New Quantum Mechanics 21: Micro as Macro

The new quantum mechanics as realQM explored in this sequence of posts offers a model for the microscopic physics of atoms which is of the same form as the classical continuum mechanical models of macroscopic physics such as Maxwell's equations for electro-magnetics, Navier's equations for solid mechanics and Navier-Stokes equations for fluid mechanics in terms of deterministic field variables depending on a common 3d space coordinate and time.

realQM thus describes an atom with $N$ electrons realQM as a nonlinear system of partial differential equations in $N$ electronic wave functions depending on a common 3d space coordinate and time.

On the other hand, the standard model of quantum mechanics, referred to as stdQM, is Schrödinger's equation as a linear partial differential equation for a probabilistic wave function in $3N$ spatial coordinates and time for an atom with $N$ electrons.  

With realQM the mathematical models for macroscopic and microscopic physics thus have the same form and the understanding of physics can then take the same form. Microphysics can then be understood to the same extent as macrophysics.

On the other hand, the understanding of microphysics according to stdQM is viewed to be fundamentally different from that of macroscopic physics, which effectively means that stdQM is not understood at all, as acknowledged by all prominent physicists.

As an example of the confusion on difference, consider what is commonly viewed to be a basic property of stdQM, namely that there is limit to the accuracy that both position and velocity can be determined on atomic scales, as expressed in Heisenberg's Uncertainty Principle (HUP).

This feature of stdQM is compared with the situation in macroscopic physics, where the claim is that both position and velocity can be determined to arbitrary precision, thus making the case that microphysics and microphysics are fundamentally different.

But the position of a macroscopic body cannot be precisely determined by one point coordinate, since  a macroscopic body is extended in space and thus occupies many points in space.  No one single point determines the position of and extended body. There is thus also a Macroscopic Uncertainty Principle (MUP).

The argument is then that if the macroscopic body is a pointlike particle,  then both its position and velocity can have precise values and thus there is no MUP. But a pointlike body is not a macroscopic body and so the argument lacks logic.

The idea supported by stdQM that the microscopic world is so fundamentally different from the macroscopic world that it can never be understood, thus may well lack logic. If so that could open to understanding of microscopic physics for human beings with experience from macroscopic physics.

If you think that there is little need of making sense of stdQM, recall Feynman's testimony:

• We have always had a great deal of difficulty understanding the world view that quantum mechanics represents. At least I do, because I’m an old enough man that I haven’t got to the point that this stuff is obvious to me. Okay, I still get nervous with it ... You know how it always is: every new idea, it takes a generation or two until it becomes obvious that there’s no real problem. I cannot define the real problem, therefore I suspect that there is no real problem, but I’m not sure there’s no real problem. (Int. J. Theoret. Phys. 21, 471 (1982).) 

It is total confusion, if it is totally unclear if there is a problem or no problem and it is totally clear that nobody understands stdQM....

Recall that stdQM is based on a linear multi-dimensional Schrödinger equation, which is simply picked from the sky using black magic ad hoc formalism, which could be anything, and is then taken as a revelation about real physics when interpreted by reversing the black magics. 

This is like scribbling down a sign/equation at random without intentional meaning, and then giving the sign/equation an interpretation as if it had an original meaning, which may well be meaningless, instead of expressing a meaning in a sign/equation to discover consequences and deeper meaning.   

New Quantum Mechanics 20: Shell Structure

Further computational exploration of realQM supports the following electronic shell structure of an atom:

Electrons are partitioned into an increasing sequence of main spherical shells $S_1$, $S_2$,..,$S_M$ with each main shell $S_m$ subdivided into two half-spherical shells each of which for $m>2$ is divided into two angular directions into $m\times m$ electron domains thus with a total of $2m^2$ electrons in each full shell $S_m$.  The case $m=2$ is special with the main shell divided radially into two subshells which are each divided into half-spherical subshells each of which is finally divided azimuthally, into $2\times 2$ electron domains for $S_2$ subshell, thus with a total of $2m^2$ electrons in each main shell $S_m$ when fully filled, for $m=1,...,M$, see figs below.

This gives the familiar sequence 2, 8, 18, 32,.. as the number of electrons in each main shell.

4 subshell of S_2

8 shell as variant of full S_2 shell

 9=3x3 halfshell of S_3

The electron structure can thus be described as follows with parenthesis around main shells and radial subshell partition within parenthesis:

• (2)+(4+4)
• (2)+(4+4)+(2)
• ...
• (2)+(4+4)+(4+4) 
• (2)+(4+4)+(8)+(2)
• ....
• (2)+(4+4)+(18)+(2)
• ...
• (2)+(4+4)+(18)+(8)

Below we show computed ground state energies assuming full spherical symmetry with a radial resolution of 1000 mesh points, where the electrons in each subshell are homogenised azimuthally, with the electron subshell structure indicated and table values in parenthesis. Notice that the 8 main shell structure is repeated so that in particular Argon with 18 electrons has the form 2+(4+4)+(4+4):

Lithium (2)+1: -7.55 (-7.48)                  1st ionisation:      (0.2)
Beryllium (2)+(2): -15.14 (-14.57)           1st ionisation: 0.5 (0.35)
Boron (2)+(2+1): -25.3 (-24.53)               1st ionisation: 0.2 (0.3)
Carbon (2)+(2+2): -38.2  (-37.7)               1st ionisation 0.5 (0.4)
Nitrogen (2)+(3+2):  -55.3 (-54.4)            1st ionisation  0.5  (0.5)
Oxygen (2)+(3+3): -75.5 (-74.8)               1st ionisation  0.5  (0.5)
Fluorine (2)+(3+4):  -99.9   (-99.5)            1st ionisation  0.5      (0.65)
Neon (2)+(4+4):   -132.4     (-128.5  )        1st ionisation 0.6        (0.8)
Sodium (2)+(4+4)+(1): -165 (-162)
Magnesium (2)+(4+4)+(2): -202  (-200)
Aluminium (2)+(4+4)+(2+1): -244 (-243)
Silicon (2)+(4+4)+(2+2): -291 (-290)
Phosphorus (2)+(4+4)+(3+2): -340 (-340)
Sulphur (2)+(4+4)+(4+2): -397 (-399)
Chlorine (2)+(4+4)+(3+4): -457 (-461)
Argon: (2)+(4+4)+(4+4): -523 (-526)
Calcium: (2)+(4+4)+(8)+(2): -670 (-680)
Titanium: (2)+(4+4)+(10)+(2): -848 (-853)
Chromium: (2)+(4+4)+(12)+(2): -1039 (-1050)
Iron: (2)+(4+4)+(14)+(2): -1260 (-1272)
Nickel: (2)+(4+4)+(16)+(2): -1516 (-1520)
Zinc: (2)+(4+4)+(18)+(2): -1773 (-1795)
Germanium: (2)+(4+4)+(18)+(2+2): -2089 (-2097)
Selenium: (2)+(4+4)+(18)+(4+2):- 2416 (-2428)
Krypton: (2)+(4+4)+(18)+(4+4): -2766 (-2788)
Xenon: (2)+(4+4)+(18)+(18)+(4+4): -7355  (-7438)
Radon: (2)+(4+4)+(18)+(32)+(18)+(4+4): -22800 (-23560)

We see good agreement even with the crude approximation of azimuthal homogenisation used in the computations.

To see the effect of the subshell structure we compare Neon: (2)+(4+4) with Neon: (2)+(8) without the (4+4) subshell structure, which has a ground state energy of -153, which is much smaller than the observed -128.5.  We conclude that somehow the (4+4) subdivision of the second is preferred before a subdivision without subshells. The difference between (8) and (4+4) is the homogeneous Neumann condition acting between subshells, tending to increase the width of the shell and thus increase the energy.

The deeper reason for this preference remains to describe, but the intuition suggests that it relates to the shape or size of the domain occupied by an electron.  With subshells electron domains are obtained by subdivision in both radial and azimuthal direction, while without subshells there is only azimuthal/angular subdivision of each shell.

We observe that ionisation energies, which are of similar size in different shells, become increasingly small as compared to ground state energies, and thus are delicate to compute as the difference between the ground state energies of atom and ion.

Here are sample outputs for Boron and Magnesium as functions of distance $r$ from the kernel along the horizontal axis :

We observe that the red curve depicting shell charge $\psi^2(r)r^2dr$ per shell radius increment $dr$, is roughly constant in radius $r$, as a possible emergent design principle. More precisely, $\psi (r)\sim \sqrt{Z}/r$ mathches with $d_m\sim m^2/Z$ and $r_m\sim m^3/Z$ with $d_m$ the width of shell $S_m$ and thus the width of the subshells of $S_m$ scaling with $m/Z$, and thus the width of electrons in $S_m$ scaling with $m/Z$.

We thus have $\sum_mm^2\sim M^3\sim Z$ and with $d_m\sim m^2/Z$ the atomic radius $\sum_md_m\sim M^3/Z\sim 1$ is basically the same for all atoms, in accordance with observation.

Further, the kernel potential energy and thus the total energy in $S_m$ scales with $Z^2/m$ and the total energy by summation over shells scales with $\log(M)Z^2\sim \log(Z)Z^2$, in close correspondence with $Z^{\frac{1}{3}}Z^2$ by density functional theory.

Recall that the electron configuration of stdQM is based on the eigen-functions for Schrödinger's equation for the Hydrogen atom with one electron, while as we have seen that of realQM rather relates to spatial partitioning. Of course, eigen-functions express some form of partitioning, and so there is a connection, but the basic problem may concern partitioning of many electrons rather than eigen-functions for one electron.



   

Quantum Mechanics as Theory Still Without Meaning

Yet another poll (with earlier polls in references) shows that physicists still today after 100 years of deep thinking and fierce debate show little agreement about the stature of quantum mechanics as the prime scientific advancement of modern physics.

The different polls indicate that less than 50% of all physicists today adhere to the Copenhagen Interpretation, as the main text book interpretation of quantum mechanics. This means that quantum mechanics today after 100 years of fruitless search for a common interpretation, remains a mystery without meaning. Theory without interpretation has no meaning and science without meaning cannot be real science.

If only 50% of physicists would agree on the meaning of the basic text book theories of classical physics embodied in Newton/Lagranges equations of motion, Navier's equation for solid mechanics, Navier-Stokes equations for fluid dynamics and Maxwell's equations for electromagnetic, that would signify a total collapse of classical physics as science and subject of academic study.

But this not so: classical physics is the role model of science because there is virtually no disagreement on the formulation and meaning of these basic equations.

But the polls show that there is no agreement on the role and meaning of Schrödinger's equation as the basis of quantum mechanics, and physicists do not seem to believe this will ever change. This is far from satisfactory from scientific point of view.

This is my motivation to search for a meaningful quantum mechanics in the form of realQM presented in recent posts. Of course you may say that for many reasons my chances of finding some meaning are very small, but science without meaning cannot be real science.

PS Lubos Motl, as a strong proponent of a textbook all-settled Copenhagen interpretation defined by himself, reacts to the polls with

• The foundations of quantum mechanics were fully built in the 1920s, mostly in 1925 or at most 1926, and by 1930, all the universal rules of the theory took their present form...as the Copenhagen interpretation. If you subtract all these rules, all this "interpretation", you will be left with no physical theory whatsoever. At most, you will be left with some mathematics – but pure mathematics can say nothing about the world around us or our perceptions.
• In virtually all questions, the more correct answers attracted visibly greater fractions of physicists than the wrong answers.

Lubos claims that more correct views, with the true correct views carried by only Lubos himself, gathers a greater fraction than less correct views, and so everything is ok from Lubos point of view. But is greater fraction sufficient from scientific point of view, as if scientific truth is to be decided by democratic voting? Shouldn't Lobos ask for 99.9% adherence to his one and only correct view? If physics is to keep its position as the king science?

Or is modern physics instead to be viewed as the root of modernity through a collapse of classical ideals of rationality, objectivity and causality?

realQM vs Hartree-Fock and DFT

I have put up an updated version of realQM (real Quantum Mechanics) to be compared with stdQM (standard QM).

stdQM is based on a linear Schrödinger equation in a $3N$ dimensional wave function with global support for an atom with $N$ electrons, which is made computable in Hartree-Fock and Density Functional Theory DFT approximations reducing the dimensionality to basically 3d.

realQM is based on a system of non-linear Schrödinger equations in $N$ 3d electron wave functions with local disjoint supports, which is computable without approximation. Evidence that realQM describes real physics is given.

Trump: End of Global Warming Alarmism

The new president of US Donald Trump expressed a clear standpoint against global warming alarmism during the presidential race:

• The concept of global warming was created by and for the Chinese in order to make U.S. manufacturing non-competitive.
• Any and all weather events are used by the GLOBAL WARMING HOAXSTERS to justify higher taxes to save our planet! They don't believe it is $\$\$\$\$$!
• This very expensive GLOBAL WARMING bullshit has got to stop. Our planet is freezing, record low temps,and our GW scientists are stuck in ice.
• It’s snowing & freezing in NYC. What the hell ever happened to global warming?
• Ice storm rolls from Texas to Tennessee - I'm in Los Angeles and it's freezing. Global warming is a total, and very expensive, hoax!

Trump says that he will end all federal clean energy development, all research on solar, wind, efficiency, batteries, clean cars, and climate science:

• I will also cancel all wasteful climate change spending from Obama-Clinton, including all global warming payments to the United Nations. These steps will save $100 billion over 8 years, and this money will be used to help rebuild the vital infrastructure, including water systems, in America’s inner cities.

This is hopeful to the world and to science. It says that you cannot fool all the people all the time, in a democracy with free debate and science. 

This is the beginning of the end of global warming alarmism including its most aggressive form led by Sweden and Germany. The weather is now celebrating Trump's victory by heavy snow fall over Stockholm...

PS Trump picks top climate skeptic to lead EPA transition:

• Choosing Myron Ebell means Trump plans to drastically reshape climate policies.
• Ebell’s views appear to square with Trump’s when it comes to EPA’s agenda. Trump has called global warming “bullshit” and he has said he would “cancel” the Paris global warming accord and roll back President Obama’s executive actions on climate change (ClimateWire, May 27).

Finally, reason is taking over...

Why are Scientists Openly Supporting Hillary?

Physicists and mathematicians such as Peter Woit, Leonard Susskind and Terence Tao have come out as strong supporters of Hillary in the presidential race, and then of course as strong opponents to Trump. This is unusual because scientists seldom (openly) take on political missions.

Why is that? Isn't science beyond politics? No, not in our time, and then not in particular climate science, which has become 100% politics. Climate scientists don't like Trump, because he says that climate science is 100% politics and not science. 

Is it the same thing with physics and math? Is a pure mathematician like Tao and a string theorist like Susskind fearing that a questioning non-opportunist Trump would be more difficult to deal with than an opportunist Hillary representing (scientific) establishment? What if Trump would question the value of string theory, as he did with climate science?

Weinberg: Why Quantum Mechanics Needs an Overhaul!

My new book Real Quantum Mechanics seems to fill a need: Nobel Laureate in Physics Steven Weinberg believes that quantum mechanics needs an overhaul because current debates suggest need for new approach to comprehend reality:

• I’m not as happy about quantum mechanics as I used to be, and not as dismissive of its critics.
• It’s a bad sign in particular that those physicists who are happy about quantum mechanics, and see nothing wrong with it, don’t agree with each other about what it means.

I hope this can motivate you to check out the new approach to quantum reality presented in the book, which addresses many of the issues raised by Weinberg.

Weinberg takes the first step to progress by admitting that quantum mechanics in its present form cannot be the answer to the physics of atoms and molecules.

Of course the witness by Weinberg is not well received by ardent believers in a quantum mechanics once and for all cut in stone by Heisenberg and Born, such as Lubos Motl.

But it may be that questioning a theory, in particular a theory supposedly being embraced by all educated, shows more brains and knowledge than simply swallowing it without any question.

PS1 I put up a comment on Lubos Reference frame, but the discussion was quickly cut by Lubos, us usual...any questioning of the dogma of Heisenberg-Bohr-Born is impossible to Lubos, but that is not in the spirit of real science and physics...

PS2 Here is my closing comment which will be censored by Lubos: It is natural to draw a parallel between Lubos defence of the establishment of QM and the defence of the Clinton establishment by Woit, Tao, Susskind et cet, (rightly questioned by Lubos) in both cases a defence with objective to close the discussion and pretend that everything is perfectly normal. Right Lobos?

PS3 Here is a link to Weinberg's talk.

Real Quantum Mechanics: New Book

I am now starting to compile RealQM into a new book and I have put up a very first version for inspection.

The book follows in the foot steps of Schrödinger with a hope that it could have made him smile:

And God Said Gravitational Potential!

The question if we live in a simulation is discussed by Ethan Siegel with reference to the 2016 Isaac Asimov Memorial Debate: Is the Universe a Simulation?

The debate connects to my app on Dark Energy at App Store allowing you to play with a cosmological model where gravitational potential $\phi$ is primordial from which mass $\rho$ is connected through the equation $\rho =\Delta\phi$ of local differentiation with the Laplacian differential operator $\Delta$.

In this model a fluctuation of $\phi$ around a zero initial state can create massiv positive and negative mass $\rho$ through the action of differentiation with the power of amplifying fluctuations, thus generating (seemingly out of nothing) universa of positive and negative mass which repel each other and drift apart.

What we can see may thus be one universe of positive mass with its negative counterpart since long gone beyond sight, but with the repellation still felt as dark energy.

Further, visible matter can be connected to $\Delta\phi$ being singular (like a delta-function), while dark matter may correspond to $\Delta\phi$ being smooth.

In the debate the idea of gravitation as primordial "operating system" is touched upon, but is not penetrated since the common view is that it is mass which is primordial from which gravitation magically is created by magic instant action at distance. Turning this common view around making gravitation/gravitational potential primordial, may open to understanding both dark matter and energy. What do you think?

Real Quantum Mechanics

Physics is wrong, from string theory to quantum mechanics. The three biggest figures in quantum mechanics, Schrödinger, Einstein and Dirac, were all quantum skeptics. (Roger Penrose in Discover Interview Sept 2009)

The approach to quantum mechanics in terms of classical realistic continuum mechanics, which I have explored in recent posts as Physical Quantum Mechanics, is now available for inspection in more precise terms in the following draft manuscript:

• Real Quantum Mechanics

Here is a sample result:

Pomperipossa om Skolmatematik: Lite för Alla eller Mycket för Några?

Det moderna digitala samhället bygger på avancerad matematik och det är av vital betydelse att det finns samhällsmedborgare som kan bära och utveckla denna kunskap.

Skolans matematik har som mål att ge alla det minimum av matematisk kunskap, som anses nödvändigt för fungera som rationell samhällsmedborgare och inga resurser skall sparas för att nå detta mål. Genom massiva stödinsatser med start i förskolan skall varje elev garanteras att uppnå miniminivån, som en del av en samhällelig "läsa-skriva-räkna garanti".

Samhället har alltså behov av ett relativt fåtal som kan använda avancerad matematik, i likhet med avancerad medicin, medan skolan fokuserar på trivial matematik att användas av alla.

Detta går inte ihop. Det samhälleliga värdet av avancerad matematik i händerna på ett fåtal är stort, i likhet med avancerad medicin. Men denna kan inte ersättas av trivial matematik som alla kan, som att veta hur man sätter på ett plåster. Man kan inte ersätta hjärttransplantationer utförda av några specialister med en massa plåster applicerade av många.

För att skolmatematiken skall få mening och motsvara samhällets behov av matematik, fordras att nuvarande obligatorium i form av lite för alla, ersätts av valfrihet utan obligatorium och minimikrav, där de som vill ges möjlighet att lära sig så avancerad matematik som möjligt utan maximibegränsning.

Nuvarande mantra om mimimum för alla, måste alltså överges. Men det kommer att sitta hårt inne eftersom en hel skolbyråkrati bygger på detta mantra.

Vidare behöver lärarutbildningen reformeras. Med minimum för alla som mantra räcker det med lärare med ett minimum av kunskap. Med maximum för några, krävs lärare som kan mer än minimum.

Det finns många saker att jämföra med, som tex pianospel vilket i likhet med matematik är svårt att  lära sig och krävande att utöva. Vilket samhälle är då att föredra? Ett samhälle där alla hjälpligt kan traggla sig igenom en pekfingervals på vita tangenter som resultat av en massiv utbildningsinsats från tidiga skolår, men ingen kan spela Chopin, eller ett samhälle där några kan glädja många genom riktigt spel och kanske några inte ens kan pekfingervalsen eftersom det finns så mycket annat som kan vara mer givande, som att lära sig bemästra olika datorspel? Astrid Lindgren skulle säkert ha kunnat låta Pomperipossa ta sig an denna problematik, och då kanske något skulle kunna hända?

Programmering i Skolan: Trivium eller Quadrivium?

Skolverkets förslag till nya läroplaner för grundskolan med programmering som nytt inslag, utformat på uppdrag en Regering som registrerat vibrationer från omvärlden, verkar ha fastnat i systemet då ingen proposition till Riksdagen om programmering är i sikte.

Lika bra det, eftersom Skolverkets förslag har tillkommit efter principen att göra så lite som möjligt, dock med den stora fördelen någon vidareutbildning av lärare i det nya ämnet programmering inte behövs.

Men ute i kodstugor och på bibliotek växer en underground-rörelse fram, där barn får prova på att programmera i Scratch:

• Scratch helps young people learn to think creatively, reason systematically, and work collaboratively — essential skills for life in the 21st century.
• Scratch is a project of the Lifelong Kindergarten Group at the MIT Media Lab. 

Det informella samhället har här tagit ett initiativ till folkbildning, enligt gammal god svensk socialdemokratisk tradition, för att kompensera brister i skolans utbildning. 

Tanken med Scratch är alltså att hjälpa barn från tidig ålder att tänka kreativt och resonera systematiskt, eftersom detta anses vara väsentliga färdigheter i det vuxenliv som väntar barnen. 

Denna tanke ligger också till grund för den ökning av antalet undervisningstimmar i matematik på låg- och mellanstadium med 225 timmar, som Riksdagen beslutat: Med en ordentlig dos matematik under tidiga skolår för alla barn kommer både individ och samhälle att frodas, därför att matematik i likhet med programmering bygger på systematiskt resonerande, och det skall man lära sig i småskolan!

Men om nu systematiskt resonerande/matematik/programmering är så viktigt, och faktiskt inte så vidare enkelt ens för vuxna, vore det då inte bättre att vänta lite till dess barnen är mogna att ta emot mer än bara det enklaste? Och inte bränna det mesta krutet i förtid utan vidareutbildning av lärare, utan istället ge lärarna ordentlig vidareutbildning så att de kan förmedla något bortom det triviala? 

Klassik utbildning bestod av inledande trivium = grammatik, logik och retorik, följd av quadrivium = aritmetik, geometri, musik och astronomi. Vi ser att trivium (väsentligen språk) kom tidigt, medan quadrivium (väsentligen matematik) låg senare i utbildningen. Kanske något att tänka på även idag?

Mathematics as Magics 3: Towards a New School Mathematics

School mathematics with its 150 year history is based on an idea/fiction of unreasonable effectiveness of mathematics (according to Wigner), which when confronted with the reality of the unreasonable ineffectiveness of mathematics (according to Hamming), as any deep contradiction between ideal an reality, has resulted in a big mess and lots of frustration.

Both students and teachers are brain-washed to believe that mathematics is very powerful, while their experience is the opposite.

The idea of the unreasonable effectiveness of mathematics goes back to the declared success of analytical mathematics/Calculus of Newton's Principia Mathematica, which allowed Newton with pen-and-paper to play the role of God as monitor of the Universe.

School mathematics is supposed to bring some of this power to the people. But Principia Mathematica was very difficult to read, in fact written by Newton so as to make criticism from "little smatterers" impossible.  All efforts since then to make analytical mathematics/Calculus simple, have failed and the result is a an analytical pen-and-paper school mathematics which has resisted all efforts to be brought to the people.

This could mean the end of school mathematics, because teaching a subject experienced to be unreasonable ineffective in our society, like Latin, cannot be sustained over time.

But the computer has changed the game completely, and in fact has made mathematics fulfil the prophecy of almost god-like quality, as the basis and work horse of the digital society.

To convince young minds about the usefulness and power of mathematics + computer, it is sufficient to point at computer games such as Minecraft.

The digital society is an expression of the reasonable effectiveness of mathematics + computer and as such can be a meaningful new form of school mathematics, which can be brought to any young mind that can be trigged by a computer game.

PS Is it possible that the tremendous efforts which were made before the computer to develop school mathematics into its prominent position, were made in anticipation of the computer revolution, which would come sooner or later according the vision of Leibniz as a father of both Calculus and the computer? Yes, I guess it may well be that a collective unconscious awareness can motivate a change in society for which the true reason shows only later. Leibniz:

• It is unworthy of excellent men to lose hours like slaves in the labour of calculation which could safely be relegated to anyone else if machines were used.

Mathematics as Magics 2

The idea of the unreasonable effectiveness of mathematics in Wigner's formulation is based on the analytical solution of the two-body problem given in Newton's Principia Mathematica showing that a single planet subject to the inverse square law of gravitation from a fixed sun, will move in an elliptic (or parabolic or hyperbolic) orbit.

Newton could thus confirm Kepler's laws from a single hypothesis of the inverse square law, with  Newton as mathematician thereby convincingly playing the role God! This gave mathematics a tremendous boost into the queen of sciences with immense (seemingly magical) power, which is the basic argument behind extensive compulsory school mathematics: Learn math and play God!

But if Newton was playing with one planet, God is playing with many planets and thus solves the N-body problem of N bodies moving under mutual gravitational attraction with N any number. But already the 3-body problem has resisted analytical solution since Newton, which can be seen to signify the unreasonable ineffectiveness of analytical mathematics in Hamming's formulation.

But the N-body problem can be solved by computational mathematics for N very large, which expresses a reasonable effectiveness of mathematics + computer.

You can explore the N-body problem in the following apps for young minds:

• Dark Energy: NewMath by Claes Johnson
  https://appsto.re/se/5rkebb.i
• Mechanics1: NewMath by Claes Johnson
  https://appsto.re/se/mMzbbb.i

We learn that mathematics + computer (as NewMath) is powerful and should be taught as a subject of reasonable effectiveness, understanding that analytical mathematics alone may be unreasonably ineffective.

Mathematics as Magics 1

Why does the subject of mathematics have such a prominent position in basic school forcing children through lengthy math hours during 9 years, while the fact is that most adults have forgotten most of their own school mathematics and get by very well with a bare minimum of simple arithmetics?

As an expression of this prominent position the total math hours on basic school in Sweden (grades 1-9) has recently been expanded from 900, in two steps with 120 hours in 2013 and additional 105 hours in 2016 to a total of 1.125 math hours, out of a total of about 6.685 in all subjects, thus roughly 1/6 math or almost one full day a week of math for 9 years.

The logic of the expansion is presented to be (i) math is important for both individual and society and (ii) the result of all the math hours invested is close to zero for many students (not even simple arithmetics mastered), from which the conclusion is drawn that (iii) more hours are required.

Of course the logic is a bit weak: if 900 hours gives no result, why would 1125 give better result?
But maybe this is irrelevant, since anyway most children when adults will not miss whatever math they missed to learn in school. But if 900 gives no result you could as well argue that cutting down to a half would give the same result and that would save hours to something more meaningful.

But this is not the way the argument goes. It is instead: mathematics is very important for both individual and society and thus no effort should be spared for the purpose of mathematical enlightenment of the minds of all young people of a nation (like Sweden or China). Of course we can expect another expansion in 2019 and so on until the school day is filled with math!

But why is mathematics viewed to be so important, when most people have little use of more than a bare minimum of arithmetics? Who is selling this idea? How come that it is so uncritically embraced by just about everybody? Take any subject and claim that it should be expanded by 225 hours and see if you can succeed! Math is the unique subject for which this is possible.

Let us see what answer we can find in the book Is God a Mathematician? by Mario Livio:

• A few years ago, I was giving a talk at Cornell University. One of my PowerPoint slides read: “Is God a mathematician?” As soon as that slide appeared, I heard a student in the front row gasp: “Oh God, I hope not!”
• My rhetorical question was neither a attempt to define God for my audience nor a shrewd scheme to intimidate the math phobics. Rather, I was simply presenting a mystery with which some of the most original minds have struggled for centuries—the apparent omnipresence and omnipotent powers of mathematics.
• What is it that gives mathematics such incredible powers? Or, as Einstein once wondered: “How is it possible that mathematics, a product of human thought that is independent of experience [the emphasis is mine], fits so excellently the objects of physical reality?”
• This sense of utter bewilderment is not new. Some of the philosophers in ancient Greece, Pythagoras and Plato in particular, were already in awe of the apparent ability of mathematics to shape and guide the universe, while existing, as it seemed, above the powers of humans to alter, direct, or influence it. 
• The English political philosopher Thomas Hobbes (1588–1679) could not hide his admiration either. In Leviathan, Hobbes’s impressive exposition of what he regarded as the foundation of society and government, he singled out geometry as the paradigm of rational argument: 
• “Seeing then that truth consisteth in the right ordering of names in our affirmations, a man that seeketh precise truth had need to remember what every name he uses stands for, and to place it accordingly; or else he will find himself entangled in words, as a bird in lime twigs; the more he struggles, the more belimed. And therefore in geometry (which is the only science that it hath pleased God hitherto to bestow on mankind), men begin at settling the significations of their words; which settling of significations, they call definitions, and place them in the beginning of their reckoning”. 
• Millennia of impressive mathematical research and erudite philosophical speculation have done relatively little to shed light on the enigma of the power of mathematics. If anything, the mystery has in some sense even deepened.
• Physics Nobel laureate Eugene Wigner (1902–95) was equally dumbfounded: (a success that Wigner dubbed “the unreasonable effectiveness of mathematics”): “The miracle of the appropriateness of the language of mathematics to the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning”. 
• The person who presents what may be the most extreme and most speculative version of the “mathematics as a part of the physical world” scenario is an astrophysicist colleague, Max Tegmark of MIT. Tegmark argues that “our universe is not just described by mathematics—it is mathematics”. 

We here find the idea of a mystery of an unreasonable effectiveness of mathematics as a positive answer to the question Is God a mathematician? We see this idea expressed by physicists (Wigner and Tegmark), while many mathematicians would take the (opposite) position of the British mathematician Godfrey Harold Hardy (1877–1947):

• Hardy was so proud of the fact that his work consisted of nothing but pure mathematics that he emphatically declared: “No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.”  

Hardy's view of the uselessness of mathematics is supported by the mathematician and computer scientist Hamming:

• “There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology.” 

We thus find a range of views concerning the usefulness or effectiveness of mathematics from unbounded admiration to controlled skepticism. Most adults would say that they believe that math is very important, as something they learned in school, while they have themselves found very little use math beyond (if they can) helping their children through the frustrations of school math.

We see that the idea of math as something of god-like quality gets stronger as we move away from pure mathematicians (Hardy), over physicists (Wigner, Tegmark) to ordinary people. A similar pattern may be found in religion with a priest as covert non-believer and a community of strong believers.

The less you know of mathematics, the more powerful you tend to believe it is. To convey this idea is in fact an explicitly stated goal of Swedish school mathematics, and in this respect the education is very successful: When finishing school all students, independent of success in math, are fully convinced that math is very powerful and important and in addition very beautiful!

Mathematics is thus viewed as truly magical by many, which makes rational reasoning about school mathematics very difficult, or simply impossible.  How to be rational about magics?

I will continue with some examples of the magical character of math....and eventually I will land on a standpoint of reasonable effectiveness,  between Wigner's unlimited optimism of unreasonable effectiveness and Hamming's deep pessimism of unreasonable ineffectiveness.

Why Does Trump Not Play the Climate Card?

Trump is known to be skeptic to climate alarmism, according to The Guardian as the world's only national leader to express such an insight. It may come from a sound gut feeling that something is very fishy with that story.

Trump could tell the American people, and the people of the world, the truth that climate alarmism has no real scientific support and battling a non-existent climate enemy at a projected cost of 2-10% of GNP, thus is not needed. He could then say that these enormous resources could better be used for a meaningful purpose such as improving living conditions for the all the poor people of the world. That it would be more than enough to eliminate poverty.

But Trump is not playing such a strong climate card, which could tilt the game to his favour. Why?

Is it that this card would be harassed as being so much worse than any of the politically incorrect cards he has been playing, you name it, so bad that it simply cannot be played? Would freeing resources to eliminate poverty thus be so incredibly politically incorrect?

Or is Trump just waiting to play the card, to get maximal effect?

Hur kunde KI gå på Macchiarini?

Hur kunde KI's högsta vetenskapliga kompetens tro att ett plaströr bestruket med stamceller inopererad på en patient som ersättning för en defekt luftstrupe, skulle utvecklas till en normal fungerade luftstrupe i en helt ny helt oprövad och fullständigt spekulativ form av regenerativ bioteknik?

Jo, det gick till så här enligt Sten Heckschers utredning, där prefekten Felländer-Tsai ger svaret:

• Det är sällsynt med kirurger som inte tvekar inför att skära bort sjuka luftrör på folk och ersätta dessa med decellulariserade nekrograft som hottats upp med stamceller och tillväxtfaktorer. 
• Men det är banbrytande och platsar i Lancet och kan på sikt leda till nya behandlingsparadigm. 
• Jag känner så väl igen fenomenet, har erfarenhet av liknande personer och inom vår verksamhet gjordes ju Sveriges första njur- och levertransplantation (under intressanta former...), till stor nytta för mänskligheten och patienterna (även om vissa dog/dör på operationsbordet). 
• Det fordras en särskild personlighet för att eviscerera folk in vivo. 
• Gränsen mellan succé och fiasko är dock hårfin och succén hänger naturligtvis på en mängd randvillkor och också andra nyckelpersoner som kan balansera det hela.

En vild chansning alltså, ju vildare desto bättre, på decellulariserade nekrograft som hottats upp med stamceller och tillväxtfaktorer utförd av en läkare känd för att eviscerera folk in vivo, som skulle kunna vara banbrytande och göra KI inte bara till utdelare av Nobelpris, utan även till mottagare.  

En vild chansning alltså, som blev verklighet genom ett stark rekommendationsbrev från 14 av KI's mest framstående stamcellsforskare att anställa Macchiarini att utföra sin vivisektion på KI med dödlig utgång, forskare som på något sätt måste ha trott på Macchiarinis nya helt oprövade regenerativa teknik.

Ingen av dessa 14 undertecknare framträder i media, men deras världsledande stamcellsforskning fortsätter mot nya höjdpunkter med stora statliga anslag till utvalda "centers of excellency". Rektorer har fått avgå, men ingen av de forskare som låg bakom rektorernas fatala beslut. Rektor Harriet Wallberg säger i media ungefär att hon blev lurad att först anställa Macchiarini, och så var det nog. Rektor Hamsten kanske också tycker att han blev lurad att förlänga anställningen (2 ggr). Rektorer är ju oftast bara passiva reagenter på underliggande krafter från aktiva forskare eller finansiärer.

Det var inte inkorrekt formalia som ledde till katastrofen, vilket Heckscher menar, utan bristande omdöme hos ledande forskare. Formalia i all ära, men det är till slut omdöme med mer eller mindre korrekt formalia, som är avgörande.   

Om KI representerar svensk forsknings kronjuvel, vad säger detta om övrig svensk forskning?

  

Climate vs Chaos and Turbulence

Both climate alarmists and skeptics like to suggest deep understanding by expressing that global climate is a non-linear chaotic system and as such is unpredictable (as discussed by Kip Hansen in a recent sequence of posts):

• The climate system is a coupled non-linear chaotic system, and therefore the long-term prediction of future climate states is not possible. (IPCC TAR WG1, Working Group I: The Scientific Basis)
• It is my belief that most climate variability and even climate change could simply be the result of chaos in the climate system. (Roy Spencer)

But to simply say that a chaotic system is unpredictable is not the entire story. It is true that point values in space/time of a chaotic system are unpredictable, due to strong pointwise sensitivity to pointwise perturbations, but mean values of a chaotic system typically are predictable.

It is certainly impossible to predict the daily temperature of a specific city within one degree one year ahead, but meaningful monthly temperatures are routinely reported in tourist guides.

The book Turbulent Incompressible Fluid Flow presents the following analysis of turbulence as prime example of chaos:

1. Point values are unpredictable due to local exponential instability.
2. Mean values are predictable due to cancellation of instability effects.

It may thus well be possible (with a high degree of certainty) to predict that the global mean temperature will be the same 100 years from now, within a degree up or down.

For the Lorenz system, as a key example of a chaotic system, it is impossible to predict in which lobe a trajectory will be long ahead in time, but the total time spent in each lobe is observed to become nearly equal over long time. About the weather in Scandinavia, we know for sure that it will be variable with alternating low and high pressures, with sunshine following rain and vice versa as a result of the dynamics. 

Vilka Kunskaper i Matematik Kommer att Behövas i Yrkeslivet?

Huvudargumentet för att öka den redan omfattande undervisningstiden i matematik i den obligatoriska grundskolan, vilket Riksdagen beslutat enligt föregående bloggpost, formuleras på följande sätt:

1. För de enskilda eleverna är det av stor vikt att de får de kunskaper i matematik de kommer att behöva i yrkeslivet eller för fortsatta studier. 
2. Att de har sådana kunskaper är viktigt även för samhället i stort.

Eleverna skall alltså bibringas de kunskaper i matematik de kommer att behöva i yrkeslivet. Men detta  är ju en synnerligen kryptisk närmast logiskt cirkulär formulering.  

Men vilka kan då dessa kunskaper vara? Ett sätt att närma sig denna fråga är att kartlägga vilken matematik som idag de facto används inom olika yrkesgrupper, lämpligen genom en bred enkätundersökning hos dagens yrkesverksamma. Har en sådan undersökning gjorts, och vad var i så fall resultatet?

Ännu Mer Undervisningstid i Matematik!

Riksdagen har sagt ja till Regeringens förslag att ytterligare utöka den totala undervisningstiden i matematik i grundskolan med 105 tim från 1020 tim till 1125 tim, detta efter att tiden ökades med 120 tim 2013. Den totala undervisningstiden i alla ämnen är 6785 tim vilket innebär att var sjätte skoldag, eller nästan en hel dag varje vecka, skall ägnas matematik under alla grundskolans 9 år. Lagrådsremissen bakom beslutet argumenterar på följande sätt :

1. Matematik är ett av tre ämnen som krävs för behörighet till samtliga nationella program i gymnasieskolan. 
2. Grundläggande kunskaper i matematik är också en förutsättning för att klara många högskoleutbildningar.
3. För de enskilda eleverna är det av stor vikt att de får de kunskaper i matematik de kommer att behöva i yrkeslivet eller för fortsatta studier. 
4. Att de har sådana kunskaper är viktigt även för samhället i stort.
5. Mycket tyder dock på att svenska elevers matematikkunskaper försämrats under 2000-talet.
6. Som redovisas i promemorian finns det internationell forskning som stöder sambandet mellan utökad undervisningstid och kunskapsresultat.
7. Någon förändring av kursplanen och kunskapskraven i matematik med anledning av utökningen av undervisningstiden är inte avsedd.

Logiken förefaller vara att om ytterligare tid ägnas åt en kursplan/undervisning med dokumenterat dåligt resultat, så kommer resultaten att förbättras. 

Vem kan ha hittat på ett så befängt förslag? Sverker Lundin ger i Who wants to be scientific , anyway? en förklaring: Matematik (eller vetenskap) har blivit modernitetens nya religion när den gamla nu har lagt sig att dö, en religion som ingen vuxen egentligen tror på och mycket få utövar, men en religion som det blivit klädsamt och politiskt korrekt att bekänna sig till i modernitetens tecken, men då bara i "interpassiv" form med försvarslösa skolelever som mottagare av predikan. 

I detta narrspel är finns det aldrig tillräckligt med ritualer för att uppvisa sin fasta tro, och timantalet i matematik kommer således att fortsätta att öka, medan resultaten fortsätter att sjunka och det bara  blir viktigare och viktigare både för de enskilda eleverna och samhället i stort att de kunskaper i matematik som behövs i skolan också lärs ut i skolan.

De nya 105 timmarna skall företrädesvis tillföras mellanstadiet, medan de 120 som tillfördes 2013 avsåg främst lågstadiet. Detta speglar en utbredd förställning att något fundamentalt har gått snett i den tidiga matematikundervisningen, oklart dock vad, och att om bara detta tidiga misstag, oklart vad, undviks eller snabbt rättas till genom extra timmar, så kommer allt att gå så mycket bättre. Men en ensidig jakt på att undvika det första misstaget, oklart vilket det är, kommer naturligtvis medföra att det inte blir mycket tid över till förkovran i senare årskurser, men det kanske inte gör så mycket...

New Quantum Mechanics 19: 1st Excitation of He

Here are results for the first excitation of Helium ground state into a 1S2S state with excitation energy = 0.68 = 2.90 -2.22, to be compared with observed 0.72:

New Quantum Mechanics 18: Helium Ground State Revisited

Concerning the ground state and ground state energy of Helium the following illumination can be made:

Standard quantum mechanics describes the ground state of Helium as $1S2$ with a 6d wave function $\psi (x1,x2)$ depending on two 3d Euclidean space coordinates $x1$ and $x2$ of the form

• $\psi (x1,x2) =C \exp(-Z\vert x1\vert )\exp (-Z\vert x2\vert )$,       (1)

with $Z =2$ the kernel charge, and $C$ a normalising constant. This describes two identical spherically symmetric electron distributions as solution of a reduced Schrödinger equation without electronic repulsion potential, with a total energy $E =-4$, way off the observed $-2.903$. 

To handle this discrepancy between model and observation the following corrections in the computation of total energy are made, while keeping the spherically symmetric form (1) of the ground state as the solution of a reduced Schrödinger equation:  

1 . Including Coulomb repulsion energy of (1) gives  $E=-2.75$.

2. Changing the kernel attraction to $Z=2 -5/16$ claiming screening gives $E=-2.85$.

3. Changing Coulomb repulsion by inflating the wave function to depend on $\vert x1-x2\vert$ can give  at best $E=-2.903724...$ to be compared with precise observation according to Nist atomic data base $-2.903385$ thus with an relative error of $0.0001$. Here the dependence on $\vert x1-x2\vert$ of the inflated wave function upon integration with respect to $x2$ reduces to a dependence on only the modulus of $x1$. Thus the inflated non spherically symmetric wave function can be argued to anyway represent two spherically symmetric electronic distributions.

We see that a spherically symmetric ground state of the form (1) is attributed to have correct energy, by suitably modifying the computation of energy so as to give perfect fit with observation. This kind of physics has been very successful and convincing (in particular to physicists), but it may be that it should be subject to critical scientific scrutiny.

The ideal in any case is a model with a solution which ab initio in direct computation has correct energy, not a  model with a solutions which has correct energy only if the computation of energy is changed by some ad hoc trick until match.

The effect of the fix according to 3. is to introduce a correlation between the two electrons to the effect that they would tend appear on opposite sides of the kernel, thus avoiding close contact. Such an effect can be introduced by angular weighting in (1) which can reduce electron repulsion energy but at the expense of increasing kinetic energy by angular variation of wave functions with global support and then seemingly without sufficient net effect. With the local support of the wave functions meeting with a homogeneous Neumann condition (more or less vanishing kinetic energy) of the new model, such an increase of kinetic energy is not present and a good match with observation is obtained.

New Quantum Mechanics 17: The Nightmare of Multi-Dimensional Schrödinger Equation

Once Schrödinger had formulated his equation for the Hydrogen atom with one electron and with great satisfaction observed an amazing correspondence to experimental data, he faced the problem of generalising his equation to atoms with many electrons.

The basic problem was the generalisation of the Laplacian to the case of many electrons and here Schrödinger took the easy route (in the third out of Four Lectures on Wave Mechanics delivered at the Royal Institution in 1928) of a formal generalisation introducing a set of new independent space coordinates and associated Laplacian for each new electron, thus ending up with a wave function $\psi (x1,...,xN)$ for an atom with $N$ electrons depending on $N$ 3d spatial coordinates $x1$,...,$xN$.

Already Helium with a Schrödinger equation in 6 spatial dimensions then posed a severe computational problem, which Schrödinger did not attempt to solve.  With a resolution of $10^2$ for each coordinate an atom with $N$ electrons then gives a discrete problem with $10^{6N}$ unknowns, which already for Neon with $N=10$ is bigger that the total number of atoms in the universe.

The easy generalisation thus came with the severe side-effect of giving a computationally hopeless problem, and thus from scientific point meaningless model.

To handle the absurdity of the $3N$ dimensions rescue steps were then taken by Hartree and Fock to reduce the dimensionality by restricting wave functions to be linear combinations of products of one-electron wave functions $\psi_j(xj)$ with global support:

• $\psi_1(x1)\times\psi_2(x2)\times ....\times\psi_N(xN)$    

to be solved computationally by iterating over the one-electron wave functions. The dimensionality was further reduced by ad hoc postulating that only fully symmetric or anti-symmetric wave functions (in the variables $(x1,...,xN)$) would describe physics adding ad hoc a Pauli Exclusion Principle on the way to help the case. But the dimensionality was still large and to get results in correspondence with observations required ad hoc trial and error choice of one-electron wave functions in Hartree-Fock computations setting the standard.

We thus see an easy generalisation into many dimensions followed by a very troublesome rescue operation stepping back from the many dimensions. It would seem more rational to not give in to the temptation of easy generalisation, and in this sequence of posts we explore such a route.

PS In the second of the Four Lectures Schrödinger argues against an atom model in terms of charge density by comparing with the known Maxwell's equations for electromagnetics in terms of electromagnetic fields, which works so amazingly well, with the prospect of a model in terms of energies, which is not known to work.

New Quantum Mechanics 16: Relation to Hartree and Hartree-Fock

The standard computational form of the quantum mechanics of an atom with N electrons (Hartree or Hartree-Fock) seeks solutions to the standard multi-dimensional Schrödinger equation as linear combinations of wave functions $\psi (x1,x2,...,xN)$ depending on $N$ 3d space coordinates $x1$,...,$xN$ as a product:

• $\psi (x1,x2,...,xN)=\psi_1(x1)\times\psi_2(x2)\times ....\times\psi_N(x_N)$ 

where the $\psi_j$ are globally defined electronic wave functions depending on a single space coordinate $xj$.

The new model takes the form of a non-standard free boundary Schrödinger equation in a wave function $\psi (x)$ as a sum:

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

where the $\psi_j(x)$ are electronic wave functions with local support on a common partition of 3d space with common space coordinate $x$.

The difference between the new model and Hartree/Hartree-Fock is evident and profound.  A big trouble with electronic wave functions having global support is that they overlap and demand an exclusion principle and new physics of exchange energy.  The wave functions of the new model do not overlap and there is no need of any exclusion principle or exchange energy.

PS Standard quantum mechanics comes with new forms of energy such as exchange energy and correlation energy. Here correlation energy is simply the difference between experimental total energy and total energy computed with Hartree-Fock and thus is not a physical form of energy as suggested by the name, simply a computational /modeling error.

New Quantum Mechanics 15: Relation to "Atoms in Molecules"

Atoms in Molecules developed by Richard Bader is a charge density theory based on basins of attraction of atomic kernels with boundaries characterised by vanishing normal derivative of charge density.

This connects to the homogeneous Neumann boundary condition identifying separation between electrons of the new model under study in this sequence of posts.

Atoms in Molecules is focussed on the role of atomic kernels in molecules, while the new model primarily (so far) concerns electrons in atoms.

New Quantum Mechanics 14: $H^-$ Ion

Below are results for the $H^-$ ion with two electrons and a proton. The ground state energy comes out as -0.514, slightly below the energy -0.5 of $H$, which means that $H$ is slightly electro-negative and thus by acquiring an electron into $H^-$ may react with $H^+$ to form $H2$ (with ground state energy -1.17), as one possible route to formation of $H2$. Another route is covered in this post with two H atoms being attracted to form a covalent bond.

The two electron wave functions of $H^-$ occupy half-spherical domains (depicted in red and blue) and meet at a plane with a homogeneous Neumann condition satisfied on both sides.

New Quantum Mechanics 13: The Trouble with Standard QM

Standard quantum mechanics of atom is based on the eigen functions of the Schrödinger equation for a Hydrogen atom with one electron, named "orbitals" being the elements of the Aufbau or build of many-electron atoms in the form of s, p, d and f orbitals of increasing complexity, see below.

These "orbitals" have global support and has led to the firm conviction that all electrons must have global support and so have to be viewed to always be everywhere and nowhere at the same time (as a basic mystery of qm beyond conception of human minds). To handle this strange situation Pauli felt forced to introduce his exclusion principle, while strongly regretting to ever have come up with such an idea, even in his Nobel Lecture:

• Already in my original paper I stressed the circumstance that 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. 
• Of course in the beginning I hoped that the new quantum mechanics, with the help of which it was possible to deduce so many half-empirical formal rules in use at that time, will also rigorously deduce the exclusion principle. 
• Instead of it there was for electrons still an exclusion: not of particular states any longer, but of whole classes of states, namely the exclusion of all classes different from the antisymmetrical one. 
• The impression that the shadow of some incompleteness fell here on the bright light of success of the new quantum mechanics seems to me unavoidable. 

In my model electrons have local support and occupy different regions of space and thus have physical presence. Besides the model seems to fit with observations. It may be that this is the way it is.

The trouble with (modern) physics is largely the trouble with standard QM, the rest of the trouble being caused by Einstein's relativity theory. Here is recent evidence of the crisis of modern physics:
The LHC "nightmare scenario" has come true.

Here is a catalogue of "orbitals" believed to form the Aufbau of atoms:

And here is the Aufbau of the periodic table, which is filled with ad hoc rules (Pauli, Madelung, Hund,..) and exceptions from these rules:

 

New Quantum Mechanics 12: H2 Non Bonded

Here are results for two hydrogen atoms forming an H2 molecule at kernel distance R = 1.4 at minimal total energy of -1.17 and a non-bonded molecule for larger distance approaching full separation for R larger than 6-10 at a total energy of -1. The results fit quite well with table data listed below.

The computations were made (on an iPad) in cylindrical coordinates in rotational symmetry around molecule axis on a mesh of 2 x 400 along the axis and 100 in the radial direction. The electrons are separated by a plane perpendicular to the axis through the the molecule center, with a homogeneous Neumann boundary condition for each electron half space Schrödinger equation. The electronic potentials are computed by solving a Poisson equation in full space for each electron.

PS To capture energy approach to -1 as R becomes large, in particular the (delicate) $R^{-6}$ dependence of the van der Waal force, requires a (second order) perturbation analysis, which is beyond the scope of the basic model under study with $R^{-1}$ dependence of kernel and electronic potential energies.

%TABLE II. Born–Oppenheimer total, E

%Relativistic energies of the ground state of the hydrogen molecule

%L. Wolniewicz

%Citation: J. Chem. Phys. 99, 1851 (1993); 

for two hydrogen atoms separated by a distance R bohr

 R    energy

0.20 2.197803500 

0.30 0.619241793 

0.40 -0.120230242 

0.50 -0.526638671 

0.60 -0.769635353 

0.80 -1.020056603 

0.90 -1.083643180 

1.00 -1.124539664 

1.10 -1.150057316 

1.20 -1.164935195 

1.30 -1.172347104 

1.35 -1.173963683 

1.40 -1.174475671 

1.45 -1.174057029 

1.50 -1.172855038 

1.60 -1.168583333 

1.70 -1.162458688 

1.80 -1.155068699 

2.00 -1.138132919 

2.20 -1.120132079 

2.40 -1.102422568 

2.60 -1.085791199 

2.80 -1.070683196 

3.00 -1.057326233 

3.20 -1.045799627 

3.40 -1.036075361 

3.60 -1.028046276 

3.80 -1.021549766 

4.00 -1.016390228 

4.20 -1.012359938 

4.40 -1.009256497 

4.60 -1.006895204 

4.80 -1.005115986 

5.00 -1.003785643 

5.20 -1.002796804 

5.40 -1.002065047 

5.60 -1.001525243 

5.80 -1.001127874 

6.00 -1.000835702 

6.20 -1.000620961 

6.40 -1.000463077 

6.60 -1.000346878 

6.80 -1.000261213 

7.00 -1.000197911 

7.20 -1.000150992 

7.40 -1.000116086 

7.60 -1.000090001 

7.80 -1.000070408 

8.00 -1.000055603 

8.50 -1.000032170 

9.00 -1.000019780 

9.50 -1.000012855

10.00 -1.000008754 

11.00 -1.000004506 

12.00 -1.000002546

New Quantum Mechanics 11: Helium Mystery Resolved

The modern physics of quantum mechanics born in 1926 was a towering success for the Hydrogen atom with one electron, but already Helium with two electrons posed difficulties, which have never been resolved (to be true).

The result is that prominent physicists always pride themselves by stating that quantum mechanics cannot be understood, only be followed to the benefit of humanity, like a religion:

• I think I can safely say that nobody understands quantum mechanics. (Richard Feynman, in The Character of Physical Law (1965))

Text books and tables list the ground state of Helium as $1S^2$ with two spherically symmetric electrons (the S) with opposite spin in a first shell (the 1), named parahelium.  The energy of a $1S^2$ state according to basic quantum  theory is equal to -2.75 (Hartree), while the observation of ground state energy  is -2.903. To handle this apparent collapse of basic quantum theory, the computation of energy is changed by introducing a suitable perturbation away from spherical symmetry which delivers the wanted result of -2.903, while maintaining that the ground state still is $1S^2$.

Of course, this does not make sense, but since quantum mechanics is not "anschaulich" or  "visualisable" (as required by Schrödinger) and therefore cannot be understood by humans, this is not a big deal.  By a suitable perturbation the desired result can be reached, and we are not allowed to ask any further questions following the dictate of Dirac: Shut up and calculate.

New Quantum Mechanics resolves the situation as follows:

The ground state is predicted to be a spherically (half-)symmetric continuous electron charge distribution with each electron occupying a half-space, and the electrons meeting on at plane (free boundary) where the normal derivative for each electron charge distribution vanishes. The result of ground state energy computations according to earlier posts shows close agreement with the observed -2.903:

Notice the asymmetric electron potential and the resulting slightly asymmetric charge distribution with polar accumulation. The model shows a non-standard electron configuration, which may be the true one (if there is anything like that).