A Three-dimensional Multi-Electron Wave Function

Consider a wavefunction $\psi$  for an atom with $N$ electrons as a sum of $N$ functions $\psi_1(x)$,…,$\psi_N(x)$, all depending on a common three-dimensional space coordinate $x$ (plus time):

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

with associated energy as the sum of kinetic energy, attractive kernel potential energy and repulsive interelectron energy:

• $E(\psi )= \frac{1}{2}\int\vert\nabla\psi\vert^2dx - \int\frac{N\psi^2}{\vert x\vert}dx+\sum_{j\neq k}\int\int\frac{\psi_j^2(x)\psi_k^2(y)}{2\vert x-y\vert}dxdy$,

under the normalization

• $\int\psi_j^2dx =1$ for $j=1,...,N$,

where $\psi_j(x)$ represents the distribution of electron $j$.

The ground state is determined as the state of minimal energy determined as the solution of a non-linear system of equations in three space dimensions expressing minimality.  We see that minimization favors atomistic wavefunctions $\psi (x)=\sum_j\psi_j(x)$ built from electronic wave functions $\psi_j$ with disjoint supports, which makes the interelectronic repulsion energy small without cost of kinetic energy.

The ground state of Helium thus will have its two electrons separated into two half-spheres with corresponding wave functions $\psi_1(x)$ and $\psi_2(x)$ meeting smoothly at a common separation surface. It is possible that this is the origin of the Zweideutigkeit or two-valuedness expressed in Pauli's exclusion principle, which Pauli did not like because it was ad hoc without rationale.

The  sequence of posts on Quantum Contradictions explores atomic ground states based on the above wave function with surprisingly good correspondence with observations, see also Many-Minds Quantum Mechanics.

We compare with standard quantum mechanics with multi-dimensional wave functions $\psi (x_1,…,x_N)$ depending on $N$ three-dimensional space coordinates $x_1$,…,$x_N$, typically in the form of a Slater determinant as a linear combination of products of $N$ functions $\psi_1$,…,$\psi_N$, each function separately depending on three space coordinates,  thus based on wavefunctions depending on altogether $3N$ space coordinates. Such multi-dimensional wave functions defy direct physical interpretation and are also impossible to compute for atoms with several electrons and thus do not belong to science. Yet they are supposed to be fundamental to atomistic physics.

The standard view is that macroscopic and microscopic (atomistic) physics are fundamentally different,  because microscopic physics demands a multi-dimensional wave function, while macroscopic physics is described by systems of three-dimensional functions. If also microscopic physics can be described by systems of three-dimensional functions, as indictated above, then there will be no fundamental difference between macroscopic and microscopic physics and a major obstacle for progress can be eliminated.

Computations based on wavefunctions of the above form are under way and will presented when available.  For simple hand calculations see here and here.

PS1 For Helium with two electrons at distance $\frac{1}{2}$ from the kernel and mutual distance $1$ as an approximate ground state configuration energy in the above model, we get $E = -3$, to be compared with the observed $-2.903$.

For Lithium with two electrons at distance $\frac{1}{3}$ from the kernel and mutual distance $\frac{2}{3}$ together with a third electron at distance 1 from an effective kernel of charge +1, we get $E = -8$, to be compared with the observed $-7.5$. The ground state energy of three electrons at distance $\frac{1}{3}$ from the kernel and mutual distance $\frac{1}{2}$, we get $E = -7.5$ indicating that the configuration with two electrons in an inner shell and one in an outer shell has smaller energy and thus is the actual ground state configuration for Lithium, thus obtained without reference to Pauli's exclusion principle.

PS2 Recall that the standard quantum mechanics is formulated in terms of a multi-dimensional wave function $\psi (x_1,x_2,…,x_N)$ depending on $N$ three-dimensional space coordinates $x_1$,…$x_N$, altogheter on $3N$ space coordinates, which is devastating because both physical interpretation and computational determination is impossible. To reduce the dimensionality typically an Ansatz is made as Slater determinants of three-dimensional wave functions $\psi_i$ as linear combinations of products of the form (subject to permutations of the coordinates):

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

leading to a set of one-electron wave equations coupled by complex exchange-correlation terms which are very difficult to determine. The above Ansatz with a sum instead of products of three-dimensional wave functions may offer more computationally managable and thus more useful models.

PS3 For Beryllium with 4 electrons, we get $E=-14$ from 2 electrons at distance $\frac{1}{4}$ from the kernel with mutual distance $\frac{1}{2}$, together with $E = -\frac{2}{3}$ from 2 electrons of width $\frac{1}{2}$ at distance $\frac{1}{4} + \frac{1}{2}$ from an effective charge of +2, which gives altogether $E = -14.667 which is exactly what is observed!!

PS4 For N electrons distributed over one shell at distance $\frac{1}{N}$ to the kernel assuming the average distance between any pair of electrons is $\frac{1}{N}$, we get $E = -\frac{N^2}{2}$, which is much larger than the observed $E \approx - N^2$ and thus is not the ground state configuration.  A multi-shell distribution in the model gives better agreement with observations and so the model may capture the real shell structure (without resort to any Pauli exclusion principle).

PS5 Note that the above model allows discontinuous electron distributions (joining smoothy) without cost of kinetic energy which favors electron separation. We compare with Hartree models as systems of one-electron models with continuous electron distributions for which separation requires kinetic energy cost and a resort to Pauli's exclusion principle is necessary to prevent more than two electron distributions to overlay.

PS6 To find the ground state, we can use time-stepping of the parabolic system

• $\frac{\partial\psi_j(x,t)}{\partial t} = \Delta\psi (x,t) + \frac{N\psi (x,t)}{\vert x\vert}-\sum_{k\neq j}\int\frac{\psi_k^2(y,t)}{2\vert x-y\vert}dy\,\psi_j(t,x)$ for $t > 0$, $j=1,…,N$,

with successive normalization to $\int\psi_j^2(x,t)\, dx=1$ after each time step and $\psi =\sum_{k=1}^N\psi_k$.  Further

• $V_k\equiv\int\frac{\psi_k^2(y,t)}{2\vert x-y\vert}dy$,

can be computed by solving $-\Delta V_k = 2\pi\psi_k^2$.

      

Lennart Bengtsson, IPCC Alarmism and The Royal Swedish Academy

The Global Warming Policy Foundation (GWPF) announces in a press release:

• Professor Lennart Bengtsson, one of Sweden’s leading climate scientists, has joined the GWPF’s Academic Advisory Council.

Judith Curry comments:

• The whole concept behind IPCC is basically wrong. (Lennart Bengtsson)
• I regard this as a very interesting and significant event, that runs counter to the near universal trend in academic circles to attempt to ignore and marginalize organizations and individuals that are skeptical of UNFCCC/IPCC global warming science, impacts, and/or solutions.

LB has thus withdrawn his earlier support of IPCC climate alarmism as expressed in the Statement on the Scientific Basis of Climate Change by The Royal Swedish Academy of Sciences authored by LB. 

LB has thus turned into a skeptic of climate alarmism led by his conscience as scientist. By the same logic LB now has to withdraw also his Statement in the name of The Royal Academy in support of IPCC.  

Let us hope that LB has the courage to do so. When that is done current Swedish climate politics will have no scientific basis and will collapse, the sooner the better.

PS LB now uses Der Spiegel to propagate his message as converted skeptic.

What is the Difference between Macroscopic and Microscopic Physics?

Multiscale modeling with simple ordered microscale (atom) and ordered macroscale (car) connected by complex intermediate scales.

The standard view is that classical physics is useful to describe the macroscopic world, like the flow of air around an airplane or the formation of a galaxy,  while the microscopic world of atoms and molecules requires a fundamentally different form of physics named quantum mechanics.

The standard view is thus that the world is divided into a macroscopic world and a microscopic world supposedly governed by different physics. The standard view is that we as human beings with experience from the macroscopic world cannot fathom the microscopic world because it is so fundamentally different from the macroscopic world.

But is it really reasonable from a scientific point of view to divide the world this way? What is the dividing line? How does the microscopic world interact with the macroscopic world?
Is it possible to tell if a mathematical model in its typical form of a differential equation describes microscopic or macroscopic physics?

None of these questions has a good answer and it is then natural to seek the origin of the idea that microscopics is so different from macroscopics. The standard wisdom says that microscopics is described by quantum mechanics and macroscopics by classical mechanics and quantum mechanics is fundamentally different from classical mechanics and therefore macroscopics is fundamentally different from microscopics.

The standard wisdom according to the Copenhagen interpretation of quantum mechanics is that the microscopic world is a strange world without causality and predictability functioning like a microscopic game of roulette. This strange idea comes from the insistence to describe microscopics by a multi-dimensional linear Schrödinger equation, which because of its many dimensions can only be given a probabilistic interpretation and not a physical realistic interpretation.

The multi-d linear Schrödinger equation is an ad hoc model which is not derived from basic principles and thus is accepted as a true mystery beyond comprehension of human minds and as such necessarily a correct description of microscopics.

But suppose, we do not take the incomprehensible (and uncomputable) linear multi-d Schrödinger equation as description of microscopics, because this lacks rationale. Suppose we seek instead a description in the form of field equations in three space dimensions plus time, in the form of Hartree models as non-linear coupled systems of one-particle Schrödinger equations, which have the same principal form as macroscopic continuum models.

Then there will be no fundamental difference between microscopics and macroscopics and all the problems arising from postulating such a difference will disappear. This must be a good case for Ockham's razor.

Note that claiming that microscopics functions like a game of roulette is contradictory, because a game of roulette requires microscopics, which leads to an infinite regression of microscopics upon microscopics. When I present this argument, which makes perfect sense to a classical physicist, in a discussion with a modern physicist, I get no response as if this argument is beyond what is allowed in modern physics. Is this reasonable?    

The Principal Difference Between Laws and Principles of Physics

There are laws of physics and there are principles of physics. Examples of principles are:

• Principle of Relativity: Physical laws take the same form in all systems of reference (Einstein).
• Principle of Special Relativity: The speed of light is the same for all observers (Einstein).
• Principle of Equivalence:  Inertial and gravitational mass are equal (Einstein).
• Pauli's Exclusion Principle: No two particles with the same quantum numbers can be at the same position in space and time.
• Principle of CPT Symmetry: Physical laws are invariant under charge conjugation, parity transformation and time reversal.

Examples of laws are:

• Newton's 2nd Law: $F = am$.
• Hooke's Law: $\sigma = E\epsilon$. 
• Gauss' Law: $\nabla E =\rho$.
• Faraday's Law: $\frac{\partial B}{\partial t}+\nabla\times E =0$.
• Ampere's Law: $\frac{\partial E}{\partial t}-\nabla\times B =J$.

To seek to identify the difference between laws and principles, if any, recall how Einstein introduces the Special Theory of Relativity in his 1905 annus mirabilis article On the Electrodynamics of Moving Bodies: 

• The same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.
• We will raise this conjecture (the purport of which will hereafter be called the “Principle of Relativity”) to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. 

We see here how Einstein introduces principle as synonomous to postulate as stipulation or regulation or rule or dictate telling physicists how to view certain aspects of physics.

On the other hand, a physical law expresses a quantitative relation between different physical entities like the stress-strain relation of Hooke's law. 

We conclude:

• Principles belong to modern physics.
• Laws belong to classical physics (mechanics and electromagnetics). 
• Laws are expressed in quantitative mathematical formulas. 
• Principles are expressed in words.
• Laws are statements about physical reality (which may or may not be true).
• Principles express stipulations which are to be respected by physicists, which are not statements about physical reality which can be true or false, but definitions which are true by construction. 
• Principles are man-made and need no justification.
• Laws relate to physical reality and require justification.
• A principle should be viewed with critical suspicion.
• A law may be viewed with admiration. 

One may wonder why physics as principle as judiciary law as regulation as dictate, has become the trademark of modern physics, when modern society is democracy and not dictatorship. 

Quantum Mechanics as Gift from God More Intelligent than Man

The Conclusion after 100 pages of Do We Really Understand Quantum Mechanics? Strange Correlations, Paradoxes and Theorems by F. Laloe, is:

• Quantum mechanics is, with relativity, the essence of the big conceptual revolution of the physics of the 20th century. 
• Now, do we really understand quantum mechanics? 
• It is probably safe to say that we understand its machinery pretty well; in other words, we know how to use its formalism to make predictions in an extremely large number of situations, even in cases that may be very intricate. 
• Heinrich Hertz, who played such a crucial role in the understanding of electromagnetic waves in the 19th century (Hertzian waves), remarked that, sometimes, the equations in physics are “more intelligent than the person who invented them” [182]. 
• The remark certainly applies to the equations of quantum mechanics, in particular to the Schrödinger equation, or to the superposition principle: they contain probably much more substance that any of their inventors thought, for instance in terms of unexpected types of correlations, entanglement, etc. 
• It is astonishing to see that, in all known cases, the equations have always predicted exactly the correct results, even when they looked completely counter-intuitive. 
• Conceptually, the situation is less clear. 
• Nevertheless, among all intellectual constructions of the human mind, quantum mechanics may be the most successful of all theories since, despite all efforts of physicists to find its limits of validity (as they do for all physical theories), and many sorts of speculation, no one for the moment has yet been able to obtain clear evidence that they even exist. Future will tell us if this is the case; surprises are always possible!

Laloe illuminates the fact that modern physicists (and nobody else) do not understand the modern physics of quantum mechanics, and do not even pretend to do so,  as a conceptual revolution away from classical physics based on understanding. The argument is that the linear Schrödinger equation must be more intelligent than Schrödinger, since Schrödinger admittedly could not understand it and nobody else has ever claimed to understand it either. 

If the difference between science and religion is that science is all about understanding, while religion leaves understanding to divinity, modern physics appears to be more religion than science.

But it is hard to understand that an equation that cannot be solved, always predicts exactly the correct results! It is more easy to believe that any observation made can be claimed to fit exactly with the equation, since checking is impossible. It would be more convincing if observation was somewhat different from theory.

No, We Don't Understand Quantum Mechanics, But There Is Hope.

                                                            Yes, QM is a strange world.

The Preface of book Do We Really Understand Quantum Mechanics by Franck Laloe supplemented by an article with the same title, tells the truth about quantum mechanics:

• In many ways, quantum mechanics QM is a surprising theory... because it creates a big contrast between its triumphs and difficulties. 
• On the one hand, among all theories, quantum mechanics is probably one of the most successful achievements of science.  The applications of quantum mechanics are everywhere in our twentyfirst century environment, with all sorts of devices that would have been unthinkable 50 years ago. 
• On the other hand, conceptually this theory remains relatively fragile because of its delicate interpretation – fortunately, this fragility has little consequence for  its efficiency. 
• The reason why difficulties persist is certainly not that physicists have tried to ignore them or put them under the rug!
• Actually, a large number of interpretations have been proposed over the decades, involving various methods and mathematical techniques. 
• We have a rare situation in the history of sciences: consensus exists concerning a systematic approach to physical phenomena, involving calculation methods having an extraordinary predictive power; nevertheless, almost a century after the introduction of these methods, the same consensus is far from being reached concerning the interpretation of the theory and its
  foundations. 
• This is reminiscent of the colossus with feet of clay.
• The difficulties of quantum mechanics originate from the object it uses to describe physical systems, the state vector (wave function) $\Psi$.
• Without any doubt, the state vector is a curious object to describe reality!

The message is that QM a formidable achievement of the human intellect which is incredibly useful in practice, but like a colossus with feet of clay has a main character flaw, namely that it is a curious way to describe reality and as such not understood by physicists. 

There are two ways the handle if a physical theory is not understood because it is so curious, either the theory is dismisssed as being seriously flawed or the curiosity is chosen as a sign that the theory is correct and beyond questioning by human minds.

The reason QM is so mysterious is that the wave function $\Psi =\Psi (x_1,x_2,…,x_N)$ for an atom or molecule with $N$ electrons depends on $N$ independent three-dimensional space variables $x_1$, $x_2$,…, $x_N$, together with time, thus is a function in $3N$ space dimensions plus time and as such has no direct real physical meaning since real physics takes place in $3$ space dimensions. 

The wave function $\Psi$ is introduced as a the solution to a linear multi-dimensional linear wave equation named Schrödinger's equation of the form

• $i\frac{\partial\Psi}{\partial t}+H\Psi = 0$,

where $H$ is a Hamiltonian operator acting on wave functions. The mysticism of QM thus originates from Schrödinger's equation and is manifested by the fact that there is no real derivation of Schrödinger's equation from basic physical laws. Instead, Schrödinger's equation is motivated as a purely formal manipulation of classical Hamiltonian mechanics without physical meaning. 

The main trouble with QM based on a linear multi-d Schrödinger equation is thus the physical interpretation of the multi-d wave function and the accepted answer to this enigma is to view 

• $\vert\Psi (x_1,…,x_N)\vert^2$ 

as a probability distribution of a particle configuration described by the coordinates $(x_1,…,x_N)$ representing human knowledge about a physics and not physics itself. Epistemology of what we can know is thus allowed to replace ontology of what is.

The linear multi-d Schrödinger equation thus lacks connection to physical reality. Moreover, because of its many dimensions the equation cannot be solved (analytically or computationally), and the beautiful net result is that QM is based on an equation without physical meaning which cannot be solved. No wonder that physicists still after 100 years of hard struggle do not really understand QM. 

But since Schrödinger's linear multi-d equation lacks physical meaning (and neither can be solved) there is no compelling reason to view it as the foundation of atomistic physics. 

It appears to be more constructive to consider instead systems of non-linear Schrödinger equations in $N$ three-dimensional wave functions $\psi_1(x),…,\psi_N(x)$ with $x$ a 3d space coordinate,  in the spirit of of Hartree models, as physically meaningful computable models of potentially great practical usefulness. 

Sums of such wave functions then play a basic role and have physical meaning, to be compared the standard setting with $\Psi (x_1,…,x_N)$ in the form of Slater determinants as sums of muli-d products $\psi (x_1)\psi (x_2)…\psi (x_N)$ of complicated unphysical nature. 

    

Omodern Matematikundervisning Utan Ansvariga Matematiker

Matematikinstitutionerna vid KTH och Chalmers skickar varje år en ny larmrapport om ytterligare försämrade matematikkunskaper hos nyantagna teknologer och nu var det dags igen:

• Kraftigt ras i högskolestudenterns mattekunskaper.

Med larmrapporten friskriver sig högskolematematikerna från sitt ansvar att se till att landets matematikutbildning är modern och funktionell, genom att skylla på skolmatematiken:  

• De högskolelärare som SvD pratar med är överens om att studenternas svaga grundkunskaper gjort att utbildningsnivån vid högskolorna sänkts undan för undan.
• Visst har vi anpassat nivån, men det är inget folk vill tala högt om. 
• För svag matteundervisning i grundskola och gymnasium, i kombination med en för generös betygsättning, ligger bakom problemen.

Men skolmatematiken är en (förenklad) variant av högskolematematiken och anledningen att skolmatematiken inte längre fungerar är att högskolematematiken är omodern och inte motsvarar datorsamhällets nya möjligheter och behov.  

När jag försöker få högskolematematikerna att bära sitt ansvar och modernisera utbildningen möts jag av oförstående och uppgivenhet och mitt öppna brev till Svenska Matematikersamfundet och Nationalkommitten för Matematik leder ingenstans. Se också mitt inlägg i kommande maj-nummer av SMS-Bulletinen.

Extremism of Modern Physics as Bluff Poker Physics

Modern physics has been driven into an increasingly extremist position with focus on extremely small or large spatial or temporal scales or extremely large energies. When problems were met on a certain (extreme) scale, the study was directed to yet more extreme scales and energies, as in a steadily increasing bet in a game of poker with little on hand to never get called. When LHC does not deliver, then the bet is raised to a new bigger more powerful LHC...

When Einstein was pressed about the meaning of his special theory of relativity, he increased the bet to general relativity and when pressed about the meaning of general relativity he jumped the bet to cosmology...

When physicists after the introduction of quantum mechanics faced questions about the electronic structure of atoms and molecules, they turned to the three orders of magnitude smaller proton and neutron forming atomic kernels, and then to the quarks forming the proton and neutron and then ultimately to string theory on scales 15 orders of magnitude smaller than the proton in an ulitmate attempt to find the origin of gravitation acting on cosmological scales. In each case the problems met on one scale were met by resort to smaller or larger scales, steadily increasing the bet and preventing a call.

Today cosmology is directed to multiversa and inflation after Big Bang as the next step after Einstein's cosmology of general relativity supposedly all originating from string theory.  But this may be the last possible bet and a call is approaching anticipated as a crisis in physics.

Crisis in Physics vs Computational Physics

The May14 issue of Scientific American asks the following questions:

• Does Physics Have a Problem?
• Crisis in Physics?

These questions naturally present themselves because modern theoretical physicists have driven themselves to search for the truth on scales which are either too small (string theory) or too big (cosmology) to be assessed experimentally. But theory without experiment may well be empty theory and that may be the meaning of the crisis. Of course, advocates of string theory like Lubos, forcefully denies that there is a crisis in physics. But there are other blog voices…and leading physicists show little hope..

But modern physicists have a new tool to use and that is computational physics, which offers an experimental laboratory without the scale limits of a physical laboratory. 

Computational physics needs computable models, but both quantum mechanics and general relativity are based on models which are not computable, and so there is a lot of work to be done. The question is if modern theoretical physicists have the right training to do this work.     

Wanted: Constructive Physics

                                     Wanted: Constructive version of Schrödinger's equation!

The book Constructive Physics by Y.I. Oshigov has an important message:

• Only in the rebuilding of the gigantic construction of the modern physics in the constructive manner can open doors to the understanding of the complex processes in the sense of exact sciences.
• The modern situation in physics looks like a crisis, and the genealogy of this crisis is the same as for the crisis in mathematics in the first third of the 20th century: this is the crisis in the axiomatic method.
• Today we possess the more exact kit of instruments of the constructive mathematics: algorithms must replace formulas.
• (The multidimensional wave function) harbors serious defects….it does not allow the computation of such functions already for a small number of particles, for example 10, let alone for the more complex systems.
• This complexity barrier is principal. We should not think then that the quantum theory for many bodies gives such reliable answers to questions as it was the case in one particle case.

In short, quantum mechanics based on Schrödinger's equation for a wave function in $3N$ space dimensions for $N$ particles (electrons or kernels) must be given a new constructive form. A real challenge! My answer is given as Many-Minds Quantum Mechanics.