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 corresponding to an energy of $-2.175$ Hartree is that of Orthohelium, compared to $-2.903$ for the ground state, while Parahelium has 0.3 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:

Lines with higher frequency can be obtained by freezing the boundary of the inner electron and let the outer electron take on Hydrogen like excited states from a shielded kernel. Details to come.

PS The ground state of Parahelium with the 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).