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.

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.  

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. 

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.