Triumph of Mathematics: Newton and Schrödinger

In the previous post we discussed the danger of being carried away by mathematical beauty into the fantasy land of modern physics in the form of Standard Quantum Mechanics, Standard Model and String Theory. Compare with Sabine Hossenfelder's Lost in Math and Max Tegmark's Mathematical Universe.

Let us recall the greatest successes of mathematical thinking about physics: 

• Newton's Theory of Gravitation                                    (N)
• Schrödinger's equation for the Hydrogen Atom.          (S)
• Maxwell's equations for electromagnetics.                   (M)

Newton's Theory of Gravitation is mathematics because it is captured in the following equations

•  $\Delta\phi =\rho$                                                         (N1)
• $F=-\nabla\phi$                                                         (N2)

where $\rho$ is mass density, $\phi$ is gravitational potential and $F$  is gravitational force all depending on a Euclidean space coordinate $x$ with corresponding Laplacian $\Delta$ and gradient $\nabla$ differential operator. 

These equations can be derived mathematically from a principle of conservation of energy and force. A point mass at $x=0$ then comes with a gravitational potential $-\frac{1}{\vert x\vert}$ and gravitational force scaling with $\frac{1}{\vert x\vert^2}$ as Newton's power two law. 

Newton's Model (N1) + (N2) capturing all of celestial mechanics through the differential operators $\Delta$ and $\nabla$, is the most formidable success of mathematical thinking all times. There was simply no  alternative for a celestial Creator, which allowed humans to get insight into the creation process by pure mathematical thinking. Amazing, right?  

Schrödinger's equation formulated in 1926 describes the charge density $\psi$ of a Hydrogen atom as the minimiser of the total energy $E_{tot}=E_{kin}+E_{pot}$ with 

• $E_{kin}=\frac{1}{2}\int\vert\nabla\psi\vert^2dx$
• $E_{pot}= -\int\frac{\psi^2}{\vert x\vert}dx$ 

under the side condition 

• $\int\psi^2dx =1$, 

as the solution of the eigenvalue problem 

• $-\frac{1}{2}\Delta\psi +V\psi = E\psi$                                  (S)

where $V(x)=-\frac{1}{\vert x\vert}$ is the electric Coulomb potential of the proton kernel, and $E$ is an eigenvalue. The connection between Coulomb potential and charge density is the same as between gravitational potential and mass density.

This model is not derived from basic principles like Newton's model, but is itself a first principle.

The success is that its eigenvalues match the observed spectrum of the Hydrogen atom to high precision, with a smallest eigenvalue $-\frac{1}{2}$ as the minimum of  $E_{tot}$ attained by a charge distribution finding the optimal combination of 

• being compressed around the kernel making $E_{pot}$ small
• paying a compression cost of $E_{kin}$

and so taking the simple form 

• $\psi (x) = \exp (-\vert x\vert )$

as a density decaying exponentially with the distance the kernel. 

Schrödinger's model of the ground state of a Hydrogen atom with a proton kernel surrounded by an electron charge density, can be compared with Newton's model of a Sun surrounded by an orbiting planet finding a balance of kinetic energy of motion and gravitational potential energy. 

The "compression energy" $E_{kin}$ of the electron then corresponds to the kinetic energy of the planet scaling with $\frac{1}{2}p^2$ with $p$ momentum . This gives a formal connection between $\nabla$ and $p$ as motivation of the name kinetic energy given to $E_{kin}$. 

The electron of a Hydrogen atom cannot be a point particle orbiting the kernel like a planet orbiting a sun, because a moving electron would radiate energy and so collapse into the kernel. The electron thus must have extension is space and it makes sense to associate $\nabla\psi$ with some form of "compression" or "strain" as in an elastic body. 

It is thus possible to argue that (S) is a most natural model of a Hydrogen atom as a kernel surrounded by an electron charge density finding an optimal solution to small potential energy at a compression cost. One can argue that the Creator had no choice when complementing celestial mechanics with Hydrogen atoms. 

Sum up so far: The macroscopic world of mechanics captured by (N) and the microscopic world of the Hydrogen atom captured by (S) as the result of pure mathematical thinking without need of observational input (Kant a priori), is a formidable success. Add to that (M).

But the success is not complete as concerns Schrödinger's equation since only the Hydrogen atom is covered. What then about a Schrödinger equation for atoms or molecules with many electrons? 

This was the question confronting Schrödinger and the world of modern physics in 1926 and the route that was taken has come to form modern physics all the way into our time. The idea was again to rely on pure mathematical thinking and so generalise (S) formally from one to many electrons simply by adding a new spatial coordinate for each new electron to arrive at a Schrödinger equation for an atom/molecule with $N$ electrons taking form in $3N$ spatial dimensions. No physics was involved in the generalisation, only mathematical formality. The mathematician von Neumann took control with his Mathematical Foundations of Quantum Mechanics 1932 leading physics into a strange universe of wave functions evolving in Hilbert spaces according to von Neumann Postulates (without physics), and physicists had no choice but to follow. 

The resulting Schrödinger equation was a multi-dimensional monster which did not describe any physics and in addition showed to be uncomputable except for very small $N$. The equation was easy to write down on a piece of paper just increasing the number of spatial dimensions, but real physics in three space dimension was present only in the case $N=1$. To save the situation the model was given a statistical interpretation by Max Born as all possibilities instead of particular realities as Standard Quantum Mechanics StdQM. Schrödinger never accepted StdQM.

RealQM offers a different generalisation in the original spirit of Schrödinger in the form of non-overlapping electron charge densities interacting by Coulomb potentials each coming with compression cost in total energy minimum of ground state. RealQM is a deterministic model of physics in three space dimensions and as such readily computable. 

Final sum up: (N) and (S) represent astounding complete successes of pure mathematical thinking, while StdQM appears as a monumental failure of too much belief in power of mathematical formality. RealQM gives hope to continued success of more modest mathematical thinking. The idea of a mathematical universe is great but should be combined with modesty according to the Greeks.

Paul Dirac claimed that very advanced mathematics was required to design the universeas, maybe as an excuse that his equation for the electron was very complicated beyond human understanding leading to Dirac's famous "Shut up and calculate". 

I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space any more.  (von Neumann)

Mathematical Idealism of Quantum Mechanics 1926-2025

Mysterium Cosmographicum by Kepler (1596) describing the Solar system as being based on five Platonic solids as perfect expression of mathematical idealism. 

This is a follow up of the post on The Tragedy of Schrödinger and His Equation seeking the origin of the present crisis of modern physics disputed by few, as the departure from classical physics taken in 1926 when generalising Schrödinger's equation for the Hydrogen atom with one electron  to atoms with more than one electron as the foundation of modern physics. 

While Schrödinger's equation for one electron had the form of well understood classical deterministic continuum mechanics in 3 space dimensions, the Schrödinger equation for $N>1$ electrons took a completely new form as a linear differential equation in $3N$ spatial variables never seen before, which represented a formal mathematical generalisation which was easy to write down, but did not have any physical meaning. 

Mathematics thus presented a formal generalisation of Schrödinger's equation from one to many electrons, which modern physicists felt obliged to accepts because it looked so neat. But the generalisation was purely formal mathematical and so the physics had to be put in afterwards. That showed to be very difficult and so developed into the basic trauma of modern physics. 

The leading physicist Max Born took on the challenge and came up with the idea of giving solutions to Schrödinger's equation a statistical meaning thus changing atom physics into a casino of electrons without physics.

But physicists unable to come up with something better, were stuck with the linear $3N$ dimensiosnal Schrödinger equation in the hands of mathematicians like von Neumann which has taken over the scene as Standard Quantum Mechanics StdQM. 

The trouble with mathematics is that without proper understanding trivialities can be mistaken to be deep truths. Physicists were thus blinded by the mathematical formalism and the lack of physical meaning was rationalised as being natural because an atom was not really of this world, just a possibility in a world of statistics. 

RealQM offers a different generalisation of Schrödinger's equation for the Hydrogen atom, into a non-linear system of one-electron equations in the form of classical deterministic continuum mechanics with direct physical meaning in terms of Coulombic interaction between non-overlapping one-electron charge densities and atomic kernels. 

Paul Dirac (Nobel Prize in Physics 1933) was one of the key leading physicist who was carried away by the beauty of the Schrödinger equation for any number of electrons:

• The general theory of quantum mechanics is now almost complete, the imperfections that still remain being in connection with the exact fitting in of the theory with relativity ideas. 
• The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.
•  It there­ fore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.

This was an extension into the microscopic world of atoms of Laplace's grand vision for a macroscopic world fully described by mathematics: 

• We may regard the present state of the universe as the effect of its past and the cause of its future.
•  If an intelligence were to know, at a given instant, all the forces that govern nature and the positions of all its components, and if it were vast enough to analyze this data, then it could comprehend the entire past and foresee the entire future. 
• For such an intelligence, nothing would be uncertain; the future, just like the past, would be fully determined.

The extension represented a heroic break from classical physics of macroscopics into the modern physics of microscopics, but it came with severe caveats: Solutions of Schrödinger's  equation 

• lack physical meaning in a classical sense
• are uncomputable for many electrons. 

Great effort has been allocated to come to grips with these aspect for now 100 years and will certainly continue since no real relief has been attained. It is here RealQM comes in as a fresh restart from classical continuum mechanics. RealQM is a computable and parameter free model of real physics in 3d space and time. 

We may compare StdQM with the Pythagorean view of a World built from relations between natural numbers like the harmonics of the major musical scale created from overtones of a vibrating string with frequencies relating like 3 to 2 for the fifth tone. This was a grand vision based on a formal mathematical idea of the world as simply relations between natural numbers. The physics had to be put in afterwords which worked well for the vibrating string but in general new physics had to be brought for each new case. And then came the shock that ended the Pythagorean Society: The length of the diagonal of a square with side 1 is $\sqrt 2$, which is not a quotient of two natural numbers. Pythagoras grand formal mathematical idealism without physics did not capture real physics. Nor did the Platonic solids.

The same scheme is repeated with StdQM: A mathematical model is created from grand formal mathematical idealism without physics and the physics has to be put in afterwards in the form of new ad hoc ingredients such as

• electron spin
• Pauli Exclusion Principle
• Aufbau Principle
• antisymmetric wave functions 
• Hund's Rule
• Madelung Rule
• Octet Rule
• Spin-Orbit Coupling
• Russel-Saunders Coupling
• jj Coupling
• Lande's Interval Rule
• Selection Rule for Electronic Transitions
• Zeeman Effect
• Stark Effect
• Exchange Interaction
• Hyperfine Structure

while the 1926 Schrödinger equation itself remains intact. The idealism is even more extreme in the Standard Model and of course String Theory. 

Another aspect of the idealism of StdQM comes to expression as the now accepted factum that building a quantum computer by StdQM promised to be possible, lacks realism.

Orthohelium Alternative Electron Configuration

In the previous post we let RealQM compute the first line in the spectrum of Helium from ground state as Orthohelium assuming that one electron is excited from 1S to 2S and obtained an energy of 0.728 Hartree in accordance with observation as a line easy to observe as the difference between $-2.175$ and $-2.903$ Hartree. 

We compare assuming instead excitation to the 2nd eigenvalue of Schrödinger's equation for Helium in the form given by RealQM. You can run the code here. We obtain the same energy of 0.728 Hartree in agreement with observation. 

RealQM thus gives the same first line in the spectrum of Helium with two different electron configurations.

Note that in Standard Quantum Mechanics StdQM the first exited states of Helium comes on two forms, as Orthohelium with the the two electrons having the same spin as a triple state, and Parahelium with different spin as a singlet state. RealQM corresponds to Orthohelium since in RealQ there is only same spin. 

Orthohelium is much easier to observe (more stable) than Parahelium and so appears to represent more solid physics. 

The fact that RealQM gives correct 1st line of Helium as Orthohelium with two different electron configurations is another piece of evidence that RealQM captures physics. Both configurations have in the RealQM electric dipoles varying over time and so radiate, with the 1S to 2S weaker.

The electron configuration of Orthohelium in StdQM is very complex, and as such probably with less physics. 

Recall that DFT being focussed on ground states has to struggle with excited states in heavy software. It may show to be remarkable that the 3-line code of RealQM can outperform DFT.

 

RealQM Spectrum of Helium: 1st Excited State as Orthohelium

We now explore the spectrum of Helium delivered by RealQM. The spectrum of an atom primarily reflects energy differences between the ground state and excited states, but also between excited states.

Standard Quantum Mechanics StdQM presents the first exited state of the Helium atom with two electrons to be a singlet state with the two electrons having opposite spin named Parahelium, but there is also a triplet state with the electrons having same spin named Orthohelium. 

The first line in the spectrum of Helium (smallest energy jump from ground state) corresponding to an energy of $-2.175$ Hartree is that of Orthohelium, compared to $-2.903$ for the ground state, while Parahelium has 0.03 higher energy of $-2.145$ Hartree. 

Let us see what RealQM delivers. Since in RealQM all electrons have the same spin, RealQM connects to Orthohelium giving the first line in the spectrum. We let RealQM model this state with one electron around the kernel and the other electron in an excited state outside. You can here run the code to find that RealQM gives an energy of $-2.175\pm 0.005$ Hartree depending on iteration stop criterion. 

We thus see that RealQM gives a result in agreement with observation of the 1st line in the spectrum of Helium in the form of a very simple arrangement of the two electrons: An inner electron around the kernel and an outer electron around the kernel + inner electron. Run code to see. The corresponding triplet state of StdQM is very complicated:

Orthohelium corresponds to the first line in the Lyman series for Hydrogen (with drop of wave length from 122 nm to 62.6 nm), with details on further lines to come.

PS The ground state of Helium with its two electrons overlapping with opposite spin in StdQM, is in RealQM modeled with the electrons split into halfspaces without overlapping. See this code. The transition to excited state in RealQM is thus from a state of half space split electrons into a state with spherically symmetric electrons, one inner and one outer, which appears to require a quite precise excitation input.

Difference Between Principles and Laws of Physics

The Standard Model of Particle Physics as Epistemology          

In physics there are Principles such as

1. Principle of Relativity
2. Principle of Equivalence
3. Principle of Conservation of Energy
4. Pauli Exclusion Principle 

And there are Laws such as

• Newton's Law of Gravitation
• Coulombs Law
• Gauss's Law
• Faraday's Law
• Ampère's Law
• Ohms Law 
• Hooke's Law
• Fourier's Law
• Boyle's Law
• Dalton's Law
• Ideal Gas Law
• ...

We see that there appears to be many more Laws than Principles. What in fact is the difference between a Principle and a Law of physics?

We may have an intuitive idea of a physical laws as describing a relation between physical quantities like Newton's Law of Gravitation expressing a relation between matter/mass and gravitational force, or the Ideal Gas Law expressing a relation between pressure, density and temperature in a gas. Physical laws typically involve parameters or constants as numbers such as the Gravitational constant $G$ and the gas constant $\gamma$. Laws express ontology of physical reality of what is.

But what about Principles? Inspecting the above list of Principles we meet a different situation, which rather is an expression of agreements between scientists how to view physics, that is as man-made epistemology,  Let us go through the list of Principles:

1. Principle of Relativity: Laws of physics are to have the same form in all coordinate systems.
2. Principle of Equivalence: Inertial mass is the same as gravitational mass.
3. Principle of Conservation of Energy: Energy can be transformed but total energy is constant.
4. Pauli Exclusion Principle: No two electrons with same spin can occupy the same position. 

We see no parameters and make the following observations: 

1. Principle of Relativity: Mathematical formality. Absurd or trivial. Your pick.
2. Principle of Equivalence: Agreement to relate inertial mass to gravitational mass as primordial,
3. Principle of Conservation of Energy: Agreement that nothing comes for free. Except Big Bang.
4. Pauli Exclusion Principle: Agreement to justify Standard Quantum Mechanics StdQM.

The picture seems pretty clear: Physical Laws express physics as ontology, while Principles are man-made agreements as epistemology. Another difference concerns satisfaction: A Law of Physics can be more or less true/precise as a being quantitative, while a Principle is absolute qualitative.

There are also Postulates which have the same nature as Principles. The basic Postulate of StdQM is that a physical system is described by a wave function satisfying a Schrödinger equation, which is not derived from Laws of physics. Another basic Postulate is Pauli's Exclusion Principle. 

Connecting to the previous post on StdQM, we understand that the name Pauli Exclusion Principle indicates that physicists do not view it as a law of physics to be respected by physical electrons, but rather as an agreement among physicists how to make sense of StdQM as epistemology.

On the other hand RealQM, as an alternative to StdQM, is based on Coulomb's Law for charge densities as ontology.

Recall that there is a fundamental difference between ontology of physics as what exists (without presence of humans), and epistemology as what humans say about physics. 

In classical deterministic mechanistic physics there was a clear distinction between ontology as mechanics without humans and epistemology as human observation and understanding.

Einstein's Special Theory of Relativity mixed up physics and human observation, which became manifest in StdQM with the Observer taking an even more active role through measurement deciding physics. Today the confusion is total as an expression of the crisis of modern physics. 

In Ancient Greek physics ontology and epistemology was deeply intertwined in a battle between idealism and materialism, and the Scientific Revolution had to wait for 2000 years to emerge from instead a fruitful cooperation of materialism and idealism in the form of Calculus. Today we are full swing back to idealism in the form of StdQM and String Theory.

Human Rights Principles have the form of agreements such as Universal Declaration of Human Rights adopted by UN General Assembly in 1948. Respect is not guaranteed.

Electron Spin: Weak as Physics Strong by Theory

This is a continuation of previous post. ChatGPT informs that atoms can be divided into diamagnetic with paired electrons (no net spin) and paramagnetic with unpaired electrons (with net spin) with opposite non-zero reactions to a magnetic field. According to Standard Quantum Mechanics StdQM. 

But these effects/reactions are incredibly small and can only be observed in laboratory settings with very strong magnetic fields of 1-20 Tesla, while the Earth's magnetic field is about $10^5$ times weaker. The magnetic field gradient in the Stern-Gerlach experiment viewed as evidence of electron spin, is very steep about 10-100 Tesla/m. 

We learn that electron spin is manifested as a very small magnetic effect. Yet electron spin serves a fundamental role in StdQM by dividing matter into bosons like photons and fermions like electrons with different spin characteristics. 

In particular, Pauli's Exclusion Principle PEP for electrons/fermions is formulated in terms of spin allowing two electrons with different spin to share common space, which is forbidden/excluded for same spin. 

PEP serves a fundamental role in StdQM to theoretically explain the periodic table and chemical reactions. StdQM theory collapses without PEP. 

Electron spin has a very weak magnetic effect, much weaker than Coulombic attraction/repulsion between electric charges, but yet it governs the world of atoms and molecules according to StdQM theory.

We are thus led to the following apparent contradiction:: 

• Electron spin has a very weak physical effect, which according to theory is very strong.  

We compare with RealQM where electron spin has no role to play which goes along with a very small physical effect. There are also forms of DFT where spin has a minor if any role. 

Recall that science explaining a very large effect of something very small, like a tornado in Mexico from a butterfly flap in Amazonas, has a very difficult task. It requires a very precise mathematical model allowing to accurately simulate the far away effect of a butterfly flap to identify it as the true origin of the tornado. This is an impossible task. To use electron spin to explain the world of atoms and molecules faces a similar difficulty.

The Mysterious Two-Valuedness of Spin Quantum Mechanics

Once Schrödinger in 1926 had formulated his partial differential equation for the Hydrogen atom with one electron with an eigenvalue spectrum in full agreement with observation, the next challenge was the Helium atom with two electrons: How to generalise from one to many electrons? 

The way to to do this was not clear and the simplest option was followed: Make a formal mathematical generalisation with a stroke of a pen, just add a new 3d spatial coordinate for each new electron to form Schrödinger's multi-dimensional wave equation in $3N$ spatial dimensions (plus time) for an atom/molecule with $N$ electrons, and then seek to live with that equation. The trouble still haunting modern physics is that the physical meaning of Schrödinger's equation is still hidden if any at all, despite intense efforts over 100 years.  

For the Helium atom with two electrons this gives a six-dimensional wave equation, with the ground state appearing as having minimal energy. But what is the electron configuration of that state? The idea then came up, from the success for the Hydrogen atom, to view the ground state of Helium to be composed of two spherically symmetry Hydrogen-type wave functions with the electrons so to speak on top of each other.  To make that possible in view of the Coulomb repulsion between electrons, Wolfgang Pauli suggested to assign the electrons different values of "spin" as "spin-up" and "spin-down" and then postulate a Pauli Exclusion Principle PEP proclaiming that two electrons with different "spin" can share spatial domain. 

The ground state of Helium was thus declared to be a $1S^2$ state with two identical spherically symmetric electron charge distributions with different spin, which gave a rough fit with observation. 

Pauli himself viewed PEP to be a mistake, but the physics community happily adopted the idea of a two-valuedness of quantum mechanics in the form of "spin-up" and "spin-down", which is now firmly implanted in Standard Quantum Mechanics StdQM.

In RealQM, as an alternative to the formal generalisation of StdQM into many electrons, the two-valuedness of Helium takes a different form as a split of the two electrons to be restricted to half-spaces meeting at a plane through the kernel. This a physical split of charge distribution to be compared with the formal split of StdQM into "spin-up" and "spin-down".

In RealQM the separating plane gives the charge distribution a direction in space, which is lacking with only "spin-up" and "spin-down".

The previous post takes up possible physical effects of the RealQM electron split in the form of diamagnetism. 

RealQM presents a physical origin to the observed two-valuedness of He, which is independent of any PEP. There is no PEP in RealQM because it serves no need, and so can be dispensed. 

Pauli would have been very satistfied with this message, but quantum mechanics has continued to cling to PEP as the correct expression of two-valuedness. 

Since all atoms have an innermost shell of two electrons, and maybe also an outermost, RealQM for any atom carries a form of two-valuedness, which is not based on two-valued spin.

RealQM with electrons split into two half-spaces gives a ground state energy which fits better with observations than the $1S^2$ configuration with split spin. Does that say anything?

 

Why is a Helium Atom DiaMagnetic?

There seems to be a consensus of Standard Quantum Mechanics StdQM, supported by observation, that the Helium atom He is diamagnetic and so can react to an external magnetic field, even though its $1S^2$ spherically symmetric ground state has zero intrinsic magnetic moment. 

To explain the apparent contradiction, the idea of StdQM is to say that an external magnetic field can induce a magnetic moment by somehow changing He from its ground state with zero magnetic susceptibility into a new ground state with non-zero magnetic susceptibility. The physics of this change of ground state is however not well explained.

In RealQM, as a new alternative to StdQM, the ground state of He consists of two half-lobes of electron charge density meeting at a separating plane through the kernel, which forms a non-spherical symmetric charge distribution with separation in the normal direction to the plane as asymmetry, which can generate a non-zero electric dipole moment. 

The next question in the optics of RealQM is if the Helium atom with non-zero electric dipole moment can be affected by a magnetic field?  

The answer is yes, if the charge distribution with electric dipole moment is rotating, then alignment with an external magnetic field can occur as an expression of diamagnetism. 

It is thinkable that the the half-lobes of charge density of He according to RealQM are rotating around an axis parallel to the separating plane and so give an effect of diamagnetism.

It thus seems possible that RealQM can explain the diamagnetism of He in ground state from asymmetric charge distribution with electric dipole. 

In StdQM He in ground state has a spherically symmetric charge distribution and the explanation of diamagnetism appears more farfetched.

Check out asymmetry of He in ground state running this code. 

Stern-Gerlach Experiment with He?

The Stern-Gerlach experiment with Silver atoms with one outermost $1S$ electron is supposed to be the definite experiment showing that electrons have spin in two-valued form as $+\frac{1}{2}$ and $-\frac{1}{2}$.

Standard Quantum Mechanics StdQM predicts that a noble gas like Helium in ground state with its two electrons of different spin in a $1S^2$ spherically symmetric configuration with spin $0=\frac{1}{2}-\frac{1}{2}$, will not give any result in a Stern-Gerlach experiment. 

StdQM theory has been so convincing that no Stern-Gerlach experiment with a noble gas is reported in the literature. ChatGPT informs that if such an experiment gave a positive result like with Silver, then the whole theory of StdQM would have to be rewritten. 

But no experiment like that has evidently been performed. Why? That would be a good test of the validity of the theory, right?

If we now turn to RealQM, we have that the two electrons of Helium in ground state occupying two half-spaces separated by a plane through the kernel with a combined electron charge distribution, which is not spherically symmetric with charge concentration on both sides of the plane with polarisation effect. 

It is thus according to RealQM thinkable that Helium could give a positive result in the Stern-Gerlach experiment. What do you think?

Does Helium He Form Molecule He2?

This is an update of previous post on the same theme.

The Hydrogen atom H with one electron forms a molecule H2 with substantial binding energy of 0.17 Hartree.  

What then about the Helium atom He with two electrons? We know from school that He is viewed to be a noble gas and as such would not be expected to form a He2 molecule with any binding energy. 

Experiments gives clear evidence of existence of H2 but not so of He2. 

Theory in the form of Standard Quantum Mechanics StdQM gave no clear answer for a long time, but in 1997 computations were published by Komasa and Rychlewski showing very weak binding energy (0.00004) at a kernel distance of 5.6 Bohr (compared to 1.4 Bohr for H2), which must be the same as no-binding.

Testing RealQM on a coarse $50^3$ mesh gives (run this code and vary distance D) results, which are qualitatively in accordance with the above results by StdQM, in the sense that a no-binding is indicated by the following numbers with D kernel distance, $E$ total energy and $\Delta E$ energy difference in Hartree with positive value indicating very weak no-binding 

• D=12      $E=-5.806$
• D=9.6     $\Delta E = 0.013$
• D=8        $\Delta E = 0.014$
• D=6.4     $\Delta E = 0.015$
• D=4.8    $\Delta E = 0.021$
• D=3.2    $\Delta E = 0.043$

These values are to be compared with $\Delta E = -0.17$ with strong indication of bonding for H2 at distance 1.4 Bohr. Compare with this code for He atom on $100^3$ mesh.

Both StdQM and RealQM thus indicate no-binding of two He atoms to He2 molecule at distance smaller than 12 Bohr. 

On the other hand He can form weak He2 Dimer binding by van der Waals forces at a much bigger distance of 100 Bohr. 

RealQM does not include effects of Pauli repulsion, since there is no use of a Pauli Exclusion Principle for non-overlapping one-electron densities as the building blocks of RealQM. The above results by StdQM contradict strong presence of Pauli repulsion for He2.

The reason RealQM gives substantial binding for H2 but not He2, is that the two electrons of He occupy different half spaces separated by a plane through the kernel, and with these planes perpendicular to the axis between He kernels, the two outer electrons are prevented from entering the region between the kernels to form a bond.  

Another aspect is that the kernel repulsion range for He is about 4 times that of H, because of scaling with charge squared, while the electron range is smaller for He than for H, which means that decrease of energy by electron-kernel attraction with decreasing kernel distance is countered by increase of kernel-kernel repulsion with no net decrease of total energy and so no-binding for He. 

RealQM thus appears to capture the no-binding of He2 in a qualitative sense on a coarse mesh. If this is really the case, it is remarkable. 

PS The standard explanation that noble gasses like He do not want to form molecules is that such atoms have an outer "full shell" which does not invite to either covalent or ionic bond, which may have some truth but also is vague.