The Ad Hoc Invention of Spin by Pauli

The basic idea of Quantum Mechanics QM of giving a two-valued quality of spin as spin-up and spin-down of electrons, came from the need to explain the shell structure of atoms with $2n^2$ electrons in shell $n=1,2,3,...$ with the factor 2 accounting for the two values of spin. 

This led to the Pauli Exclusion Principle PEP allowing two electrons with different spin to occupy the same region of space in an atom, but not three. The two electrons of the Helium atom could then have identical overlapping spatial presence in agreement with the principle of QM of electrons as being identical and interchangeable. Without PEP the Helium atom would appear as a mystery, as well as other atoms.

Pauli received the Nobel Prize in Physics in 1945 for his PEP, but his Nobel Lecture expressed deep dissatisfaction with PEP, because it was an ad hoc invention without convincing physics with only purpose to explain an observed apparent factor 2. But the Nobel Committee resisted and gave him the Prize despite his protests. 

RealQM is an alternative to QM based on non-overlapping one-electron charge densities where there is no role for spin and PEP. The two electrons of Helium will thus be separated into two half-spaces meeting at a common dividing plane with continuity of non-zero charge density. This separation introduces a two-valued quality of geometric nature: one electron occupying one half-space and the other electron occupying the other half space, thus as electrons with identity from spatial occupation. RealQM explains the factor 2 in the shell structure, without resort to spin.

The presence of spin in QM was one of the aspects which made Schrödinger protest against unphysical  aspects of QM and leave the field he had created in 1926 with his model for the Hydrogen atom with one electron, since the generalisation by Born-Heisenberg-Dirac to many electrons took unphysical form. It did not help that he received the Nobel Prize in Physics in 1933 for his model. 

Today QM is viewed to be settled once and for all, but the protests of Pauli and Schrödinger are still as alive as ever. RealQM follows the spirit of Pauli and Schrödinger. 

Pauli Exclusion Principle vs Periodic Table

In 1925 Wolfgang Pauli introduced a 4th quantum number into the budding new physics of Quantum Mechanics QM in order to explain the observed two-valued periodicity of the Periodic Table PT with $2\times n^2$ electrons in shell $n=1,2,3,...$ giving the sequence $2, 8, 18, 32,..$. 

Pauli was unhappy with his "two-valuedness" or as an ad hoc pick without physics. He was comforted a bit by Uhlenbeck and Goudsmit who named it spin with two values "spin-up" and "spin-down" still without physics. 

What emerged was Pauli's Exclusion Principle PEP stating that two electrons with different spin can occupy the same atomic orbital, but not with same spin, which quickly came to serve a fundamental role in QM. But Pauli was still unhappy with PEP when he for the “discovery of PEP” in 1945 received the Nobel Prize in Physics, because PEP lacked physics and so could be “discovered” as physics, just imagination. What in fact prevented two electrons with same spin to occupy the same orbital? It was like a law prohibiting same-sex marriage, unphysical and no longer valid.

Today PEP is enforced asking wave functions to be anti-symmetric motivated by a cocktail of Lorentz invariance, locality and stability taken from (relativistic) Quantum Field Theory QFT.

The argument chain is: 

• QFT implies anti-symmetry. 
• Anti-symmetry implies PEP.
• PEP implies two-valuedness/spin.
• Two-valuedness/spin is observed in PT. 
• Conclusion: QFT, antisymmetry, PEP and two-valuedness/spin is all confirmed. 

But the logic is the incorrect logic of confirming an assumption by observing a consequence, as noted by Aristotle. 

Back to the PT: The actual periodicity observed is 2, 8, 8, 18, 18, 32, 32,.. with repetition of periods, and this is not explained by PEP not really by QM either, as remarked by Eric Scerri as authority of PT.

How then to explain the actual periodicity? Let us take a look at Real Quantum Mechanics RealQM as an alternative to the Standard Quantum Mechanics StdQM of above with anti-symmetric wave functions.

In RealQM electrons appear as charge densities with non-overlapping supports and the arrangement of electrons in an atom becomes a packing problem. It starts with the two electrons of Helium packed to occupy two half-spheres meeting at a common separating plane. 

This arrangement serves as origin of two-valuedness with the next shell to be filled consisting of two half-shells each one allowing a natural division into $2\times 2$ subdomains, the next one into $3\times 3$ subdomains altogether forming the original sequence of periods $2\times n^2$. 

The period doubling can then be explained as the result of electron packing where the next shell to be filled after 2 and 8 is not wide enough to allow division into $3\times 3$ only $2\times 2$ et cet.

It appears thus that RealQM can give an explanation of the periodicity of PT based on solid physics of packing of electron charge densities. 

Pauli passed away in 1958, and since then there is nobody questioning PEP by asking for physics. Maybe there is still reason to do so? To explain the repeated periods of PT?

PS A $n\times n$ subdivision of a half-shell reflects eigenfunction configuration of a vibrating square membrane, which connects the the orbitals of StdQM given by the eigenfunctions of the Hydrogen atom. 

 

Conversation with Leading Chemist about RealQM

Quantum chemists idea of a H2O molecule in agreement with RealQM

In a discussion with a leading theoretical chemist about RealQM as a new model of atom physics in the form of classical continuum mechanics, I have met skepticism. I take this as evidence that RealQM offers a new perspective on Quantum Mechanics QM. RealQM may be wrong, but at least appears to offer something genuinely new, which after all may not be so wrong.  

Of course, something new about atom physics proposed by a mathematician/civil engineer with special interest in finite element methods for continuum mechanics, will be viewed with deep skepticism by a quantum chemistry expert educated in Standard Quantum Mechanics StdQM. 

RealQM describes atoms/molecules in terms of atomic kernels surrounded by non-overlapping one-electron charge densities interacting by Coulomb potentials with electron densities meeting with a Bernoulli free boundary with continuity and zero normal derivate (Neumann natural boundary condition) as a distinct new feature. That is all: Coulomb and Bernoulli as a parameter free model + a Laplacian giving an electron "kinetic energy" in StdQM. 

RealQM can be viewed as a form of structural mechanics ideally suited for finite element methods. See this talk on Structural Mechanics of Atoms and Molecules with kinetic energy appearing as a form of elastic compression energy as measure of density gradient.

The discussion came to land on in particular Pauli's Exclusion Principle PEP, as the cornerstone of StdQM, which states that 

• No two electrons with the same spin can occupy the same position in space.                       (1)
• Electrons with different spin can do so, thus at most two since spin as only two values.   (2)                                             

RealQM can be viewed to satisfy (1) since electron charge densities do not overlap, while (2) is empty because RealQM does not involve spin as if all electrons have the same spin. See also this post. 

But this was not appreciated by the leading chemist, who claimed that RealQM does not include the Pauli Repulsion Force PRF, which is supposed to be the force required to guarantee satisfaction of PEP, and so RealQM must be fundamentally wrong! Any physics without PRF must be wrong. Period.

Is this a valid conclusion? What is in fact PRF? Is it a new force beyond the four fundamental forces of physics? ChatGPT says no; PRF is not a new force but an emergent phenomenon required to prevent two electrons with the same spin from occupying the same position. 

In structural mechanics two elastic bodies are prevented from occupying the same portion of physical space by meeting with contact forces, which ultimately are Coulomb forces. Non-penetration is then realised by some form of elastic spring force acting on the boundary as a real physics. In contrast, rigid bodies have no boundary springs and in fact are unphysical in the sense that non-penetration is a mathematical stipulation without means of physical realisation.

Similarly, PEP is a stipulation as mathematics without real physics, and the stipulation is fulfilled by assuming that the wave functions of StdQM are anti-symmetric as a purely mathematical property. The mathematical stipulation is satisfied by imposing mathematical structure, but the physics is missing. Anti-symmetric wave functions is a human construct without physics.

PRF is not a new physical force but an emerging fictitious force appearing in the mathematics, like fictitious forces in classical physics emerging from choice of coordinate system. 

In RealQM there are no fictitious forces, only Coulomb forces, but what about the Bernoulli free boundary condition? Well, continuity is realised by moving the free boundary and the Neumann condition comes from energy minimisation. No PRF is needed and no PRF emerges. 

RealQM thus has a mathematical form without the fictitious PRF emerging in StdQM. Is this a reason to reject RealQM? It is like rejecting the choice a non-rotating coordinate system in classical mechanics without fictitious forces. Why do that?  

I will seek to resume the discussion with the leading chemist to see if RealQM after all could have a chance in an upcoming review process.   

RealQM offers a new methodology for computational simulation of atoms and molecules in the spirit of classical structural mechanics with orders of magnitude smaller cost than computational realisations of StdQM as the current state of the art. 

Let me recall the following reaction to something new by established science, which I have met personally: If it is correct, then it is not new, an implication which can also be expressed: If it is new, then it is incorrect. In other words, anything which is new and correct can be refuted. At least in a mature field like StdQM without innovation since long with the last one in the form of DFT prepared in the 1960s.

Pauli Unhappy with his Exclusion Principle

A corner stone of standard Quantum Mechanics stdQM is the Pauli Exclusion Principle PEP stated by Pauli himself above. Pauli was awarded the 1945 Nobel Prize in Physics for the 

• discovery of the Exclusion Principle, also called the Pauli Principle.

In his Nobel Lecture Pauli recalls the basic problem he resolved by PEP:

• The series of whole numbers 2, 8, 18, 32... giving the lengths of the periods in the natural system of chemical elements, was zealously discussed in Munich, including the remark of the Swedish physicist, Rydberg, that these numbers are of the simple form $2*n^2$ for $n=1,2,3...$.
• The question, as to why all electrons for an atom in its ground state were not bound in the innermost shell, had already been emphasized by Bohr as a fundamental problem.

But Pauli did not believe that PEP was a scientifically convincing resolution: 

• 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.

Pauli would probably say the same today: PEP is a mystery appearing as an ad hoc explanation of the fact that electrons in atoms organise into shells according to $2*n^2$ for $n=1,2,3...$ and so form the periodic table. The factor 2 would then be an expression of the 2-valuedness of spin. 

RealQM presents a different model of an atom where electrons appear as non-overlapping charge densities  which occupy a certain volume of space and the shell system of the periodic table resolves an energy minimisation packing problem. The 2 electrons of a Helium atom are in RealQM separated by a plane which gives the 2-valuedness a direct geometric meaning, which is passed on to atoms with more electrons. In RealQM there is no need of a PEP. This would have made Pauli happy and maybe another Nobel Prize.  

Pauli and Bohr eagerly studying physics of a spinning top.