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?