Physical Computable Quantum Mechanics: Towards The Periodic Table

The basic mathematical of quantum mechanics is Schrödinger's wave equation in terms of a wave function $\psi (x_1,x_2,...,x_N;t)$ depending on $N$ three-dimensional space coordinates $x_1$,...$x_N$ for an atom/ion with $N$ electrons, together with a time coordinate $t$. The interpretation of the wave function, thus depending on altogether $3N$ space dimensions plus time,  according to the so-called Copenhagen interpretation by Bohr-Heisenberg-Born, is that $\psi (x_1,x_2,...,x_N;t)$ is the probability distribution at time $t$ of finding electron $j$ at position $x_j$ for $j=1,...,N$.

Gerard t'Hooft, Nobel Prize in Physics 1999, summarizes the state of quantum mechanics 73 years after the formulation by Schrödinger of his equation, in the book In Search of the Ultimate Building Blocks:

• Quantum mechanics works beautifully, there is little doubt about that.
• However, a very peculiar question presented itself: what do these equations actually mean? What is it they are describing?
• It seems as if electrons can "exist" at different places simultaneously.
• The rules of quantum mechanics work so well that it has become very hard to refute them. Ingenious tricks discovered by Heisenberg, Dirac and many others further improved and streamlined the general rules.
• But Einstein and Schrödinger always had serous objections to the Copenhagen Interpretation. Maybe it works all right, but where exactly is the electron, at the point $x$ or $y$, in reality? 
• The history books say that Bohr has proved Einstein wrong. But others, including myself, suspect that, in the long run, the Einsteinian view might return.
• Probabilities and statistics are mistreated a great deal, even by physicists. Some have uttered, for instance, the theory that all possibilities for certain events are being realized in "parallel worlds"....To my sober mind this is nonsense.

I agree with t'Hooft/Einstein/Schrödinger: Quantum mechanics may work if handled with proper "tricks", but the $3N$-dimensional wave function as a probability distribution cannot be the right thing to describe electron configurations. The reason is double: First, a probability distribution has no direct physical meaning and thus cannot describe the way electrons actually behave. Second, a $3N$ dimensional wave function is uncomputable as soon as $N$ is not very small like 2.  

The first "trick" used to get a computable wave function introduced by Hartree, was to make an Ansatz as a product of $N$ wave functions $\psi_j(x_j)$  each depending on a three-dimensional space coordinate (plus time):

• $\psi (x_1,...,x_N)=\psi_1(x_1)\psi_2(x_2)...\psi_N(x_N),$ 

which reduces Schrödinger's $N$-electron wave equation in $3N$ space dimensions, to a system of one-electron wave equations in 3d. 

I have explored another "trick", which possibly can be elevated from trick to fundamental physical model, with the wave function instead expressed as a sum 

• $\psi (x)=\psi_1(x)+\psi_2(x)+...\psi_N(x)$

in terms of $N$ functions $\psi_1(x),...,\psi_N(x)$ depending on a a common 3d space coordinate $x$ (plus time), with non-overlapping supports $\Omega_j$ filling 3d space. Each wave function $\psi_j(x)$ can then be interpreted as the "charge density" of electron $j$, following the original idea of Schrödinger, subject to the normalization

• $\int_{\Omega_j}\vert\psi_j(x)\vert^2dx =  1.$

The time-independent ground state $\psi (x)$ is then determined as the continuous differentiable function $\psi (x)=\psi_1(x)+\psi_2(x)+...\psi_N(x)$ based on a non-overlapping partition of 3d space into subdomains $\Omega_j$ serving as the supports of the $\psi_j$, which minimizes the total energy functional

• $\frac{1}{2}\int\vert\nabla\psi\vert^2 dx-\int\frac{N\psi^2}{\vert x\vert}dx+\sum_{k\neq j}\int_{\Omega_j}\int_{\Omega_k}\frac{\psi_j^2(x_j)\psi_k^2(x_k)}{2\vert x_j-x_k\vert}dx_jdx_k.$ 

This is a free-boundary electron (or charge) density formulation keeping the individuality of the electrons, which can be viewed as a "smoothed $N$-particle problem" of interacting non-overlapping "electron clouds" under Laplacian smoothing. Preliminary computation (see Quantum Contradictions and Physical Quantum Mechanics) with this model shows a surprisingly good agreement with observations. More precise computations will be reported shortly.

My dream is to rediscover the periodic table from this model. It is not impossible that the dream can come true.

For example, the ground state of Helium in this model consists of two disjoint (contacting) half-spherical electron clouds with energy in close agreement with observation.
This is to be compared with the postulated ground state according to the standard Schrödinger equation considered to be two overlaying spherical electron configurations named as $1s2$, which however has wrong energy and thus is not the true ground state, and thus asks for perturbative correction.

PS1 In the series Quantum Contradictions 1-29 we considered a model closely related to (1) with each one-electron wave function $\psi_j$ a smooth function defined in 3d space satisfying a one-electron wave equation with the Laplacian smoothing acting on the sum of the wave functions,  and not on each individual wave function, with the effect of reducing contributions to the smoothing energy from angular variation. In this case the one-electron wave functions have overlapping support, while being largely separated by repulsion and loosing individuality under overlapping Laplacian smoothing.

PS2  Computing an approximation of the ground-state energy $E$ of Helium using the two normalized separated half-spherical wave functions

• $\psi_1(x)=\sqrt{\frac{2\alpha^3}{\pi}}\exp(-\alpha\vert x\vert )$ for $x=(x_1,x_2,x_3)$ with $x_3\ge 0$
• $\psi_2(x)=\sqrt{\frac{2\alpha^3}{\pi}}\exp(-\alpha\vert x\vert )$ for $x$ with $x_3\le 0$,

 we find with $\alpha =27/16$ the following value:

• $E=-2.902\pm 0.002$,

to be compared with the observed $-2.903$. The standard 1s2 ground state of two overlaying fully spherical wave functions gives a best value of -2.85 asking for so called perturbation correction, which effectively introduces electron separation. We have thus found evidence that the ground state of Helium is not the standard 1s2 state with two overlaying spherical wave functions, without or with perturbation correction, but instead consists of two non-overlapping half-spherical wave functions. If this conclusion indeed shows to be  correct, then electronic wave functions will have to be recomputed for all atoms. Mind-boggling!

PS3 For Li+ we get E = -7.32 (-7.28), for Be2+ we get = -13.72 (-13.65) and for B3+ we get -22.12 (-22.03) with measured value in parenthesis....more to come...