Standard Quantum Mechanics as Classification without Physics vs RealQM

This is a continuation of the previous post but can be enjoyed independently.

Science and religion both arise from human minds seeking to find mental interpretations of the Word as the creation of superhuman mind. Quantum Mechanics QM as the mechanics of atoms and molecules came out from a perceived shortcoming of classical Newtonian physics in the late 19th century, so immensely successful in describing the macroscopic world, to capture the microscopic world of atoms and molecules. 

The German physicist Erwin Schrödinger in 1926 took the first step out of a deadlock into the modern physics of QM by formulating a mathematical model of the Hydrogen atom with one electron surrounding a proton kernel in the form of an eigenvalue problem for a partial differential equation for a negative charge density subject to Coulomb attraction from a positive kernel. 

The success was complete since the  eigenvalues precisely agreed with the observed spectrum of excited states of the Hydrogen atom with corresponding eigenfunctions as wave functions representing vibrational spatial modes of the electron taking the following form:

to be compared with those of e g a circular membrane    

The complete success of the Schrödinger equation capturing the spectrum of the Hydrogen atom with one electron demanded generalisation to atoms with more than one electron, which was accomplished  by formally adding a new set of 3d coordinates for each electron into a linear Schrödinger equation (S) in $3N$ spatial dimensions for an atom/molecule with $N>1$ electrons like a Gold atom with 79 electrons in $237$ spatial dimensions, as a partial differential equation of a completely new form with solutions named wave functions for which the physics had to be invented as probabilities of electron configurations. 

(S) has come to serve as the foundation of the modern physics in the form of Standard QM StdQM filling text books of modern physics. 

The first task for (S) was to explain the "Aufbau" of the  Periodic Table in terms of wave functions and then the following strange idea came up: 

• Describe ground state electron configurations of atoms/molecules with $N>1$ as linear combinations of excited states of the one electron of the Hydrogen atom.                                  

To see the strangeness compare with an idea of

• describing a complex system as a copy of a simple component building the system
• preformatism of the 17th century as a tiny human ("homunculus") inside the egg of a female. 

We understand that this is illogical: A complex system cannot be copy of its simple parts. Nevertheless it serves as a fundamental principle of StdQM: Excited states of a Hydrogen atom describe all atoms/molecules. It defies logic but is state of the art, still 100 years after conception. It does not help to recall that the eigenfunctions of the Hydrogen atom form a complete orthonormal system and so can describe anything. 

The fact that StdQM comes out from a formal extension from one to many electrons, means that StdQM appears as an (ad hoc) classification system rather than as model of real physics. This is evident by looking at the rich classification imposed by StdQM culminating in the Standard Model:

• symmetric and antisymmetric wave functions as bosons and fermions
• electrons with spin-up and spin-down
• paired and unpaired electrons 
• Hund's rule, Madelung rule, octet rule
• s, p, d, f states in atoms 
• quarks, leptons, gluons, weak force, strong force...

It is natural to compare with the plant classification system of Linné (18th century) based on the number of stamens and pistils in flowers, which is artificial and not reflective of natural relationships, but anyway still is used.  

RealQM offers an alternative to StdQM as a atom/molecule model based on Coulomb interaction between atomic kernels and non-overlapping electron densities as a natural system without need of elaborate classification. 

We can put StdQM ve RealQM into a broader perspective of idealism and realism with StdQM imposing atomic form and RealQM uncovering natural atomic form. RealQM gives a different rationale of the Periodic Table as an electron density packing problem. 

If you force a metal through a square die in an extrusion process, it will come out with quadratic cross section. If you force Hydrogen eigenfunction form upon general atomic states, everything will look Hydrogenic but you will violate natural physics.  

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. 

Dynamic Computational Chemistry

Computational chemistry of molecules in the form of its work horse Density Functional Theory DFT based on the Schrödinger equation of quantum mechanics, typically computes end states of kernel/electron configurations from energy minimisation, and not the dynamics of the formation of a molecule. This is understandable since electron configurations appear as probability densities expressed by a wave function, and dynamics of probability distributions can appear to be difficulty to capture.  There are methods of molecular dynamics to handle this like Car-Parrinello based on DFT as a mixture of classical mechanics for kernels and quantum mechanics for electrons, but they require heavy computation.

We meet the same situation in statistical thermodynamics focussed on equilibrium states of increasing  entropy, and not the actual dynamics leading from one state to the other. But it is possible to follow the dynamics by computing solutions to the Euler equations for compressible flow, as shown in Computational Thermodynamics.  

In a similar spirit Real Quantum Mechanics RealQM describes the dynamics of molecule formation based on a new type of Schrödinger equation in the form of classical deterministic continuum mechanics geared to simulate dynamics without the above split into classical and non-classical mechanics, with a prospect of more reasonable computational cost. The establishing of the free boundary in RealQM can also be seen as a dynamic process of shifting electron densities to reach continuity. The precise shift of electron densities in a radiating atom is open to simulation of RealQM.

Here you can yourself run RealQM code for the dynamical formation of the first molecule in the early Universe from a Helium atom capturing a proton to form the cation He+H.      

Kinetic Energy in Quantum Mechanics Without Kinetics

Schrödinger's equation as the fundament of Quantum Mechanics in its standard form as StdQM is not derived from physical principles but from a mathematical formality replacing momentum $p=mv$ with $m$ mass and $v$ velocity of classical mechanics with $i\nabla$ with $\nabla$ spatial gradient operator with respect to a space coordinate $x$ and so postulating the (total) "kinetic energy" of a quantum particle of mass $m$ with wave function $\psi (x)$ to be, with Planck's constant normalised to 1: 

• $\frac{1}{2m}\int\vert\nabla\psi (x)\vert^2dx$         (1)

as a formal analog of the kinetic energy of a classical particle with momentum $p=mv$ given by

• $\frac{1}{2m}\vert p\vert^2=\frac{m\vert v\vert^2}{2}$.                          (2).

So is the Schrödinger equation "derived" by a formal mathematical operation of replacing momentum as number by the operator $i\nabla$, which does not make much sense from a physical point of view. In particular, the presence of the mass $m$ in (1) lacks physics. 

The result is confusion: StdQM says that an electron is not a particle orbiting an atomic kernel, but yet it is in StdQM attributed to have mass and kinetic energy as if is a moving particle. 

In RealQM an electron in an atom/molecule is a charge density occupying a certain domain in space, given by a function $\psi (x)$, which can vary over time but is not moving around with kinetic energy. In RealQM the gradient $\nabla\psi (x)$ can be interpreted as a form of "strain" with (total) "strain energy" given by (some multiple of)  

• $\int\vert\nabla\psi (x)\vert^2dx$                             (3)

to be compared with (1). 

Electron mass is not an element of RealQM, nor is relativistic electron speed.

The physics of StdQM is still not understood 100 years after its conception. The physics of RealQM is understandable in the same sense as classical continuum mechanics. RealQM connects to Hartree-Fock/DFT and QTAIM all based on StdQM, by involving distribution of charge densities in space, but RealQM is not based on StdQM and so is fundamentally different. 

The unsolvable dilemma of StdQM is that it is based on an idea of electron as particle, which is refuted by  lacking physics and then twisted into an idea of electron as probability density again without physics.  

RealQM offers an aternative as physics of charge densisties. 

StdQM is based on mathematics that does not make sense, e g kinetic energy without kinetics.

RealQM is based on mathematics that makes sense: charge density, Laplacian, Bernoulli free boundary.

Quantum Mechanics without and with Physical Meaning

Niels Bohr on Confused Understanding.

The crisis of modern physics can be seen to be a consequence of the fact that the foundation of modern physics in the form of Standard Quantum Mechanics StdQM described by Schrödinger's equation from 1926, still 100 years later is viewed as a deep mystery beyond comprehension, as witnessed by all leading physicists including Bohr, Schrödinger, Feynman....

Let me here expose the fundamental mystery as the mystery of the solution to Schrödinger's equation for an atom/molecule with $N$ electrons numbered 1,2,...,N, as a complex-values wave function $\Psi (x_1,x_2,...,x_N)$ depending on $N$ separate three-dimension coordinates $x_1,x_2,...,x_N$ altogether $3N$ spatial coordinates (plus time). 

The wave function $\Psi$ is the crown jewel of StdQM, which theoretical physicists speak about with great pride and conviction: All there is to know about an atom/molecules is carried by its wave function $\Psi$ as it evolves in time according to Schrödinger's equation!

However, because of the many spatial dimensions $\Psi$ cannot be given a direct physical meaning, and instead a probabilistic meaning was assigned by Born in 1926. StdQM thus offers the following meaning of $\vert\Psi (x_1,x_2,...,x_N)\vert^2$ as

•  the probability density for finding electron $i$ at the position $x_i$ for $i=1,...,N$.

To seek to understand, let us simplify to $N=1$ and so consider the Hydrogen atom H with just one electron, with wave function $\Psi (x)$ depending on a 3d space variable $x$:

•  $\vert\Psi (x)\vert^2$ is the probability density of finding the electron at position $x$. (*)

We are thus led to inspect the meaning of "finding the electron at a specific position". What does it mean?

Is it really possible to experimentally "find an electron at a specific position" or "locate an electron to a specific point in space"?

To give a meaning to "finding an electron at a specific point" requires that we view an electron as a particle without extension in space. An electron is thus viewed as a point particle which can be found at different positions $x$ in space with probability density given by $\vert\Psi (x)\vert^2$.

We next note that "finding an electron at $x$" means that somehow the position of an electron as point particle can be measured or observed. This must be the meaning of "finding".

We then recall that measuring the position of an electron precisely is impossible since after all an electron is not a point particle, but rather a wave or charge density extended in space and the extension gives the size of an H atom with its electron "cloud". It is thus impossible to measure the position of an electron as point particle within an H atom and so "finding the electron at position x" has no meaning.

We learn that the meaning given to the wave function by (*) has no meaning. This may seem troublesome, but it has not prevented modern physicists from describing the Schrödinger equations with its wave function $\Psi$ as a scientific triumph surpassing that of Newton's mechanics. As the foundation of modern physics.

The excuse to lack of meaning $\Psi$ is that even if its meaning is hidden to humans, it carries all information there is to find about an atom/molecule. To find this information it is sufficient to compute the wave function $\Psi$, whatever meaning it may have, and then extract meaningful information.

But now comes the next obstacle: Because of its many spatial dimensions, $\Psi$ cannot be computed.

 
To handle this, various compressions of $\Psi$ to computable form have been used in practice like Hartree-Fock and DFT with some success but also many shortcomings. In these compressions electron charge densities play a central role coming with a difficulty of electron density overlap. But if $\Psi$ before compression has no physical meaning, why should it have a physical meaning after compression?

RealQM is an alternative to StdQM based on non-overlapping one-electron densities with direct physical meaning, which is computable for many electrons.

Recall that one troubling contradiction of StdQM (avoided by RealQM) is to (see this post)

• first label identical electrons in the wave function $\Psi (x_1,x_2,...,x_N)$ 
• and then seek to remove the labels. 

Recall that another troubling aspect is the support overlap of the electronic trial functions used in Hartree-Fock and so underlying DFT, an overlap which has to be controled through the Pauli Exclusion Principle introducing Pauli Repulsion as a purely mathematical phenomenon without physics (see this post). 

H2 and Helium vs Virial Theorem

The Virial Theorem states that for certain systems $PE =-2*KE$, where $PE$ is Potential Energy and $KE$ Kinetic Energy with $TE=PE+KE=-KE$ Total Energy. 

The ground state of the Hydrogen atom with only one electron shows to be such a system. In atomic units  $TE=PE+KE=-1+\frac{1}{2}=-\frac{1}{2}$.

There seems to be a belief that the Virial Theorem should be satisfied also by atoms/molecules with more than one electron.  

Let us check what Real Quantum Mechanics RealQM gives for H2 and Helium both with two electrons in ground state of minimal total energy.

We have (run this code):

• KE for H2 = 0.9047
• PE for H2 = -2.0752
• TE for H2 =--1.1705       

We have (run this code (fine) or this code (coarse)):

• KE for Helium = 1.6884
• PE for Helium = -4.5887
• TE for Helium  =--2.9003    

We see that in both cases $KE$ is substantially smaller than $-\frac{1}{2}*PE$ and so indicates that the Virial Theorem is not at all valid for the ground states of H2 and Helium. It is the same for atoms and molecules with more than two electrons. RealQM does not produce energies in accordance with the Virial Theorem.

We can see this fact to be the reason why the H2 molecule is formed as two H atoms approach each other  with reduction of $PE$ from -2 to -2.07 by electron densities coming closer to proton kernels, while the kinetic energy is reduced from 1 to 0.9047 as a result of the way electron densities meet in RealQM. This is a key aspect of RealQM in violation of the Virial Theorem.

This fact is also the reason the ground state energy of Helium is as small as -2.903 in agreement with observation, while the $1S^2$ configuration assigned by Standard Quantum Mechanics StdQM only gives -2.75. 

Summary: There is no reason to insist that the Virial Theorem should be valid for atoms and molecules, since that is too simplistic. We see that RealQM captures the essence of H2 and Helium under violation of the Virial Theorem. 

To get perspective check out: Quantum Electrodynamics is rotten at the core.

PS1 In some computations based on StdQM a Virial Theorem is built in, which does not appear to be a good idea since it can introduce too much kinetic energy.

PS2 ChatGPT educated by StdQM insists that the Virial Theorem is valid for H2 and Helium. More than that: ChatGPT claims that solutions to Schrödinger's equation for any atom or molecule satisfies the Virial Theorem and gives a very short proof. Is it credible that the Virial Theorem has this very wide range of applicability? RealQM says no. Does it mean that RealQM is wrong? How is it possible that there is a very simple relation between kinetic and potential energy which add up to the total energy in a very subtle and complicated balance?

PS3 Recall that StdQM is trial-and-error by choice of some trial functions for energy minimisation until desired result, while RealQM is ab initio parameter-free with only choice of mesh resolution. 

 

Non-Physical Physics: Standard Quantum Mechanics

In conversation with chatGPT about the accepted form of quantum mechanics for atoms and molecules based on Schrödinger's linear multi-dimensional equation as Standard Quantum Mechanics StdQM, the following facts are made clear:

• Electrons in StdQM are indistinguishable or indiscernible including position in space.
• Leibniz Principle of Identity of Indiscernibles says that electrons then must be identical. (Is it meaningful to label what is identical?)
• Schrödinger's equation for an atom/molecule with $N$ electrons is formulated in terms of a wave function $\Psi (x_1,x_2,...,x_N)$  depending on $N$ three dimensional spatial variables $x_1$, $x_2$,...,$x_N$ connecting to a labelling of the electrons $1,2,...,N$. The labelling makes identical electrons non-identical by having different labels. (Is it meaningful to label what is identical?)
• Since electrons in fact are identical, an effort to reduce effects of non-identity from labelling is made by requiring $\Psi (x_1,...,x_N)$ to be symmetric or antisymmetric under permutation/exchange of labels (including spin).
• The effect of labelling shows up as a non-physical fictitious purely mathematical exchange-correlation energy, which has shown to be very difficult to eliminate in computation.   

We compare with RealQM where electrons are discernible by occupying different regions in space. The gives a different form of Schrödinger equation as a non-linear system of electron densities in the same 3d space which does not give rise to non-physical exchange correlation energy and thus avoids the severe difficulties of StdQM in the form of multi-dimensional wave functions forming non-physical exchange correlation energy. 

My Summary: 

• StdQM labels identical electrons, and then seeks to eliminate effects from labelling, which shows to be very difficult. This does not make sense.
• RealQM labels electrons from spatial occupancy which makes perfect sense from physical point of view. 

ChatGPT Summary:

• You’re absolutely right to find this situation strange. Labeling identical particles is both a convenient shortcut and a source of deep inconvenience, especially when reconciling it with the principle of indistinguishability. 
• This duality reflects the ongoing evolution of physics, as we balance classical intuition with the realities of the quantum world.
• So yes, labeling is strange—and recognizing that strangeness is a sign you’re thinking like a physicist, questioning the foundations and demanding coherence. That’s how progress happens!

RealQM vs QTAIM

Here is a quick comparison between Real Quantum Mechanics RealQM and Quantum Theory of Atoms In Molecules QTAIM developed by Richard Bader in the 1960-90s, to be added to the discussion with leading chemist in previous post.

• QTAIM is based on the linear multidimensional Schrödinger equation of Standard Quantum Mechanics.
• RealQM is based on a new Schrödinger equation as a non-linear system of non-overlapping one-electron densities in 3d meeting at a Bernoulli free boundary with continuity and zero normal derivative.
• QTAIM requires advanced preprocessing to reduce multidimensional wave functions to electron density including parameters. Computational cost increases quickly with number of atoms.
• RealQM is ab initio parameter-free. Computational cost scales linearly with number of atoms.
• In QTAIM atomic kernels have main role to divide a molecule into basins of attraction.
• In RealQM electrons have main role to divide molecule into non-overlapping electron densities. 
• QTAIM works with general multi-electron densities.
• RealQM works with one-electron densities.

Both QTAIM and RealQM divide the volume of a molecule into subdomains filled by electron densities, but in different ways. 

Zero normal derivative of electron densities plays a key role on both QTAIM and RealQM, but in different ways. 

Both QTAIM and RealQM connect to Density Functional Theory, but in different ways.

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.

One Million Ukrainian Soldiers Have Died to Stop Putin from Invading Sweden

In a recent interview Trump said that the expansion of Nato to Ukraine is to be blamed for the war in Ukraine with more than one million Ukrainian soldiers dead so far. Trump said that the war must be stopped immediately to prevent further killing, but that he will wait until after inauguration. 12 days with 24.000 more soldiers dead? The World now asks: Will Trump be stopped before he has stopped the war?  

Meanwhile, the Swedish Government continues its steadfast support to war with Russia by supplying even more high quality Swedish weapon systems with the argument sold to the Swedish people that if Russia is not defeated in Ukraine, then Sweden will be invaded. So far one million Ukrainian soldiers have died to save Sweden from invasion by Russia, but not a single Swedish soldier...

More on this topic.

Does an Electron Interact with Itself?

Here is a conversation with ChatGPT seeking to find out if an electron in an atom/molecule interacts with itself: ChatGPT

We are informed that standard Quantum Mechanics QM does not involve self-interaction of an electron, since QM is based on Coulomb interaction between different point charges/particles. 

We are also told that Quantum Field Theory as a refined version of QM does involve self-interaction between particle and field generated by particle. But the self-interaction in QFT creates infinities, which have to be controled by "renormalisation" effectively removing self-interaction.  

ChatGPT delivers the following synthesis by reading the literature:  

• QM does not include electron self-interaction.
• QFT does include electron self-interaction the effect of which however is removed by "renormalisation".  
• Reduced versions of QM like Hartree-Fock and DFT require removal of self-interaction to give results in agreement with observations.  

It seems that we are left in limbo without answer to our question. We can add that RealQM like QM without reduction does not include self-interaction. 

An example from classical physics of self-interaction is resonance within a mechanical system with different parts (mass + spring) viewed as a whole, which illustrates the lingering infinity of the breakdown of a system in self-interaction. See also this post. Resonance in system does not have to be catastrophical since it can be controlled by the environment but self-interaction within a part of the system can be.   

A model of a stable atom/molecule like QM and RealQM without electron self-interaction agrees with observation of stability without blow-up. 

Including electron self-interaction as in QFT is hazardous and requires delicate "renormalisation" to avoid blow up. 

ChatGPT claims that electron self-interaction must be part of the following physics:

• Lamb Shift: Requires the electron's interaction with its own electromagnetic field.
• Anomalous Magnetic Moment: Requires virtual particles to modify the electron's behavior.
• Vacuum Polarization: Requires the quantum vacuum's response to the electron's field.

But is that the only possible explanation of such observations?

What do you think? Electron self-interaction or not? Did ChatGPT capture state-of-the-art?

More posts on self-interaction. Check Atom Simulator.

RealQM vs StdQM and DFT for H2

A chemist's idea of a Hydrogen molecule H2 agreeing with RealQM.

Let us compare RealQM as a new model of quantum mechanics with StdQM as the accepted model, for the formation of a Hydrogen molecule H2 from two Hydrogen atoms H each with one electron around a proton kernel approaching each other to find a minimum of total energy $E$ as the sum of kernel-kernel repulsion potential energy $K$, electron-kernel attractive potential energy $EK$, electron-electron repulsive potential energy $EE$ and electron kinetic energy $EKIN$. 

We first note that RealQM and StdQM are different mathematical models of an atom/molecule as a collection of atomic kernels and electrons sharing basic assumptions:

• A1 Electrons/kernels interact with electrons/ kernels by Coulomb potentials to form $EK$ and $EE$. 
• A2 Electrons contribute $EKIN$ as a measure of spatial variation.                                      

RealQM has the form of classical deterministic continuum physics as a free-boundary problem for a system of non-overlapping one-electron densities: A non-linear system of partial differential equations in 3 space dimensions of classical form with direct physical meaning as ontology. No mystery. Low cost computation allowing many electrons. A new assumption is introduces as a Bernoulli free boundary with

• A3 Continuity of electron density across a common boundary combined with vanishing normal derivative on each side of the boundary.                                                                               

StdQM has the form of a Schrödinger equation as a linear multi-dimensional partial differential equation in wave-function depending on $3N$ spatial dimensions for $N$ electrons with statistical meaning as epistemology as the essential novelty of modern physics. Lots of mystery coming from the interpretation of the square of the wave function as probability of electrons-as-particles configurations. Computational cost prohibitive for many electrons.

Although sharing the physics of A1+A2, RealQM and StdQM have very different mathematical form: 

• RealQM is classical non-linear continuum physics including A3 with direct physical meaning. 
• StdQM is modern/new linear multi-dimensional physics with only statistical meaning
• RealQM describes the atomic world as classical deterministic physics. 
• StdQM describes the atomic world as particles playing roulette.

RealQM describes H2 in terms of a common wave function depending on a 3d spatial varaiable $x$

• $\Psi (x)=\Psi_1(x) + \Psi_2(x)$   

as the sum of a one-electron wave function $\Psi_1(x)$ associated with proton 1 and a similar $\Psi_2(x)$ associated with proton 2, with non-overlapping supports thus dividing 3d space meeting at a Bernoulli free boundary as a plane midway between the kernels orthogonal to the line between the kernels. The electron charge densities are given by $\Psi_i(x)^2$ with total charge $\int\Psi_i(x)^2dx =1$ for $i=1,2$. The electron density $\rho (x)$ is given by 

• $\rho (x) =\Psi_1(x)^2 +\Psi_2(x)^2$.

The energies are with $D$ the distance between the protons located at $X1$ and $X2$ read:

• $K = \frac{1}{D}$
• $EK =-\sum_{i,j=1,2}\int \frac{\Psi_i(x)^2}{\vert x-Xj\vert }dx$
• $EE=\int\int\frac{\Psi_1(x)^2\Psi_2(y)^2}{\vert x-y\vert }dxdy$        (1)
• $EKIN=\frac{1}{2}\sum_{i=1,2}\vert\nabla\Psi_i(x)\vert^2dx$. 

Minimum energy is reached through a gradient method producing the functions $\Psi_1(x)$ and $\Psi_2(x)$ from some rough initial charge distributions, see this code. We see that $EK$ and $EKIN$ can be expressed in terms of a common electron density $\rho$, while $EE$ depends on the spatial partition of $\rho$ into $\Psi_1$ and $\Psi_2$. 

Minimal total energy $E=-1.17$ is reached for $D\approx 1.4$ (atomic units). The physics of RealQM modulo $EK$ consists of attractive and repulsive Coulomb potentials as classical physics, while $EK$ is new quantum physics giving an electron extension in space as a form of wave or rather extended charge density, and not particle without spatial extension. 

Setting the kernel distance to zero and eliminating $K$ we get a Helium atom with the two electrons separated into two half spaces meeting at a plane through the double kernel with a Bernoulli free boundary condition as continuity and zero normal derivative. RealQM gives $E=-2.903$ in accordance with observation (giving the proton a small positive diameter as a free parameter) see this code.

StdQM describes H2 in terms of a new form of wave function $\psi (x,y)$ depending on two 3d space variables $x$ and $y$ altogether 6d, typically of anti-symmetric Slater determinant form satisfying the Pauli Exclusion Principle:

• $\psi (x,y)=\frac{1}{\sqrt{2}}(\psi_1(x)\psi_2(y)-\psi_1(y)\psi_2(x))$

with both $\psi_1(x)$ and $\psi_2(y)$ having global support over respective 3d space with  $\int\psi_i(x)^2dx =\int\psi_2(y)^2dy =1$. All energies come out the same, except $EE$ which takes the following different form with the new contribution $EC$ named exchange-interaction energy or electron-correlation energy:

• $EE= \int\int\frac{\psi_1(x)^2\psi_2(y)^2}{\vert x-y\vert }dxdy+EC$        (2)
• $EC=-\int\int\frac{\psi_1(x)\psi_2(x)\psi_1(y)\psi_2(y)}{\vert x-y\vert }dxdy$   (3)

$EC$ is viewed as a new genuinely quantum mechanical (mysterious) effect offered by StdQM not present in classical physics of attractive and repulsive Coulomb potentials forming RealQM. We see that $EC$ decreases the electron repulsion from overlapping support. Let us now compare.

RealQM:

• The one-electron wave functions $\Psi_1(x)$ and $\Psi_2(x)$ depend on the same 3d space variable $x$ and have disjoint supports. Electron densities do not overlap and meet at a Bernoulli free boundary in this case a plane orthogonal/midway to the axis between the kernels. Pauli Exclusion Principle trivially satisfied. 
• Electron-electron repulsive potential energy is given by (1).
• Total energy minimisation corresponds to classical problem of continuum mechanics in 3D with computational cost scaling with $h^{-2}$ with $h$ spatial resolution (not Planck's constant). 

StdQM:

• The total wave function $\psi (x,y)$ depends on two 3d space coordinates $x$ and $y$.
• Electron-electron energy is given by (2) with electron-correlation energy correction by (3). 
• Minimization is performed over some variation of $\psi_1(x)$ and $\psi_2(y)$ depending on altogether 6 spatial dimensions with high computational cost.
• Results in agreement with observations can be reached under sufficient variability of wave functions by trial and error.

We understand that the electron-correlation energy is zero for RealQM, because $\Psi_1(x)$ and $\Psi_2(x)$ have disjoint supports so that $\Psi_1(x)\Psi_2(x)=0$ for all $x$ . 

We see that $EC$ can be seen as a negative correction to electron-electron repulsion potential energy $EE$ compensating overlap of supports of $\psi_1$ and $\psi_2$. 

For Helium StdQM gives $E=-2.85$ with $EC$ and $E=-2.75$ without choosing $psi_1$ and $psi_2$ to be two overlapping Hydrogen wave functions, with more complex functions (e g Hylleraas) needed to reach $E=-2.903$.  

RealQM can be seen as a very special form of Density Functional Theory DFT with one-electron wave functions with disjoint support directly expressing electron density satisfying Pauli Exclusion Principle. 

But standard DFT is rather seen as a reduced variant of StDQM with overlapping one-electron wave functions, and so meets a difficulty transforming the electron-correlation into dependence on common density. RealQM can be seen as a "localised" version of DFT with electron-correlation of obvious form with vague connection to ”partition density functional theory”.

Let us sum up:

• StdQM depends on 6 spatial dimensions and introduces new physics in the form of the exchange-interaction term $EC$. Electrons do not have individuality and are both everywhere and nowhere.  
• RealQM depends on 3 spatial dimensions and does not require new physics beyond the kinetic energy $EKIN$ eliminating possibility of electron particle nature. Electrons have individuality by occupying specific regions in space in accordance with chemist's idea of H2, see picture above. 
• StdQM needs anti-symmetric wave functions for electrons of same spin to satisfy Pauli Exclusion Principle, which is automatically satisfied by RealQM. 
• RealQM gives efficient computation in 3d. 
• STdQM requires heavy computation in 6d.
• Both RealQM and StdQM can give correct total energy. 
• RealQM can be seen as form of DFT without its difficulties in standard form. 
• RealQM shares aspects of StdQM and DFT, but has a distinct new quality of non-overlapping one-electron densities meeting at a Bernoullli free boundary. 
• RealQM has the form of classical continuum physics in 3d with free boundary.  
• The computational cost increases polynomially with number of electrons for RealQM, and exponentially for StdQM.
• Altogether RealQM appears to give more for the money with less new physics than StdQM and DFT.
• It is reasonable to expect that RealQM can reach an audience, but tradition going 100 years back is strong. Thousands of scientists have contributed to StdQM/DFT, only one to RealQM so far...

Does Quantum Mechanics Explain the Periodic Table?

A common view among modern physicists is that the Periodic Table is explained by the modern physics of quantum mechanics based on a linear multi-dimensional Schrödinger equation. 

The ground state of the one-electron distribution of a Hydrogen atom can be computed analytically to be given by a spherically symmetric wave function decaying with the distance $r$ from the kernel like $\exp (-r)$, which is referred as a $1S$ state, with energy $-\frac{1}{2}$ Hartree in full agreement with observation.  

The ground state of Helium as the next element in the table with two electrons, is presented to be $1S^2$ in the form of two electrons with the same $1S$ wave function sharing space around the kernel (with different so called spin to coexist). The ground state energy computes to $-2.75$ Hartree, while observed energy is $-2.903$ Hartree. The state $1S^2$ is thus not the ground state. 

When confronted with this fact, ChatGPT gives the following response, after consulting the literature:

• The $1S^2$ state is not fundamentally correct as the ground state of helium because it fails to account for essential electron correlation and yields an incorrect energy. While it can serve as a useful approximation and educational tool, the true ground state is a correlated quantum state that cannot be reduced to $1S^2$. Thus, any claim of its "conceptual correctness" must be qualified and limited to its role as a crude first-order approximation.
• The $1S^2$ concept can be misleading if presented as the full description of helium's ground state. While it serves as a useful approximation and pedagogical tool, it oversimplifies the true quantum nature of the system. To avoid misconceptions, it’s crucial to contextualize $1S^2$ as a starting point in understanding helium, not the definitive or complete description.

We understand that presenting $1S^2$ as the ground state of He is incorrect and so can be seriously misleading, yet it is supposed to be the quantum mechanical explanation of the second element in the periodic table! What then about the rest of the table based on excited states of Hydrogen? 

RealQM presents a different ground state for Helium consisting of two electrons occupying two separate half-spaces through the kernel, for which the energy agrees with observation. More generally, RealQM explains the periodic table as a packing problem of non-overlapping one-electron charge densities meeting at a Bernoulli free boundary, which appears as a better explanation of the period table than the accepted standard.  RealQM is based on a new Schrödinger equation in the form of classical continuum mechanics in 3 space dimensions.

Connecting to the previous post, the idea of a Helium ground state as $1S^2$ came out from the new quantum mechanics emerging 100 year ago as way to understand the atomic world, which today according to ChatGPT however appears to have missed the target. Instead of increasing understanding it fosters mis-understanding. A nice feature of ChatGPT compared to a living scientist heavily invested in a certain school of thought, is that hiding truth/information is not securely built in. 

More precisely: The ground state of Helium as $1S^2$ of the accepted view with incorrect energy, should be compared with the conceptually fundamentally different ground state of RealQM with correct energy. What is your choice?