onsdag 12 februari 2025

Sweden vs Russia New Match

In the Foreign Policy Debate today in the Swedish Parliament Swedish Foreign Minister Maria Stenergard opened by forcefully declaring:

  • Sweden is in the midst of long term confrontation with Russia.
  • Support to Ukraine is the Government's top foreign policy priority. 
  • Our task is inescapable. 
  • Sweden never stands alone.
  • We will constrain Russias ability to do us harm, particularly through our support to Ukraine.
  • We are guided by our belief in a free and strong Ukraine.
  • Sweden has increased its support to Ukraine each year of the war, totalling 70 billion SEK.
  • Russia's goal is to impose a sphere of influence with series of vassals and satellites.
  • The war is determined on the battle field.
  • For Sweden support to Ukraine is a moral obligation and an indispensable investment in our own security.
  • Swedish land forces are now part of Nato's forward forces in Latvia.
  • It is up to Ukraine if and when negations will start.
  • Our unequivocal support....to make Ukraine's position stronger.
  • Pressure on Russia's war economy must increase.
  • Our military support to Ukraine must be strengthened... now its 18th military support package the largest to date.
  • The only sustainable peace is one that Ukraine achieves through strength.
  • Negotiations from a position of weakness would only further Russian aggression.
  • Sweden will continue military support to Ukraine as long as it takes.
So Sweden is again at war again with Russia to be added to this long list of wars from 1164 to 1809 when Sweden lost Finland in the Finnish War between the Kingdom of Sweden and Russian Empire.  

But this is forgotten by the Swedish Government, which instead lives in a fantasy of repeating the glorious victory in the battle at Narva in 1700 when a Swedish army of 10.000 soldiers lead by warrior King Karl XII overpowered a Russian army of 40.000. And Sweden won Hockey WM against Russia in 1957 and 1987.

Short story: The Swedish Government is not well informed, because of lack of Intelligence?

söndag 9 februari 2025

Computational Chemistry vs Solid Mechanics

Computational Chemistry CC may take up to 50 % of supercomputer resources, while Computational Solid Mechanics CSM may take less than 10%. 

The world of molecules built from atoms and electrons can be seen as a microscopic analog of a macroscopic world built from beams of steel such as a bridge,  and so we may expect to find CC as a microscopic analog of CSM, at least in a perspective of classical continuum mechanics.  

But this is not what the above numbers show: CC requires vastly more computational work than CSM. Why is that?

Consider a macroscopic object like a bridge composed of $N$ elements, which could be the finite elements in a discretisation of the object in CSM based on Navier's equations of elasticity. The computational work can scale with $N$, that is linearly in $N$ since each element interacts with just a few neighbouring elements. Finite element codes with $N=10^6$ can be run on a lap top.

The situation in CC is vastly different. This is because CC is based on Schrödinger's equation as the basic model of Standard Quantum Mechanics StdQM, which for a molecule with $N$ electrons involves $3N$ spatial dimensions, a full 3d space for each electron. This means that the computational work increases exponentially with $N$ which makes even $N=10$ beyond the power of any thinkable computer, and so only simplified versions of Schrödinger's equation are used in practice. Full solutions named wave functions then appear as pieces of conversation, which have no precise quantitative form. 

In Density Functional Theory DFT as the current work horse of CC, Schrödinger's equation is draconically reduced into a 3d equation in a single electrons density. The CSM analog would be to compress a complex bridge construction into one simple beam, where all element individuality is erased. 

Electron individuality is thus destroyed in DFT, asking for some form of recovery as electron exchange-correlation, which has shown to be difficult to realise.

RealQM is new methodology starting from a new form of Schrödinger's equation i terms of a collection of a non-overlapping one-electron densities keeping individuality of electrons by spatial occupation. RealQM can be seen as a refined form of DFT with one-electron densities maintaining individuality. 

RealQM in principle scales linearly with $N$ just like CSM. Below you can compare a bridge and a molecule and ask yourself why the molecule in CC with StdQM requires vastly more computational work than the bridge in CSM, and so get motivated to take a look at RealQM for which the work is comparable.



Molecule.


Bridge.



fredag 7 februari 2025

First Molecule HeH+ by RealQM and DFT

Crosscut 3d showing one electron (red) moved from He++ kernel to H+ kernel and remaining electron (yellow) around He++ kernel. Note free boundary developed between electrons starting from initial vertical cut through He++ kernel. Run code below to follow dynamics.

This is a follow up on previous post on the first molecule formed after a Big Bang when one of the two electrons of a Helium atom He joins with an approaching proton H+ to form a helium hydrid ion molecule HeH+ (or rather He+H) built by a cation He+ and a Hydrogen atom H. The energy count in Hartree is as follows:

  • Energy of He atom = -2.903
  • Energy of He+ and H separated = -2.000 - 0.500 = -2.500
  • Energy E of HeH+ molecule = -2.592 
  • Dissociation Energy of HeH+ into He+ and H = 0.092  observed
  • Energy for formation FE of HeH+ from He and H+ = 0.311 
Let us compare RealQM and DFT as concerns prediction of the observed FE = 0.311. Notice the difference between He plus H+ and He+ plus H. Check by noticing that -2.903 = - 2.500 - 0.311 - 0.092. Notice that the bulk of FE is supplied by exterior forcing to make H+ approach the He++ kernel. 

RealQM code gives FE = 0.301 from E = -2.602. You can follow the transfer of one electron from He to H by running the code starting from two electron half-lobes around the He kernel with supports displayed on red and yellow. You can test a different location of H+ vs He electron split by running this code. In both cases see how one electron dynamically shifts from He to H+ forming a molecule of He+ and H starting from He and H+. 

DFT gives according to chatGPT:

DFT Functional

Predicted Dissociation Energy (Hartree)

Error Trend

LDA (Local Density Approximation)

~ -0.35

Overbinds HeH⁺ (too stable)

GGA (PBE, BLYP)

~ -0.33 to -0.34

Still overestimates bond strength

Hybrid (B3LYP, PBE0)

~ -0.30 to -0.32

Closest to exact (-0.311)

High-Accuracy (CCSD(T), FCI)

-0.311

Exact value

  • LDA and GGA functionals overestimate binding, leading to a more negative dissociation energy (~ -0.34 to -0.35 Hartree).
  • Hybrid functionals (B3LYP, PBE0) improve accuracy, but they still may predict a slightly too strong bond.
  • Post-HF methods (CCSD(T), FCI) match experimental values (-0.311 Hartree).

We see that the precision with standard DFT is not better than RealQM, rather the opposite. It is not clear that DFT can model the dynamics of the shift of one electron from He to H+.

Notice that we are here dealing with the simplest possible problem in quantum mechanics, a molecule with only two electrons as the first molecule formed in the early universe, with H2 coming only later after dissociation of HeH+ into He+ and H (and then formation of H2 under release of energy). Would you expect that DFT after 50 years of massive investment would deliver a very convincing result? Did we get that?


torsdag 6 februari 2025

Electron Affinity RealQM vs DFT

Chart of electron affinity from 0 to 0.133 Hartree with grey zero affinity.

Electron affinity is a measure of the drop in total energy in energy when a neutral atom A captures an  electron under release of energy forming a negatively charges anion A-  named negative affinity.

An atom with zero affinity has no tendency to capture another electron.  

We consider two basic cases one with zero and one with negative affinity:

  • Helium with 2 electrons filling the 1st shell as a noble gas with zero electron affinity.
  • Fluorine with negative electron affinity by filling the 2nd shell from 7 to 8 electrons.  
Observed negative electron affinities range from 0.1- 0.3 Hartree with 0.12 for Fluorine. The total energy of Fluorine is -99.7 Hartree, and so to recover a change of 0.1 Hartree in computation requires a precision of 4 correct decimals.  

Here you can run RealQM as essentially a 3-line parameter-free code in 3d with only input the kernel charge giving the following total energies in Hartree:
We see that RealQM recovers zero affinity for Helium and the trend of negative affinity for Fluorine, if not the exact value with the present resolution of a $50^3$ grid. 

Density Functional Theory DFT as a very complex code, typically gives positive affinity for Helium, and can give values in the range of 0.12 for Fluorine under suitable adjustments of the code. 

PS This is what chatGPT has to say about the role of DFT in years to come:
  • DFT will continue to be the dominant method for simulating chemical systems in the foreseeable future. Despite its limitations, it offers the best trade-off between accuracy, computational cost, and scalability. However, machine learning (ML) is emerging as a potential competitor—and in some cases, it might even surpass DFT.
The question is if RealQM can take over this role as a new form of DFT with a collection of one-electron densities instead of just one common density. The investment in DFT has been massive over a period of 50 years, while  RealQM is a spin-off of computational mechanics realised with little manpower. It is thus of interest to compare RealQM and DFT on basic tasks.

Self-Interaction in DFT vs RealQM

A main difficulty of Density Functional Theory DFT as working with a single electron density $\rho (x)$ depending on a 3d spatial coordinate $x$ representing all electrons, is that electron self-interaction is present and has to be eliminated. 

Without correction DFT gives a much too small effective net electric potential outside a neutral atom as the net potential from kernel and electrons, for which the true net potential is $-\frac{1}{r}$ with $r$ the distance to the kernel for any atom, the same for all atoms as that of the Hydrogen atom. This is the case without van der Waal dipole effects.    

Real Quantum Mechanics RealQM is a new alternative to StandardQM StdQM with DFT, works with a collection of non-overlapping one-electron charge densities. 

In RealQM  there is no self-interaction since electron Coulomb potentials contribute to the total energy only for pairs of distinct electrons

RealQM thus gives the correct effective potential $-\frac{1}{r}$ simply because for an atom with kernel charge $Z$ and $Z$ electrons, each electron interacts with $Z-1$ other electrons with net 1 as the charge in the effective potential $-\frac{1}{r}$ of the Hydrogen atom.

In StdQM the effective potential of $-\frac{1}{r}$ is viewed to be the result of incomplete shielding of the kernel by the surrounding electrons always leaving a net potential of $-\frac{1}{r}$ even if the total net charge is $0=Z-Z$. But why the shielding effect is precisely $Z-1$ is not so obvious with the typical overlapping electron orbitals used in Hartree-Fock and DFT based on Hartree-Fock. 

StdQM/DFT: 

  • works with globally overlapping electron densities without boundaries,
  • has to struggle to remove effects of non-physical electron self-interaction.
RealQM: 
  • works with nonoverlapping electron densities meeting at a free boundary,
  • has no electron self-interaction. 
A simple test of consistency of any atom model is to check if the net potential outside the atom is $-\frac{1}{r}$. RealQM directly passes this test, while basic DFT has to be modified to pass the test by eliminating non-physical effects of electron self interaction.

 

söndag 2 februari 2025

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.  


torsdag 30 januari 2025

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)







tisdag 28 januari 2025

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.

måndag 27 januari 2025

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.

 

lördag 25 januari 2025

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.