Many-Minds Quantum Mechanics

the computation is the message

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Abstract


A computational version of quantum mechanics in the spirit of the Hartree method is presented in which each electron in a multi-electron system updates its own state over time by solving its own Schrödinger equation in three-dimensional space expressing the attraction from the kernels and the repulsion from the other electrons, which defines a stable configuration of the system by computation. In this many-minds model the full many-dimensional wave-function is not computed (and thus does not exist) and Pauli’s exclusion principle is replaced by stability requirement. A parallel is made with the interaction of a group of people interacting pairwise, which can described as a set of individual states while the total multi-dimensional interaction is determined by nobody (and thus does not exist).


I was unable to give a logical reason for the exclusion principle or to deduce it from more general assumptions. I had always the feeling and I still have it today, that  this is a deficiency… From the point of view of logic, this lecture has no conclusion. (Wolfgang Pauli, Nobel Lecture 1945  On Exclusion principle and quantum mechanics

Modern Physics = End of Physics?

Quantum mechanics and theory of relativity are the pillars of modern physics, but they are incompatible 
which has prevented the development of a unified field theory and threatens to led to an end of physics
The theories of quantum mechanics and relativity are not only impossible to combine, but each theory has
its own internal difficulties, which have never been resolved. Erwin Schrödinger who formulated  the Schrödinger equation as a basis of quantum mechanics, stated to Niels Bohr in 1926:
  • If we are going to have to put up with these damned quantum jumps, I am sorry that I ever had anything to do with quantum mechanics. 

What Schrödinger objected to was more precisely the Copenhagen interpretation, with the wavefunction 

as a solution of Schrödinger’s equation being a multi-dimensional probability distribution of particle configurations, which was forcefully advocated by Bohr and Heisenberg into a credo of modern physics, despite criticism increasing with time. See the related knol Why Schrödinger Hated His Equation. 

The Multi-Dimensional Wavefunction Does Not Exist

The Schrödinger equation expresses a balance of potential and kinetic energies of a set of postively charged kernels and a set of negatively charged electrons, where the potential energies result form electrostatic attractive and repulsive forces. The ground state is characterized as the configuration 
with least total energy as the sum of potential and kinetic energies.
In a sense the Schrödinger equation is like classical Lagrange equations of motion for a system of particles interacting through attractive and repulsive gravitational forces decaying with the distance squared according to Newton’s law. But there is an important difference: Whereas a solution to Lagrange equations define the position of the particles in three-dimensional space, the wavefunction of the Copenhagen interpretation is a function depending on 3N space dimensions if the number of kernels and electrons is N,  which according to Nobel Laureate Walter Kohn is not a legitimate scientific concept if N > 100 – 1000, in other words the multi-dimensional wavefunction does not exist.

Superposition = Superstition?

The Schrödinger equation is formally linear and formally allows superposition with the sum of two solutions also being a solution. Formally, the Schrödinger cat in its closed box would thus be able to exist
in a mixture of both dead and alive states, with the possibility of collapsing to either dead or alive only upon opening of the box, which is also the idea of the quantum computer.
However, if the multidimensional wavefunction as a solution of Schrödinger´s equations does not exist, then superposition may not hold, and the Schrödinger cat would be alive at opening unless it had not died before, and thus behave just like the cats we know.
The possible non-existence of a multi-dimensional wavefunction obviously gives new perspective on quantum mechanics and puts the focus on possibly existing approximate solutions.

An Illustration


To illustrate non-existence of a complete multi-dimensional wave function, consider a group of N persons interacting with each other, each person seeking a position in balance subject to forces or influences from all the others. The complete wavefunction would then corresspond to some form of super-therapist knowledge of the full complex of interactions between all the persons. In general, there is no such super-therapist in possession of this information, but only the set views of the individuals seeking balance with respect to the others. Similarly, following Kohn, it is not legitimate to expect the existence of a complete multi-dimensional wavefunction as a solution of the multi-dimensional Schrödinger equation. 

The key question is thus the meaning of the Schrödinger equation, which is directly connected to the constructive method of computing solutions thereof. The meaning comes from the computation, and not from elsewhere.  Or as a version of MacLuhan’s the medium is the message:
  • the computation is the message.

Hartree’s Method

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 [1]. 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.

                                   Kohn                            Pauli                            Hartree

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
have used the same approach in a new formulation of the second law of thermodynamics without reference to probability.
We refer to this approach as Many-Minds Quantum Mechanics as presented in Many-MInds Relativity.
See also many-minds interpretation and [2] and work by H. D Zeh.
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:   

                         Hybrid oribitals for helium as a sum of a symmetric and non-symmetric orbitals


 

The fact that the wave functions of even simple molecules represent hybrid states was pointed out by

Linus Pauling  in the late 1920s earning him the Nobel Prize in Chemistry in 1954. Insisting that wave functions are either symmetric or anti-symmetric seems to be in contradiction to facts.

 

The two electrons of helium thus divide the territory into two hemi-spheres with a small presence at the kernel which gives a combined potential from the kernel and the electrons with a negative dip at the kernel surrounded by a positive wall from the electrons. The same pattern can be expected for the first two electrons of the next element in the periodic table Lithium, which would form an inner shell with the third electron effectively meeting a potential wall preventing penetration to the kernel, without any exclusion principle.

 

Many-Minds for Atoms and Molecules

 
The vision is to obtain the periodic table ab initio by computational solution of the Schrödinger equation without spin and Pauli´s exclusion principle, with only requirement of stability, starting with an inner shell of two electrons in a hybrid state forming a potential wall and successive electrons filling larger shells forming new walls. The same basic idea was used in the resolution of d’Alembert’s paradox based on letting physical computed turbulent solutions replace unphysical analytical unstable potential solutions.