Quantum Mechanics as Smoothed Particle Mechanics

As a follow up of the ideas in the sequence of posts on Quantum Contradictions I sketch here an approach to quantum mechanics as a form of smoothed particle mechanics which allows a deterministic physical interpretation and is computable, thus without the difficulties of the standard multidimensional wave function, which according to Nobel Laureate Walter Kohn is not a legitimate scientific concept.

Simulations using this approach are under way.

The Crisis in Modern Physics: Too Complicated

The last sequence of posts on Quantum Contradictions 1 - 20 gives examples of the crisis in modern physics recently described by the Perimeter Institute Director Neil Turok as follows:

• Theoretical physics is at a crossroads right now…In a sense we’ve entered a very deep crisis.
• You may have heard of some of these models…There’ve been grand unified models, there’ve been super-symmetric models, super-string models, loop quantum gravity models… Well, nature turns out to be simpler than all of these models.
• If you ask most theorists working on particle physics, they’re in a state of confusion.
• The extensions of the standard model, like grand unified theories, they were supposed to simplify it. But in fact they made it more complicated. 
• The number of parameters in the standard model is about 18. The number in grand unified theories is typically 100. In super-symmetric theories, the minimum is 120. And as you may have heard, string theory seems to predict 10 to the power of 1,000 different possible laws of physics. It’s called the multiverse. 
• It’s the ultimate catastrophe: that theoretical physics has led to this crazy situation where the physicists are utterly confused and seem not to have any predictions at all.
• We have to get people to try to find the new principles that will explain the simplicity.

The crisis in modern physics resulting from the confusion of modern physicists originates from the  statistical mechanics of Boltzmann used by Planck in a desperate attempt to explain blackbody radiation as statistics of quanta, which led to the quantum mechanics of Bohr and Heisenberg based on atomistic roulettes without casusality and physical reality.  

But blackbody radiation can be explained without statistics in a classical model subject to finite precision computation as exposed on Computational Blackbody Radiation, which is simple and therefore possibly correct in the spirit of the above. 

Quantum Contradictions 20: Averaged Hartree Model between Scylla and Charybdis

The present sequence of posts Quantum Contradictions 1 - 20 with the key posts 6,  9 and 12  focussing on Helium (and two-electron ions) lead to a modified Hartree model for the ground state as a system of single electron wave functions defined by minimization of the total energy as the sum of (i) kernel potential energy, (i) inter-electron potential energy and (iii) kinetic energy, where the electrons keep individuality defined by individual presence in space as expressed by the single electron wave functions as concerns (i) and (ii), while the kinetic energy is computed after angular averaging as an expression of lack of individuality.

This model gives according to 12 a ground state energy of Helium of - 2.918 for spherical wave functions with polar decentration, to be compared with the observed - 2.903. Not bad.

In this model electrons thus keep individuality as concerns potential energies but lack individuality as concerns kinetic energy as a result of polar averaging.

Helium is thus described by two electronic wave functions, defined on 3-dimensional space, of the form:

• $\psi_1(r,\theta )^2 = (1 + \beta\cos(\theta ))\exp(-2\alpha r)\times\frac{\alpha^3}{\pi}$,
• $\psi_2(r,\theta )^2 = (1 -  \beta\cos(\theta ))\exp(-2\alpha r)\times\frac{\alpha^3}{\pi}$,

where $\alpha$ and $\beta$ are positive parameters determined by total energy minimization. This corresponds to a configuration with the two electrons being (more or less) separated, with electron 1 shifted towards the North pole of a spherical atom and electron 2 towards the South pole. The kinetic energy is computed after polar averaging or summation of $\psi_1$ and $\psi_2$.

We compare with the full wave function $\psi (r_1,\theta_1,\phi_1, r_2,\theta_2, \phi_2)$ satisfying Schrödinger's linear wave equation in 6 spatial dimension, where the electrons have lost all individuality as being indistinguishable and the wave function is given a statistical meaning. We have understood that this model is unphysical and should not be used.

The averaged Hartree model is a system in 3 dimensions and as such can be given a physical meaning without statistics, while the angular averaging removes the observed unphysical nature of the original Hartree model as a classical electron cloud (or Bohr) model of the atom.

The averaged Hartree model thus can be seen as a semi-classical physical model obtained by angular averaging in a classical model, instead of the full statistics of the full quantum model necessarily introducing non-physical aspects.

The averaged Hartree model steers between the Scylla of a classical model with full electronic individuality, which does not seem to describe the atomistic world, and the Charybdis of a full multidimensional quantum model with no electronic individuality, which is an unphysical model loaded with contradictions.

PS For the hydrogen ion H- with two electrons surrounding a +1 kernel, we obtain similarly a ground state energy of - 0.531 to be compared with observed - 0.528, and with - 0.500 for Hydrogen, thus indicating that H- is a stable configuration.

Quantum Contradictions 19: Summary

Here is a summary of contradictions of text book quantum mechanics:

1. The multidimensional wave function, as the solution of the multidimensional linear Schrödinger's equation as the basic mathematical model of quantum mechanics, is not a legitimate scientific concept according to Nobel Laureate Walter Kohn, because it cannot be solved analytically nor computationally. 
2. It follows that the linear multidimensional Schrödinger equation, which is an ad hoc model invented by Schrödinger and canonized in its Copenhagen Interpretation by Bohr, Born, Dirac and Heisenberg (and then abandoned by Schrödinger), should be removed from text books.
3. Doing so eliminates the need of inventing the microscopic roulettes of the Copenhagen Interpretation in order to give the multidimensional wave function at least some physical meaning, roulettes which violate basic physical principles of causality and reality and therefore were never accepted by Einstein and Schrödinger despite strong pressure from the physics community to confess to statistics.
4. With the linear multidimensional Schrödinger equation and its statistics put into the wardrobe of scientific horrors, focus can instead be put on developing non-linear three-dimensional deterministic equations describing the atomistic world formed by interaction of positive kernels and negative electrons such as Hartree and density functional models.  The challenge is then to e.g.  explain the (shell structure) of the periodic table from such a model.
5. A tentative such model will be described in final post of this series.       

Quantum Contradictions 18: Heisenberg

The text book canon of quantum mechanics was formed by Bohr and Heisenberg in the 1920s and was named the Copenhagen Interpretation (CI) by Heisenberg in the 1950s.

Let us seek the origin and motivation behind the Copenhagen Interpretation in Heisenberg's confessional treatise The Physicist's Concept of Nature. We find the following basic beliefs of Heisenberg:

• Even in the ancient atomic theory of Democritus and Leucippus it was assumed that large-scale processes were the results of many irregular processes on a small scale.
• Thus we always use concepts which describe behaviour on the large scale without in the least bothering about the individual processes that take place on the small scale.
• Now, if the processes which we can observe with our senses are thought to arise out of the inter-actions of many small individual processes, we must conclude that all natural laws may be considered to be only statistical laws. 
• Thus it is contended that while it is possible to look upon natural processes either as determinedby laws, or else as running their course without any order whatever, we cannot form any picture of processes obeying statistical laws.
• Planck, in his work on the theory of radiation, had originally encountered an element of uncertainty in radiation phenomena. He had shown that a radiating atom does not deliver up its energy continuously, but discreetly in bundles. This assumption of a discontinuous and pulse-like transfer of energy, like every other notion of atomic theory, leads us once more to the idea that the emission of radiation is a statistical phenomenon. 
• However, it took two and a half decades before it became clear that quantum theory actually forces us to formulate these laws precisely as statistical laws and to depart radically from determinism. 
• With the mathematical formulation of quantum-theoretical laws pure determinism had to be abandoned.

We understand that Heisenberg believed that the microscopic atomic world is a roulette world of non-deterministic processes for which we cannot form any pictures but we anyway have to believe obey statistical laws. 

But atomic roulettes require microscopics upon microscopics, since a roulette is not a simple pendulum but a complex mechanical device, which leads to reduction in absurdum and thus a logical deadlock. This was understood and voiced by Schrödinger and Einstein but Bohr and Heisenberg could scream louder and took the game despite shaky shady arguments. 

But a scientific deadlock is a deadlock and so a new direction away from the quagmire of microscopic statistics must be found.        

Academic Rights Watch om "Mejlaffären på KTH"

Academic Rights Watch följer och analyserar "mejlaffären på KTH":

• Kammarrätten i Stockholm har avslagit KTH-professor Claes Johnsons begäran att få se chefernas omfattande mejlkorrespondens i samband med att Johnsons medverkan i TV-programmet Gomorron Sverige på oklara grunder ställdes in. Även KTH:s rektor förekommer i korrespondens med rubriken ”Gomorron Sverige”. Domen är olycklig då den, om den står sig i Högsta förvaltningsdomstolen, underlättar för högskolor och andra myndigheter att upprätta en skuggförvaltning bortom offentlighetens insyn.
• Johnson har överklagat till Högsta förvaltningsdomstolen. Det vore ett svårt slag för den svenska demokratin om domen i Stockholm står sig. Men vi är tillräckligt luttrade för att inte ta någonting för givet när det gäller svenska rättsinstansers tillförlitlighet i ärenden där myndighetshierarkins bevarande står mot den enskilde individens rättigheter. Vi kan inte utesluta att detta ärende hör just till den kategorin.

Det känns bra att det finns åtminstone några inom akademin som värnar om det fria ordet. 

 

Quantum Contradictions 17: David Bohm

David Joseph Bohm (20 December 1917 – 27 October 1992) was an American theoretical physicist who contributed innovative and unorthodox ideas to quantum theory, philosophy of mind, and neuropsychology. He is widely considered to be one of the most significant theoretical physicists of the 20th century. (Wikipedia)

Bohm struggled throughout his professional life to make sense out of quantum mechanics based on a deeply felt doubt in the accepted Copenhagen interpretation, as expressed in the following dialogue with David Peat:

Peat: What would you say to the prevailing belief that the mathematical formalism itself expresses the very essence of our knowledge of nature? 

Bohm: Of course, some scientists, notably the Pythagoreans, held views like this in ancient times...But this notion that the mathematical formalism expresses the essence of our knowledge about nature did not become commonly accepted until relatively recent times. 

Peat: But how did this emphasis on mathematics come about?

Bohm: It was really because quantum theory, and to a lesser extent relativity theory, were never understood adequately in terms of physical concepts that physics gradually slipped into a practice of talking mostly about the equations....To some extent this began as early as the 1920s when the astronomer Sir James Jeans proposed that God must be mathematician. Heisenberg later gave it enormous boost with his idea that science could no longer visualize atomic reality in terms of physical concepts and that mathematics is the basic expression of our knowledge of reality....Now I don't agree with these developments. In fact, I feel that the current emphasis on mathematics has gone too far.

Yes, it has certainly gone too far when what is left in physics is a mathematical equation such as Schrödinger's equation, which is not viewed to describe physical reality and thus cannot be motivated from physical principles and in addition cannot be solved neither analytically nor computationally, except in a few simple cases.

Quantum Contradictions 16: An Inconvenient Truth (about Schrödinger's Equation)

Roothaan and Weiss states in Correlated Orbitals for the Ground State of Heliumlike Systems 

(Rev. Mod. Phys. 32, 194–205 (1960)):

• For the ground, state of atoms, molecules, and solids, the explicit solution of the Schrodinger equation to obtain an accurate wave function is virtually a hopeless task. Instead, one usually chooses a convenient function with adjustable parameters, and determines the best wave function within this class by minimizing the expectation value of the energy, computed from that wave function, with respect to these parameters. The form of the function chosen represents a certain restriction, and it is essential to use physical arguments in choosing such a form; in doing so, we essentially choose a model. The success of such a wave function can, to a certain extent, be judged by comparing the calculated energy with the experimental total energy.

This statement delivers an inconvenient truth about Schrödinger's equation as the basis of quantum mechanics: The equation can be solved exactly only for Hydrogen with one electron and computational solution for many electrons is beyond the capacity of any thinkable computer. Already Helium with two electrons presents mysteries since the spherically symmetric 1s2 state presented as the ground state in text books, does not have the experimentally observed energy and the true ground state is hidden to the reader. 

Physicists thus take for granted that solutions of Schrödinger's equation describe the atomistic world, but this hypothesis is impossible to test against observation since solutions cannot be determined, except in a few elementary cases. The second pillar of modern physics, Einstein's equations of general relativity, shares the same quality. 

This is an inconvenient truth about modern physics which is not told in text books. No wonder that modern physicists no longer speak about quantum mechanics and general relativity but instead about string theory and multiverse for which no equations at all are presented.          

Quantum Contradictions 15: Electron Orbitals Cannot be Observed

                                                                Fiction or reality?

Quantum mechanics is supposed to explain the electronic shell structure of the periodic table based on solutions of the Schrödinger equation named wave functions. Eric Scerri questions this narration on the ground that the Schrödinger equation of an atom with more than one electron cannot be solved exactly and for several electrons not even computationally because of the high dimensionality of the Schrödinger equation with 3N spatial dimensions for an atom with N electrons, which is difficult already for Helium with N = 2.

Various drastic reductions of the dimensionality of the Schrödinger equation are thus being made in practice such as Hartree-Fock methods based on combinations of hydrogenic electron orbitals as exact solutions to the solvable Schrödinger equation for Hydrogen, and density functional methods, but the relation between these brute ad hoc approximate solutions and solutions to the full Schrödinger equation is unknown and so the claim that the periodic table comes out the Schrödinger equation is not well founded.

In any case computation of approximate solutions of the Schrödinger takes most of the capacity in scientific computing, and so electron orbitals and electron densities are produced in massive numbers. Eric Scerri asks Have Electron Orbitals Really Been Observed? and answers: No: only electron densities have been (can be) experimentally observed.

Direct experimental support of the validity of Schrödinger equation as a model of atomistic physics thus appears (is) impossible. Even worse, it seems (is) impossible to compare predictions by the Schrödinger equation to experimental observation because solutions cannot be computed.  Questions are inevitable:

• What is the scientific reason to believe that the Schrödinger equation describes the atomistic world?
• What is the scientific reason to believe that electrons interact precisely through the Coulomb potential of the Schrödinger equation?   

Quantum Contradictions 14: Stability of Electron Configurations vs Spin?

The stability of electron configurations is rarely studied in quantum mechanics, since the focus is put on eigenvalues and energy levels and not on often multiple eigenfunctions.

In particular, the different stability features of attractive and repulsive Coulomb forces does not seem to come into play.

In general, with experience from fluid dynamics, one may say that attraction is connected to acceleration and as such is stable, while repulsion is connected to retardation and so is unstable.

An illustration is given by a pendulum in usual hanging position (attraction) and inverted upright position (repulsion), as illustrated in the above picture. We see converging  attractive forces (in red) balancing gravitation (green) giving a stable hanging position and diverging repulsive forces giving an unstable inverted upright position.

It is natural to connect to the Helium atom with two electrons in opposition by repulsion as studied in this sequence of posts, which may be unstable by the above mechanics and thus allow a variety of non-symmetric configurations with one of the electrons dominating, which possibly could be the origin of the mysterious concept of electron spin.