## fredag 30 augusti 2013

### 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.

## torsdag 29 augusti 2013

### 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.

## måndag 26 augusti 2013

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

(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.

## torsdag 22 augusti 2013

### 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.

## onsdag 21 augusti 2013

### Rapport från Kammarrätten i Stockholm

För att få direkt information om Kammarrättens resonemang till grund för sitt avslag av min överklagan av KTHs vägran att låta mig få tillgång till epost inom KTH rubricerad Gomorron Sverige och MST/BodyandSoul, tar jag först kontakt med föredragande i ärendet Fredrika Johansson, detta efter semestrarna och efter att min överklagan till Högsta Förvaltningsdomstolen skickats in. Fredrika verkar inte ha riktigt bra koll vad gäller det lagtekniska och hänvisar mig vidare till referenten kammarrättsråd Anders Jonsson, som bär huvudansvaret för domen.

Vid ett första samtal förra veckan med Anders Jonsson visar det sig vara svårt för Jonsson att få fram handlingarna i ärendet, varför Jonsson säger att han skall läsa in sig i ärendet igen och återkomma denna vecka. Så sker varvid Jonsson säger att han fortfarande inte kunnat få fram handlingarna, denna gång eftersom de sänts till Högsta Förvaltningsdomstolen, eftersom jag överklagat. Jonsson säger att det innebär att han bara har diffusa minnesbilder att referera till som svar på de frågor jag ställer.

Min första fråga är densamma som till Fredrika Johansson, nämligen om kontakter mellan Kammarrätten och KTH i ärendet förekommit, vilken besvarats med bestämt nej av Johansson medan Jonsson säger att visst har kontakter förekommit som en naturlig del av Kammarrättens handläggning av ärendet. Speciellt har Kammarrätten via KTH fått tillgång till den av mig begärda eposten, men mer än så har kontakterna kanske inte innehållit.

Min nästa fråga är hur Kammarrätten kan veta att eposten inte kan anses såsom varande på något sätt färdigställd av KTH (eftersom den inte var avsedd för vidare bearbetning), vilket är Kammarrättens huvudmotivering för avslag, utan att Kammarrätten fått någon information från KTH (som då undanhållits mig) om att så varit fallet, dvs att den inte färdigställts. Jonsson hänvisar till domen som säger att "det inte framkommit" att så skulle ha skett, men svävar på målet vad gäller min fråga om denna slutsats grundar sig på information från KTH eller om inget framkommit eftersom Kammarrätten inget frågat. Jonsson undslipper sig dock något i stil med att "Kammarrätten måste tro på det som myndigheten (KTH) säger, och kan ju inte göra någon egen utredning". Dessutom saknar Jonsson alla dokument i ärendet och kan inte så noga komma ihåg alla turer med KTH. Jonsson säger vidare att det ju bara handlar om några epostmeddelanden om vars status man inte kan veta så mycket.

Min nästa fråga är om Kammarrätten accepterat KTHs tomma tautologiska motivering att avslå min begäran såsom varande enligt lagboken. Jonsson kan inte svara på detta. Men säger att det saknar betydelse eftersom Kammarrätten har kommit med sin egen motivering för avslag och att det torde räcka.

Min sista fråga är varför Kammarrätten i Stockholm inte explicit redovisar den avvikande domen i ett liknande ärende hos Kammarrätten i Göteborg, utan bara implicit argumenterar mot denna dom. Inget egentligt svar på detta heller.

Sammantaget har inte mitt samtal med Kammarrättens huvudaktörer i mitt ärende givit några konkreta besked. Men känslan jag får är att Kammarrätten i Stockholm agerar enligt önskemål från KTH till förmån för myndigheten KTH ("man måste ju tro på vad myndigheten säger") och nackdel för mig som enskild professor.

Nu får vi se om Högsta Förvaltningsdomstolen ger prövningstillstånd och om i så fall prövningen kan bringa mer klarhet. Erfarenheten från Kammarrätten i Stockholm inger inte förtroende för ett fungerande rättssamhälle och frågan är varför man skulle kunna förvänta sig något annat från högsta instans?

### Quantum Contradictions 13: Why Must the Electron Spin?

Wikipedia informs us:
The objective was to explain the observed doublet feature of the emission spectra of alkali metals with one valence electron in the outermost shell, from an assumption that the inner full shells (and kernel) are fully spherically symmetric. The only possibility left was then to equip the single outer valence electron with a Zweideutigkeit or two-valuedness, which was later named spin up and spin down. The same argument was used to explain the Stern-Gerlach experiment.

But the assumption that the inner full shells are spherically symmetric was never justified, and so the necessity of introducing a mysterious two-valued feature of the electron, which had to be accepted even if it was very strange, lacked rationale (but it worked).

The alternative model of this sequence of posts gives evidence of full shells starting from Helium, which are not fully spherically symmetric with a two-valuedness resulting from separation of electrons by repulsion.

### Ännu Lägre Antagningspoäng till SimuleringsTeknik KTH HT2013

Kandidatutbildningen i Simuleringsteknik vid KTH inrättades 2010 för att möta behovet av beräkningsmatematisk utbildning i dagens svenska IT-samhälle, i den nya form som reformprogrammet BodyandSoul erbjuder. Avsikten var att erbjuda en spetsutbildning på KTH med förankring i ledande KTH-forskning i beräkningsteknik, som skulle kunna attrahera studenter med specialbegåvning/intresse för avancerad datorsimulering.

Skolan för Teknikvetenskap stoppade dock detta initiativ genom att censurera min bok Mathematical Simulation Technology som presenterar BodyandSoul, och startade istället kandidatprogrammet HT2012 på en grund av traditionell matematik utan koppling till den ledande KTH-forskningen.

Rekryteringen misslyckades inför HT2012 och de 40 platserna kunde knappt fyllas trots att lägsta antagningspoäng var 13.90 och 0.70 på högskoleprovet.

Inför HT2013 är det ännu sämre: Bara 36 platser kunde fyllas, med lägsta poäng 12.52 och 0.55.

Så har då KTH lyckats förvanska en möjlig spetsutbildning med höga intagningspoäng till en omöjlig basutbildning med låga intagningspoäng.

Vilken är då rationaliteten med att landets främsta högskola satsar dyra pengar på att utbilda studenter med landets lägsta antagningspoäng?  Är det verkligen priset värt för att stoppa BodyandSoul?

PS Resultaten från KTH-test av förkunskaper i matematik inför HT2012 visar en lösningsfrekvens av 70% för Teknisk Fysik att jämföra med 25% för Simuleringsteknik.

## måndag 19 augusti 2013

### Quantum Contradictions 12: Electron Pair of Helium vs Spin

Let us return with an elaboration of post 6 in this series about the electron configuration of the ground state of Helium. Let us make an Ansatz with the distribution of the two electrons of Helium given by the following two (squared) wave functions (in standard notation):
• $\psi_1(r,\theta)^2 = (1 + \beta\cos(\theta))exp(-2\alpha r)\times\alpha^3/\pi$
• $\psi_2(r,\theta)^2 = (1 - \beta\cos(\theta))exp(-2\alpha r)\times\alpha^3/\pi$,
where $\alpha$ and $\beta$ are positive parameters to determine. This corresponds to a configuration with the two electrons being separated, with electron 1 shifted towards the upper part of a spherical atom and electron 2 towards the lower part.

Standard text book computation gives the following total energy $E$ as the sum of potential energy, radial kinetic energy and interelectronic repulsion energy in the setting of Schrödinger's equation:
• $E = - 4 \alpha + \alpha^2 + 5 \alpha/8 - \alpha\beta^2/24$
with the last term resulting from the separation weights $(1 + - \beta\cos(\theta ))$. Optimization in $\alpha$ with $\beta = 1$ gives
• $\alpha = 2 - (5/16 - 1/48)$
• $E = (41/24)^2 = - 2.918$
to be compared with the observed $E = - 2.903$. For a configuration without separation ($\beta = 0$),
$E = - 2.85$.  We thus find theoretical support of an idea of electronic Zweideutigkeit in the ground state of Helium represented by an electron pair in opposition.

In the above computation we neglected azimuthal kinetic energy based on the idea that the electron pair should be averaged over azimuthal angle to eliminate azimuthal variation.  The considered electron pair is thus to be viewed as "typical" combined with averaging over angle reducing azimuthal kinetic energy to zero.

The total (multi-dimensional) wave function $\psi(r_1, \theta_1, r_1, \theta_2)$ in the standard setting thus has a Hartree product form
• $\psi (r_1, \theta_1, r_2, \theta_2) = \psi_1(r_1, \theta_1)\times \psi_2(r_2,\theta_2)$
for which only radial kinetic energy is taken into account.

PS The angular momentum of $\psi_1$ equals $- \sin(\theta ) (\sin(\phi ), - \cos(\phi ), 0)$ and that of $\psi_2$ equals $+ \sin(\theta ) (\sin(\phi ), - \cos(\phi ), 0)$ up to the common factor $\beta\exp(-2\alpha r)\times \alpha^3/\pi$, which can be associated with a two-valued spin up/down.

The mysterious apparent Zweideutigkeit of electrons thus may simply be an effect of electron separation by repulsion.

### Quantum Contradictions 11: Theory vs Computation vs Observation

Quantum mechanics is based on an unhapppy marriage between theory and observation/measurement, with the observer necessarily affecting the observed. The wave function describing e.g. (the statistics of) of the atomic world, is thus supposed to dwell in an enormous Hilbert space beyond any comprehension and deliver a real observable as an eigenvalue of a Hermitian operator by a "collapse of the wave function" upon observation, as if the observer decides life or death of the Schrödinger cat in its box by merely opening the box and looking at the cat.

But there is a form of observation which can be done without interference with what is observed, and that is computing solutions to the equations describing the physics and thereby gaining information about the physics without so to speak having to open the box and look at the cat. In this form of computational physics observation is not restricted to eigenvalues of Hermitian operators, but can give complete information about the state of a system at a later time given the state at an initial time.

The computer thus opens new possibilities of exploring physics which were not available when quantum mechanics was formed 100 years ago and forced physics into a dead-lock with the physicist facing himself instead of physics.

## söndag 18 augusti 2013

### Quantum Contradictions 10: Symmetric and Antisymmetric Wave Functions

The wave function as solution of the Schrödinger equation as the basis of quantum mechanics, has 3N spatial dimensions for an N-particle system. As such it is a computational impossibility as noted by Nobel Laureate Walter Kohn in his Nobel Lecture 1998:
• In general the many-dimensional wave function of a system of N electrons is not a legitimate scientific concept (for N > 10 say).
But if the 3N-dimensional wave function is a computational impossibility, or monstrosity, it is also a physical monstrosity; at least if physics is viewed as some form of analog computation, which appears to be the only possibility beyond mysticism.

Nevertheless, physicists worship the 3N-dimensional wave function and believe it has a deep physical significance. In particular, it is believed that the atomic world consists of bosons represented by fully symmetric wave functions and fermions represented by fully anti-symmetric wave functions. In particular the periodic table is believed to result from the fact that electrons are fermions satisfying Pauli's exclusion principle as a consequence of anti-symmetry.

A periodic table based on bosons (such as photons) would be a trivial table with only one kind of atom.

But if the 3N-dimensional wave function is a monstrosity, then so must be the subdivision into fully symmetric and anti-symmetric wave functions and then also Pauli's exclusion principle.

In the alternative model under study in this sequence of posts, another explanation of the periodic table presents itself: In this model it is the repulsion between electrons which shapes the atomic shell structure starting from the Helium atom with two electrons repelled into opposite position around the kernel.

In other words, atoms with electrons without repulsion would be of one kind and thus correspond to bosonic atoms of one kind only capable of creating a trivial dull world.

We are thus led to the idea that it is the repulsion between electrons, which forms the atomic world of great variability creating a far from dull world of even greater variability by upscaling, and not any mystery of anti-symmetric wave functions which cannot explain e.g. the electron structure of Plutonium with N = 94.

## fredag 16 augusti 2013

### Quantum Contradictions 9: Shell Structure from Helium Scaling

To explain the observed sequence 2, 8, 18, 32,..., of the number of electrons in filled atomic shells of the periodic table, one may argue as follows (which is not the text book explanation) starting from the structure of Helium as described in a previous post:

Helium has 2 electrons filling a K-shell of width 1/2. The next element in the table is Lithium with 2 electrons filling a K-shell of width 1/3 and 1 electron in an outer L-shell of width 1 being attracted by a net charge of +1 from the kernel and the two inner shell electrons. In the model I am studying in this sequence of posts, the 3rd electron of Lithium of width 1 being attracted by a net charge of +1 finds a relative energy minimum in a second shell outside an inner shell of diameter 1/2 filled by two electrons.

More generally, seeking the number of electrons $Z$ required to fill a 2nd L-shell we compute the radius d of the 2nd shell to be
• $d_2 = \frac{1}{Z+2} + \frac{1}{Z}$
and thus the surface area $A$ of the 2nd shell to be
• $A = 4 \pi \times d_2^2$ .
In the Helium atom each electron can be viewed to fill a half-spherical surface and so $Z$ electrons of the 2nd shell of width $\frac{1}{Z}$ can be computed to require an area of
• $Z\times 2\pi\times\frac{1}{Z^2}$.
We thus obtain the following equation for a filled 2nd shell:
• $4\times\pi\times\frac{1}{(\frac{1}{Z+2} + \frac{1}{Z})^2} = 2\pi\times\frac{1}{Z}$
with approximate solution
• $2\times (\frac{2}{Z})^2 = \frac{1}{Z}$
that is $Z = 8$.

The same argument for a 3rd M-shell gives the equation
• $2\times (\frac{3}{Z})^2 = \frac{1}{Z}$
with solution $Z = 18$. More generally we get for an Nth shell the following number $Z = Z(N)$ of electrons:
•  $Z(N) = 2 \times N^2$.
It is thus possible to derive the basic formula $Z(N) = 2\times N^2$ for the atomic shell structure of the periodic table by a scaling argument starting from the 2-valuedness (or Zweideutigkeit) of the Helium electron structure.

This is an entirely different argument from the standard one based on the sequence of of excited states of Hydrogen and the notion of 2-valued spin (combined with Pauli's exclusion principle) offering the Zweideutigkeit.

The above argument is another brick in support of an electron structure without Pauli's exclusion principle, a principle which Pauli himself did not consider to be a law of physics but rather an ad hoc invention constructed to prevent the outer electron of Lithium to enter into the K-shell.

PS1 The simple shell model under study gives the following ground state energy $E$ for Neon with $N = 2$ and $2 + 8 = 10$ electrons:
• E = - 2 x 10/(1/10) + 1/(1/10)^2 + 1/(2/10) - 8 x 8/d_e + 8/2 x 1/(1/8)^2 + E_e
• d_2 = 1/10 + 1/16
• E_e = 8 x 7/2 x 1/d_e
• d_e = 2 x 1/8
where we estimate the kernel distance d_2 of the 8 outer electrons to be 1/10 + 1/2 x 1/8 and their average distance d_e to be 2 x 1/8. This gives
• E = - 95 - 393 + 256 + 112 =  - 119
where - 95 as the inner shell energy is to be compared with the observed - 94 and the total energy E = -119 with the observed - 129. With a small perturbation of d_2 and d_e we can hit this target.

PS2 For Carbon with 2 + 4 = 6 electrons we get similarly
• E = [- 2 x 6/(1/6) + 1/(1/6)^2 + 1/(2/6)] + [- 4 x 4/(1/6 + 1/8) + 4/2 x 1/(1/4)^2 + 4 x 3/2 x 1/(1/2)]
• = [- 72 + 36 + 3] + [- 55 + 32 + 12] = - 33 - 11 = - 44
to be compared with - 32.4 - 5.4 = - 39. Again the target can be hit by perturbation of the kernel distance of the 4 outer electrons and their average distance.

PS3 For Oxygen O with 2 + 6 = 8 electrons, we get similarly:
• E = [- 2 x 8/(1/8) + 1/(1/8)^2 + 1/(2/8)] + [- 6 x 6/(1/8 + 1/12) + 6/2 x 1/(1/6)^2+6 x 5/2 x 1/(1/3)]
• = [- 128 + 64 + 4] + [- 178 + 108 + 45] = - 59 - 25 =  - 84
to be compared with the observed - 59 - 16 = - 75.

## tisdag 13 augusti 2013

### Quantum Contradictions 8: Boron Mystery Resolved

With the simple model of the previous posts we get for Boron B with 5 electrons the following ground state energy E_B  in a configuration with 2 electrons in an innermost shell of radius d_1= 1/5 followed by 2 electrons in a next shell of radius d_2 = 1/3 and 1 electron in an outer shell of radius d_3=1:

• E_B = - 2 x 5/d_1 + 2 x 1/2 x 1/d_1^2 + 1/2d_1 - 2 x 3/(d_1+d_2)+ 2 x 1/2 x 1/d_2^2 + 1/2(d_1+d_2) - 1/(d_1+d_2+d_3) + 1/2 x 1/d_3^2 = - 24.0
to be compared with observed - 24.85. Again a good correspondence with a main difference coming from a too small first ionization energy of - 0.15 instead of observed - 0.3.

The same model with instead 3 electrons in a second and outer shell, gives larger energy.

The model thus suggests a 2 + 2 + 1 configuration with 3 shells rather than a standard 2 + 3 configuration with 2 shells, but its too small first ionization energy suggests that the single outer electron is about to enter the into the second shell. The next element in the table is Carbon C with 6 electrons and the question to be tackled in the next post is if the configuration is 2 + 2 + 2 or the standard 2 + 4?

PS1 Notice that the ionization energy of Boron is smaller than that of Beryllium in contradiction with the standard Aufbau principle, which signifies that Boron has an outer 5th electron subject to small attraction from the kernel which could possibly be represented as a 2 + 2 + 1 configuration.  With a slight decrease of the distance d_1 + d_2 + d_3 the 5th electron gets a correct ionization energy.

## måndag 12 augusti 2013

### Quantum Contradictions 7: Lithium Mystery Resolved

With the experience of the previous post we compute the ground state energy E_Li of Lithium as the energy of Li+ with two electrons filling a shell of radius 1/3 (= - 7.5 according to the previous post)  plus the ionization energy E_Li3 of a third electron in an outer shell of radius  d = 1 + 1/3 = 4/3, as follows:
• E_Li3 = - 1/d + 1/ 2 x 1/1 = - 0.25
• E_Li = - 7.5 - 0.25 = - 7.75
to be compared with the observed - 7.5.

We compare with the energy of a fictitious 1s3 ground state of Lithium, which is - 7.9 resulting from a radially symmetric (normalized) wave function with exponential decay exp( - 3r) in the distance r to the kernel for which the energy between two electrons is equal to 5/8 x 3 according to a standard text book computation.

The fictitious 1s3 state, which is viewed as unphysical because its energy is too small compared with the observed energy, in standard quantum mechanics is eliminated by Pauli's exclusion principle.

On the other hand, the state with two electrons filling an inner state of radius 1/3 with a third electron in an outer state of radius 4/3, has an energy in better agreement with observation and so may well be the true ground state.  It can be motivated by the fact that the width of the outer third electron equals 1 from being attracted by an Li+ ion and thus is unable to enter into the inner shell of radius 1/3.

Similarly we obtain for Beryllium Be by adding two electrons in an outer shell of width 1/2  to Be++ with energy - 14 from two electrons in an inner shell of radius 1/4 (previous post) according  to the following computation with d = 1/4 + 1/2 = 3/4:
• E_Be34 = - 2 x 2/d + 2 x 1/2 x 1/(0.5)^2 + 1/2d = - 2/3
• E_Be = - 14 - 2/3 = - 14.7
to be compared with observed - 14.8. The idea is again that the outer electrons are prevented from entering the inner shell because of too large width.

Next we will consider filling the outer shell with up to 8 electrons leading up to Neon and seek an explanation of the factor 8 as 2 x 4 with the 2 representing the 2 electrons of Helium, again without any Pauli exclusion principle (and thus no reference to spin).

### Quantum Contradictions 6: Helium Mystery Resolved

The text book ground state electron configuration of the Helium atom (with two electrons and a kernel with two protons and two neutrons) is described as a 1s2 state meaning that two spherically symmetric electrons with opposite spin have identical exponentially decreasing spatial wave functions and thus form the innermost K-shell, so to speak on top of each other without separation.

However, the energy of such configurations ranges from - 2.75 to - 2.84 (a. u.) depending on the rate of exponential decay, while the observed energy is - 2.903, and so the stated 1s2 ground state is not the true physical ground state.

To handle this contradiction, the 1s2 state is subjected to various perturbations of which those of Hylleraas from 1929 with a dependence on the interelectronic distance give the correct energy by allowing the complete wave function depending on 6 spatial dimensions to account for electron separation. The energy of the perturbed wave function with separated electrons is thus correct, but its form displaying the true electron configuration of Helium is kept as a secret to the text book reader to which only the non-true 1s2 state is presented.

The successs of the Hylleraas wave function results from allowing the two electrons to be separated and thus decrease the inter electron energy. In the simplest such model the two electrons are in opposite position at a distance of d = 1/2 (a.u.) for which configuration the total energy E can be computed as follows (as in the Rutherford model displayed above):
• E = - 2 x 2/d + 2 x 1/2 x 1/ d^2 + 1/2d = - 8 + 4 + 1 = - 3
with E the sum of kernel potential energy, electron kinetic energy and inter electron energy.

More generally, this simple model gives the following (surprisingly good) approximations of the ground state energies of the following 2 electron ions of n-electron atoms according to the formula
• E = - 2 x n/d + 2 x 1/2 x 1/ d^2 + 1/2d  with d = 1/n:
1. Li+ : - 7.5 (- 7.3)
2. Be++ : - 14 (- 13.7)
3. B+++ : - 22.5 ( - 22.2)
4. C4+ : - 33 ( - 32.8)
5. N5+ : - 45.5 (- 45.2)
6. 06+ : - 60 (- 59.8)
7. F7+ : -76.5 (- 76.3)
8. Ne8+ : -95 (- 94.9)
The Rutherford model thus appears to give a better description of the true ground state of Helium and 2-electron ions in correspondence with the Hylleraas wave function, than the postulated non-true quantum mechanical text book 1s2 state.

The challenge is now to explain why the ground state of Lithium is not 1s3 without resort to Pauli's exclusion principle. This will be taken on in a following post.

The basic idea is that the two separated electrons of Li+ filling a shell of diameter d = 1/ 3, will prevent a third electron of diameter 1 attracted by the net +1 charge of Li+ outside the shell to enter the shell, because the third electron is too wide and thus finds minimum energy in an outer shell. If this works out an explanation of the periodic table without resort to Pauli's exclusion principle may be possible.