std}$ has no interpretation in real physical space $R^3$, only a statistical invented by Born under protests from Schrödinger who never accepted $H_{std}$ as physics.

The Coulomb Hamiltonian $H_{real}$ of RealQM takes the form 

• $H_{real}= \sum_{i}(-\frac{1}{2}\Delta_i -\frac{Z}{\vert x_i\vert}) +\sum_{j<i}\frac{1}{\vert x_i-x_j\vert}$ for $i=1,2,...,N$,     

which is identical to that for StdQM above, but with a different meaning of the $x_i$ given as follows: Physical space $R^3$ is partitioned into non-overlapping domains $\Omega_i$ with $x_i$ being the coordinate $x$ in $R^3$ restricted to $\Omega_i$. The Hamiltonian $H_{real}$ acts on a wave function $\psi (x)$ appearing as a sum of one-electron wave functions $\psi_i(x)$ with $x\in\Omega_i$ thus with non-overlapping supports, all depending on the same space coordinate $x$. 

It seems that such a Hamiltonian is not described in the literature, although it appears as the most natural generalisation of the Hamiltonian for the hydrogen atom, much more reasonable than that of StdQM. Maybe the reason is the Bernoulli free boundary condition of RealQM allowing electron densities to meet with continuity and zero normal derivative at boundaries between different domains $\Omega_i$, only appears in forms of mathematical analysis of free boundary problems pursued from 1950s connecting to classical mechanics. 

Sum up:

• StdQM works with electronic wave functions with overlapping global supports (not natural).
• RealQM works with electronic wave functions with non-overlapping supports (natural).
• StdQM works with electrons without identity (not natural).
• RealQM works with electrons with identity by spatial occupancy (natural).
• StdQM is formulated in abstract multi-dimensional space (not natural).
• RealQM is formulated in real physical space (natural)
• StdQM is uncomputable (not reasonable).
• RealQM is computable (reasonable).
• StdQM cannot explain chemistry (not useful).
• RealQM has a potential to explain chemistry (possibly useful).