The basic question of computing solutions to the Schrödinger equation was addressed by Hartree
soon after its formulation in 1926. Hartree suggested to compute approximate solutions by letting each electron, assuming for simplicity that the kernels remain fixed as in the Born-Oppenheimer approximation
, seek its own balance by updating its own wave function according to the potentials form the kernels and the other electrons. The resulting set of one-electron Schrödinger equations would then be solved by iterating over the electrons, in what was referred to as Hartree’s method
Instead of seeking to solve a problem in 3N
space dimensions, which is impossible for N
large, Hartree solves N
equations in three-dimensional space, which is possible also for N
large. Hartree’s method was initially successful, but was soon modified into the Hartree-Fock method
, for a specific reason, namely:
Pauli’s Exclusion Principle and the Periodic Table of Elements
The first task of the new quantum mechanics was to determine the electron configuration of the
periodic table of chemical elements ab initio
(from scratch) by solving the Schrödinger equation. This worked fine for Hydrogen with just one electron, but already for Helium with two electrons analytical methods fell short, not to speak of all the other elements with many electrons, like Gold with 79 electrons.
We know that the periodic table is structured with the elements organized into shells at different distance to the kernel, with at most 2 electrons in the 1st shell, 8 electrons in the 2nd and 3rd shell, 18 electrons in the 4th and 5th, and 32 electrons in the 6th.
But this pattern did not seem to be present in solutions to the Schrödinger equation: It seemed that the ground state of smallest energy consisted of trivial spherically symmetric solutions with all electrons packed on top of each other around the kernel, as if there were no shells and different chemical elements with different properties depending on the electron configuration. The ground state determined by the Schrödinger equation did not seem to be the physical ground state, and to single out physical ground states the young Wolfgang Pauli
introduced his exclusion principle
preventing more than two electrons to occupy the same orbit. This eliminated the unphysical spherically symmetric packed electron distributions, and suggested that wavefunctions should be either symmetric or anti-symmetric
with electrons being anti-symmetric.
More generally, particles with anti-symmetric wavefunctions were named fermions, which by antisymmetry
never occupied the same orbit, when taking also spin into account. Particles with symmetric wavefunctions
were named bosons,
which seemed capable of forming unphysical packed
configurations with minimal energy 
. If electrons were bosons, it seemed that there would be only one kind of atom in contradiction to all experience. Thus it seemed that Pauli’s exclusion principle was needed to exclude matter from being
trivially bosonic. But the question if the minimal packed distribution was stable and could be obtained
in a constructive process was left without answer.
Seeking anti-symmetric wave functions in the form of determinants then led to the Hartree-Fock method,
which was still difficult to solve for many-electron atoms or molecules. As a remedy Kohn developed his density-functional theory
with approximate solutions in the form of a collective electron density, which gave the Nobel Prize in 1998.
Many-Minds Quantum Mechanics
Despite the relative success of density-functional theory, and because of its short-comings, we have found reasons to go back to the original Hartree method with a new requirement of accepting as physical solutions only solutions which can arise from constructive iterative computational processes mimicing a construction of an atom with electrons being added one by one and each electron seeking a balance with respect to all the others. It seems possible that in such a constructive process electrons automatically fill successive layers with 2, 8, 8,18,…electrons with each full layer acting as a potential wall preventing intrusion, even without any exclusion principle. Spherically symmetric states with possibly global minimum energy could be out of reach in such constructive processes, which could be seen as a form of instability of such states preventing physical realization. See the further discussion below.
In other words, it is possible that the exclusion principle is unnecessary, if constructability/stability is taken into account. If true this would have pleased Pauli, who was not at all happy with his principle. Like Planck in the case of black-body radiation
, Pauli introduced his principle in a moment of desperation under strong pressure to save a scientific field from collapse, because electrons with bosonic spherically symmetric states
had to be excluded, one way or the other..at any price…
Accepting the non-existence of the complete wave function makes it possible to replace a probabilistic Copenhagen interpretation of quantum mechanics by a deterministic computational interpretation. We
The central idea in both Many-Minds Quantum Mechanics and Many-Minds Relativity is to view a physical
system as consisting of many interacting individual particles/wave-functions/minds, which independently seek to establish balance with respect to the others, without any central control, like in a free market system.
We now present preliminary computational results.
Hartree’s Method for Helium
The helium atom has two electrons with ground state supposedly given by a symmetric 6-dimensional wave function being the product of two identical spherically symmetric 3-dimensional wavefunctions or orbitals. However, the corresponding energy is larger than what is observed, which effectively means that the physical ground state is neither symmetric nor antisymmetric. Computing the ground state using Many-Minds gives a hybrid orbital, which can be viewed as a combination of symmetric and antisymmetric orbitals as illustrated in the following figure with the two electrons occupying two hemispheres: