John Clauser Nobel Prize in QM 2022: Does Not Understand QM!

John Clauser shared the 2022 Nobel Prize in Physics with Alain Aspect and Anton Zeilinger for 

• groundbreaking experiments using entangled quantum states, where two particles behave like a single unit even when they are separated. Their results have cleared the way for new technology based upon quantum information.

with further motivation:

• One key factor in this development is how quantum mechanics allows two or more particles to exist in what is called an entangled state. What happens to one of the particles in an entangled pair determines what happens to the other particle, even if they are far apart (supposedly disproving Einstein's requirement of local realism).

In his Nobel Lecture John Clauser is keen to make a confession (last slide at 1.22.30 see above) that he does not understand the physical meaning of Quantum Mechanics QM expressed by Schrödinger's equation for a wave function $\Psi$ over a configuration space of 3N dimensions for an atomic system with N>1 electrons.  

To Clauser as an experimental physicist there is the 3-dimensional lab space, where his experiments are performed, and there is a 3N-dimensional configuration space supporting the wave function,  and he wants to make it very clear that he does not understand their connection. 

This is remarkable because entanglement is a direct reflection of the multidimensional nature of configuration space for $N>1$. Entanglement is built into Schrödinger's equation för any atomic system with more than one electron. But it is a property of a wave function over a configuration space and not lab space. 

Clauser gets the Nobel Prize for an experiment in lab space supposedly giving support to entanglement as an effect in configuration space, while admitting that he does not understand the connection between lab space and configuration space. 

Can you understand this? Clauser is not the first physicist to express that he does not understand QM in configuration space, in fact all do, but it is of particular weight in the context of experimental verification in lab space of an effect in configuration space. Is there a possibility of misinterpretation? 

In particular, Clauser expresses strong frustration with quantum theorists asking him as experimentalist to verify in lab space effects which can only exist in configuration space, and so he finishes by throwing out the following provoking question to his theoretical physics friends, if any left: Can two black holes be entangled?  

Clauser shows his deep belief in real physics as competitive sailor.      

For a version of Schrödinger's equation over lab space see Real Quantum Mechanics.

Corruption of Modern Physics 3: Quantum Mechanics

I don't like it (standard quantum mechanics) and I am sorry I ever had anything to do with it. 

Sabine Hossenfelder with her blog BackReAction is in trouble from expressing her opinion as physicist about particle physics as the central subject for contemporary physicists:

• They don’t like to hear that their field urgently needs to change direction, so they attack me as the bearer of bad news. 
• Everyone can see that nothing useful is coming out of particle physics, it’s just a sink of money. Lots of money. 
• And soon enough governments are going to realize that particle physics is a good place to save money that they need for more urgent things. 

This is a tough message. No wonder that Sabine is in trouble and that particle physicists are angry. But does Sabine have something important to say? Let's see. 

In recent posts I have been seeking the origin of the present crisis of physics witnessed by (not only) Sabine, in the work of Einstein on particle nature of light (origin of particle physics) and special/general relativity as a cornerstone of modern physics (filled with never resolved mysteries/contradictions).  

Let us now turn to quantum mechanics as the other cornerstone, with the crisis in full bloom from incompatibility with general relativity, which Einstein spent the last 30 years of his life to resolve in a fruitless search for a general field theory including both gravitation and electromagnetics/quantum mechanics.

Quantum mechanics is based on the (linear) Schrödinger equation for a (scalar complex-valued) wave function $\Psi (x1, x2, ..., xN,t)$ for an $N$-electron system/atom depending on $N$ three-dimensional space coordinates $x1$ to $xN$ and a time coordinate $t$, thus depending on $3N$ space coordinates and one time coordinate. Since physical reality has at most three space dimensions, the wave function can be given a direct physical meaning only for a system with one electron, that is for the Hydrogen atom. To Schrödinger as the creator of the Schrödinger equation this created deep frustration, since he required the wave function to be Anschaulich or possible to visualise (in three space dimensions). Moreover, as pointed out by Nobel Laureate Walter Kohn, already for a system with $10$ electrons the wave function is impossible to compute because its high dimensionality. This makes it possible to claim that the wave function can never be wrong.

In short, (standard) quantum mechanics as based on the (standard linear scalar) Schrödinger equation presents severe difficulties, which have never been overcome despite intense struggle by the sharpest minds over 100 years. The way out became to give up physicality/reality and give the wave function a statistical meaning as suggested by Born. In short, the (standard) wave function has no direct physical meaning and in addition is uncomputable, which physicists rationalise by saying (following Bohr) that since the wave function contains everything that can be said and correctly predicts the outcome of any experiment, it is not necessary to understand its real meaning. A physicist can handle this by confessing that he/she does not understand quantum mechanics (and nobody else either).

To sum up, modern physics is based on two theories (relativity and quantum mechanics) both loaded with unresolvable difficulties/mysteries, which together are incompatible/contradictory. No wonder a crisis has developed with ever more fanciful ingredients of multi-versa, string theory and dark matter/energy.  

 

An understandable alternative to the standard linear multi-dimensional scalar Schrödinger equation is given in Real Quantum Mechanics in the form of a non-linear system of three-dimensional scalar real wave functions. Take a look and see that you can understand!

Bohr claimed that the objective of the theory of quantum mechanics (Schrödinger equation) is to (i) predict the outcome of experiments, not to (ii) explain the outcome of experiments as the real scientific objective. This is odd and adds to the mystery of quantum mechanics. The only role prediction of an experiment can serve is to support theory (if the prediction is correct) or falsify theory (if the prediction is incorrect). Since the wave function is uncomputable and thus unknown it cannot be used to make predictions. What can be done is, knowing the outcome of an experiment, to compute/design an approximate simplified wave function by reducing dimensionality, which always (miraculously) agrees with the experiment and so shows that modern physics is in perfect shape.  For more, see 72 posts on Quantum Contradictions.

Free Will and Quantum Mechanics

Sabine Hossenfelder on BackReaction believes that quantum mechanics says that humans do not have a free will:

• Physics deals with the most fundamental laws of nature, those from which everything else derives. These laws are, to our best current knowledge, differential equations. Given those equations and the configuration of a system at one particular time, you can calculate what happens at all other times. 
• That is for what the universe without quantum mechanics is concerned. Add quantum mechanics, and you introduce a random element into some events. Importantly, this randomness in quantum mechanics is irreducible. It is not due to lack of information. In quantum mechanics, some things that happen are just not determined, and nothing you or I or anyone can do will determine them.
• Taken together, this means that the part of your future which is not already determined is due to random chance. It therefore makes no sense to say that humans have free will.

This is an expression of the monumental confusion of modern physicists caused by the idea that atoms play games of roulette attributed to quantum mechanics as being based on Schrödinger's multidimensional wave equation, something which Einstein and Schrödinger never accepted.

I have followed a different line of thought viewing physics as forms of analog computation with finite precision, which can be simulated by digital computation with finite precision presented at The World as Computation including a new approach to quantum mechanics as Real Quantum Mechanics without games of roulette. 

In this world there is room for humans with free will, although you cannot control everything in your life, due to lack of precision. If you are not an addicted gambler this may be the world for you.

Determined or Random??

PS The idea that atoms play games of roulette has corrupted modern physics into incomprehensible voodoo-science. Macroscopics as complex systems of simple microscopics can express random behavior, but random microscopics destroys cause-effect and demands contradictory microscopics of microscopics. Contradictory science is voodoo-science, even if it is text book modern physics. 

The Principles of Qspeak

Qspeak was the official language of Quantum Mechanics and had been devised to meet

the ideological needs of Modphys or Modern Physics. 

It was expected that Qspeak would have finally superseded Oldspeak (or Classical Physics, as we should call it) by about the year 2050. Meanwhile it gained ground steadily, all Party members tending to use Qspeak words and grammatical constructions more and more in their everyday speech. 

The purpose of Qspeak was not only to provide a medium of expression for the world-view and mental habits proper to the devotees of Modphys, but to make all other modes of thought impossible. It was intended that when Qspeak had been adopted once and for all and Oldspeak forgotten, a heretical thought — that is, a thought diverging from the principles of Modphys — should be literally unthinkable, at least so far as thought is dependent on words.

Its vocabulary was so constructed as to give exact and often very subtle expression to every meaning that a Party member could properly wish to express, while excluding all other meanings and also the possibility of arriving at them by indirect methods. This was done partly by the invention of new words, but chiefly by eliminating undesirable words and by stripping such words as remained of un-orthodox meanings, and so far as possible of all secondary meanings whatever. 

To give a single example. The word position still existed in Qspeak, but it could only be used in such statements as ’One cannot speak about the position of an electron’. 

It could not be used in its old sense of ’position here or there’ since position no longer existed even as concept, and were therefore of necessity nameless. Quite apart from the suppression of definitely heretical words, reduction of vocabulary was regarded as an end in itself, and no word that could be dispensed with was allowed to survive. 

Qspeak was designed not to extend but to diminish the range of thought, and this purpose was indirectly assisted by cutting the choice of words down to a minimum.

Qspeak was founded on the English language as we now know it, though many Qspeak sentences, even when not containing newly-created words, would be barely intelligible to an English-speaker of our own day:

• wave-function-collapse
• quantisation
• Hilbert-space
• mixed-state
• spin
• photon
• gluon
• lepton
• quark
• fermion
• boson
• baryon
• quit
• uncertainty-principle
• correspondence principle
• complementary principle
• exclusion principle
• superposition principle
• entanglement
• interference
• phase coherence
• plum pudding model
• hyperfine 
• bra-ket-notation
• mixed-state
• strangeness
• probability-amplitude
• matrix mechanics
• CP violation
• observables
• indistinguishable-particles
• intrinsically-identical-particles
• supersymmetry
• superpartner.

If needed The Party can use Qspeak in a form of Shut Up and Calculate, which eliminates any remaining tendency of heretical questioning.

If you think you understand Qspeak, you don't understand Qspeak. (Richard Feynman)

Those who are not shocked when they first come across Qspeak cannot possibly have understood it. (Niels Bohr)

 It is safe to say that nobody understands Qspeak. (Richard Feynman)

In physics we deal with states of affairs much simpler than those of psychology and yet we again and again learn that our task is not to investigate the essence of things—we do not at all know what this would mean; but to develop those concepts that allow us to speak with each other about the events of nature in a fruitful manner using Qspeak. (Niels Bohr)

The opposite of a correct statement is a false statement. But the opposite of a profound truth may well be another profound truth, in Qspeak. (Niels Bohr)

When asked ... [about] an underlying quantum world, Bohr would answer: There is no quantum world. There is only an abstract quantum physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about Nature, according to Qspeak. (Niels Bohr)

I think there is a moral to this story, namely that it is more important to have beauty in one’s equations in Qspeak than to have them fit experiment. (Paul Dirac)

Is Quantum Computing Possible?

• .....may or may not be mystery as to what the world view that quantum mechanics represents. At least I do, because I'm an old enough man that I haven't got to the point that this stuff is obvious to me. Okay, I still get nervous with it. And therefore, some of the younger students ... you know how it always is, every new idea, it takes a generation or two until it becomes obvious that there's no real problem. It has not yet become obvious to me that there's no real problem. I cannot define the real problem, therefore I suspect there's no real problem, but I'm note sure there's no real problem. 
• So that's why I like to investigate things. So I know that quantum mechanics seem to involve probability--and I therefore want to talk about simulating probability. (Feynman asking himself about a possibility of quantum computing in 1982)

The idea of quantum computing originates from a 1982 speculation by Feynman followed up by Deutsch on the possibility of designing a quantum computer supposedly making use of the quantum states of subatomic particles to process and store information. The hope was that quantum computing would allow certain computations, such as factoring a large natural number into prime factors, which are impossible on a classical digital computer.

A quantum computer would be able to crack encryption based on prime factorisation and thus upset the banking system and the world. In the hands of terrorists it would be a dangerous weapon...and so do we have to be afraid of quantum computing?

Not yet in any case! Quantum computing is still a speculation and nothing like any real quantum computer cracking encryption has been constructed up to date, 35 years later. But the hopes are still high...although so far the top result is factorisation of 15 into 3 x 5...(...in 2012, the factorization of 21 was achieved, setting the record for the largest number factored with Shor's algorithm...)

But what is the reason behind the hopes? The origin is the special form of Schrödinger's equation as the basic mathematical model of the atomic world viewed as a quantum world fundamentally different from the macroscopic world of our lives and the classical computer, in terms of a wave function

• $\psi (x_1,...,x_N,t)$ 

depending on $N$ three-dimensional spatial coordinates $x_1$,...,$x_N$ (and time $t$) for a system of $N$ quantum particles such as an atom with $N$ electrons. Such a wave function thus depends on $3N$ spatial variables of $N$ different versions of $R^3$ as three-dimensional Euclidean space.

The multi-dimensional wave function $\psi (x_1,...,x_N,t)$ is to be compared with a classical field variable like density $\rho (x,t)$ depending on a single 3d spatial variable $x\in R^3$. The wave function $\psi (x_1,...,x_N,t)$ depends on $N$ different copies of $R^3$, while for $\rho (x,t)$ there is only one copy, and that is the copy we are living in.

In the Many Worlds Interpretation MWI of Schrödinger's equation the $N$ different copies of $R^3$ are given existence as parallel universes or multiversa, while our experience still must be restricted to just one of them, with the other as distant shadows.

The wave function $\psi (x_1,...,x_N,t)$ thus has an immense richness through its contact with multiversa, and the idea of quantum computing is to somehow use this immense richness by sending a computational task to multiversa for processing and then bringing back the result to our single universe for inspection.

It would be like sending a piece of information to an immense cloud for complex computational processing and then bringing it back for inspection. But for this to work the cloud must exist in some form and be accessible.

Quantum computing is thus closely related to MWI and the reality of a quantum computer would seem to depend on a reality of multiversa. The alternative to MWI and multiversa is the probabilistic Copenhagen Interpretation CI, but that does not make things more clear or hopeful.

But what is the reason behind MWI and multiversa? The only reason is the multi-dimensional aspect of Schrödinger's equation, but Schrödinger's equation is a man-made ad hoc variation of the equations of motion of classical mechanics obtained by a purely formal procedure of representing momentum $p$ by a multi-dimensional gradient differential operator as $p=i\nabla$ thus formally replacing $p^2$ by the action on $\psi$ by a multi-dimensional Laplacian $-\Delta =-\sum_j\Delta_j$ with $\Delta_j$ the Laplacian with respect to $x_j$, thus acting with respect to all the $x_j$ for $j=1,...,N$.

But by formally replacing $p$ by $i\nabla$ is just a formality without physical reason, and it is from this formality that MWI and multiversa arise and then also the hopes of quantum computing.  Is there then reason to believe that the multi-dimensional $-\Delta\psi$ has a physical meaning, or does it rather represent some form of Kabbalism or scripture interpretation?

My view is that multiversa and quantum computing based on a multi-dimensional Schrödinger equation based on a formality, is far-fetched irrational dreaming, that Feynman's feeling of a real problem sensed something important,  and this is my reason for exploration of realQM based on a new version of Schrödinger's equation in physical three-dimensional space.

PS1 One may argue that if MWI is absurd, which many think, then CI is also absurd, which many think, since both are interpretations of one an the same multi-dimensional Schrödinger equation, and the conclusion would then be that if all interpretations are absurd, then so is what is being interpreted, right? Even more reason for realQM and less hope for quantum computing...

PS2 MWI was formulated by Hugh Everett III in his 1956 thesis with Wheeler. Many years later, Everett laughingly recounted to Misner, in a tape-recorded conversation at a cocktail party in May 1977, that he came up with his many-worlds idea in 1954 "after a slosh or two of sherry", when he, Misner, and Aage Petersen (Bohr’s assistant) were thinking up "ridiculous things about the implications of quantum mechanics". (see Many Worlds? Everett, Quantum Theory and Reality, Oxford University Press)

PS3 To get a glimpse of the mind-boggling complexity of $3N$-dimensional space, think of the big leaps form 1d to 2d and from 2d to 3d, and then imagine the leap to the 6d of the two electrons of Helium with $N=2$ as the simplest of all atoms beyond Hydrogen with $N=1$. In this perspective a single Helium atom as quantum computer could be imagined to have the computational power of a laptop. Yes, many dimensions and many worlds are mind-boggling, and as such maybe just phantasy.

Quantum Mechanics as Theory Still Without Meaning

Yet another poll (with earlier polls in references) shows that physicists still today after 100 years of deep thinking and fierce debate show little agreement about the stature of quantum mechanics as the prime scientific advancement of modern physics.

The different polls indicate that less than 50% of all physicists today adhere to the Copenhagen Interpretation, as the main text book interpretation of quantum mechanics. This means that quantum mechanics today after 100 years of fruitless search for a common interpretation, remains a mystery without meaning. Theory without interpretation has no meaning and science without meaning cannot be real science.

If only 50% of physicists would agree on the meaning of the basic text book theories of classical physics embodied in Newton/Lagranges equations of motion, Navier's equation for solid mechanics, Navier-Stokes equations for fluid dynamics and Maxwell's equations for electromagnetic, that would signify a total collapse of classical physics as science and subject of academic study.

But this not so: classical physics is the role model of science because there is virtually no disagreement on the formulation and meaning of these basic equations.

But the polls show that there is no agreement on the role and meaning of Schrödinger's equation as the basis of quantum mechanics, and physicists do not seem to believe this will ever change. This is far from satisfactory from scientific point of view.

This is my motivation to search for a meaningful quantum mechanics in the form of realQM presented in recent posts. Of course you may say that for many reasons my chances of finding some meaning are very small, but science without meaning cannot be real science.

PS Lubos Motl, as a strong proponent of a textbook all-settled Copenhagen interpretation defined by himself, reacts to the polls with

• The foundations of quantum mechanics were fully built in the 1920s, mostly in 1925 or at most 1926, and by 1930, all the universal rules of the theory took their present form...as the Copenhagen interpretation. If you subtract all these rules, all this "interpretation", you will be left with no physical theory whatsoever. At most, you will be left with some mathematics – but pure mathematics can say nothing about the world around us or our perceptions.
• In virtually all questions, the more correct answers attracted visibly greater fractions of physicists than the wrong answers.

Lubos claims that more correct views, with the true correct views carried by only Lubos himself, gathers a greater fraction than less correct views, and so everything is ok from Lubos point of view. But is greater fraction sufficient from scientific point of view, as if scientific truth is to be decided by democratic voting? Shouldn't Lobos ask for 99.9% adherence to his one and only correct view? If physics is to keep its position as the king science?

Or is modern physics instead to be viewed as the root of modernity through a collapse of classical ideals of rationality, objectivity and causality?

realQM vs Hartree-Fock and DFT

I have put up an updated version of realQM (real Quantum Mechanics) to be compared with stdQM (standard QM).

stdQM is based on a linear Schrödinger equation in a $3N$ dimensional wave function with global support for an atom with $N$ electrons, which is made computable in Hartree-Fock and Density Functional Theory DFT approximations reducing the dimensionality to basically 3d.

realQM is based on a system of non-linear Schrödinger equations in $N$ 3d electron wave functions with local disjoint supports, which is computable without approximation. Evidence that realQM describes real physics is given.

Weinberg: Why Quantum Mechanics Needs an Overhaul!

My new book Real Quantum Mechanics seems to fill a need: Nobel Laureate in Physics Steven Weinberg believes that quantum mechanics needs an overhaul because current debates suggest need for new approach to comprehend reality:

• I’m not as happy about quantum mechanics as I used to be, and not as dismissive of its critics.
• It’s a bad sign in particular that those physicists who are happy about quantum mechanics, and see nothing wrong with it, don’t agree with each other about what it means.

I hope this can motivate you to check out the new approach to quantum reality presented in the book, which addresses many of the issues raised by Weinberg.

Weinberg takes the first step to progress by admitting that quantum mechanics in its present form cannot be the answer to the physics of atoms and molecules.

Of course the witness by Weinberg is not well received by ardent believers in a quantum mechanics once and for all cut in stone by Heisenberg and Born, such as Lubos Motl.

But it may be that questioning a theory, in particular a theory supposedly being embraced by all educated, shows more brains and knowledge than simply swallowing it without any question.

PS1 I put up a comment on Lubos Reference frame, but the discussion was quickly cut by Lubos, us usual...any questioning of the dogma of Heisenberg-Bohr-Born is impossible to Lubos, but that is not in the spirit of real science and physics...

PS2 Here is my closing comment which will be censored by Lubos: It is natural to draw a parallel between Lubos defence of the establishment of QM and the defence of the Clinton establishment by Woit, Tao, Susskind et cet, (rightly questioned by Lubos) in both cases a defence with objective to close the discussion and pretend that everything is perfectly normal. Right Lobos?

PS3 Here is a link to Weinberg's talk.

Real Quantum Mechanics: New Book

I am now starting to compile RealQM into a new book and I have put up a very first version for inspection.

The book follows in the foot steps of Schrödinger with a hope that it could have made him smile:

Real Quantum Mechanics

Physics is wrong, from string theory to quantum mechanics. The three biggest figures in quantum mechanics, Schrödinger, Einstein and Dirac, were all quantum skeptics. (Roger Penrose in Discover Interview Sept 2009)

The approach to quantum mechanics in terms of classical realistic continuum mechanics, which I have explored in recent posts as Physical Quantum Mechanics, is now available for inspection in more precise terms in the following draft manuscript:

• Real Quantum Mechanics

Here is a sample result:

New Quantum Mechanics 19: 1st Excitation of He

Here are results for the first excitation of Helium ground state into a 1S2S state with excitation energy = 0.68 = 2.90 -2.22, to be compared with observed 0.72:

New Quantum Mechanics 18: Helium Ground State Revisited

Concerning the ground state and ground state energy of Helium the following illumination can be made:

Standard quantum mechanics describes the ground state of Helium as $1S2$ with a 6d wave function $\psi (x1,x2)$ depending on two 3d Euclidean space coordinates $x1$ and $x2$ of the form

• $\psi (x1,x2) =C \exp(-Z\vert x1\vert )\exp (-Z\vert x2\vert )$,       (1)

with $Z =2$ the kernel charge, and $C$ a normalising constant. This describes two identical spherically symmetric electron distributions as solution of a reduced Schrödinger equation without electronic repulsion potential, with a total energy $E =-4$, way off the observed $-2.903$. 

To handle this discrepancy between model and observation the following corrections in the computation of total energy are made, while keeping the spherically symmetric form (1) of the ground state as the solution of a reduced Schrödinger equation:  

1 . Including Coulomb repulsion energy of (1) gives  $E=-2.75$.

2. Changing the kernel attraction to $Z=2 -5/16$ claiming screening gives $E=-2.85$.

3. Changing Coulomb repulsion by inflating the wave function to depend on $\vert x1-x2\vert$ can give  at best $E=-2.903724...$ to be compared with precise observation according to Nist atomic data base $-2.903385$ thus with an relative error of $0.0001$. Here the dependence on $\vert x1-x2\vert$ of the inflated wave function upon integration with respect to $x2$ reduces to a dependence on only the modulus of $x1$. Thus the inflated non spherically symmetric wave function can be argued to anyway represent two spherically symmetric electronic distributions.

We see that a spherically symmetric ground state of the form (1) is attributed to have correct energy, by suitably modifying the computation of energy so as to give perfect fit with observation. This kind of physics has been very successful and convincing (in particular to physicists), but it may be that it should be subject to critical scientific scrutiny.

The ideal in any case is a model with a solution which ab initio in direct computation has correct energy, not a  model with a solutions which has correct energy only if the computation of energy is changed by some ad hoc trick until match.

The effect of the fix according to 3. is to introduce a correlation between the two electrons to the effect that they would tend appear on opposite sides of the kernel, thus avoiding close contact. Such an effect can be introduced by angular weighting in (1) which can reduce electron repulsion energy but at the expense of increasing kinetic energy by angular variation of wave functions with global support and then seemingly without sufficient net effect. With the local support of the wave functions meeting with a homogeneous Neumann condition (more or less vanishing kinetic energy) of the new model, such an increase of kinetic energy is not present and a good match with observation is obtained.

New Quantum Mechanics 17: The Nightmare of Multi-Dimensional Schrödinger Equation

Once Schrödinger had formulated his equation for the Hydrogen atom with one electron and with great satisfaction observed an amazing correspondence to experimental data, he faced the problem of generalising his equation to atoms with many electrons.

The basic problem was the generalisation of the Laplacian to the case of many electrons and here Schrödinger took the easy route (in the third out of Four Lectures on Wave Mechanics delivered at the Royal Institution in 1928) of a formal generalisation introducing a set of new independent space coordinates and associated Laplacian for each new electron, thus ending up with a wave function $\psi (x1,...,xN)$ for an atom with $N$ electrons depending on $N$ 3d spatial coordinates $x1$,...,$xN$.

Already Helium with a Schrödinger equation in 6 spatial dimensions then posed a severe computational problem, which Schrödinger did not attempt to solve.  With a resolution of $10^2$ for each coordinate an atom with $N$ electrons then gives a discrete problem with $10^{6N}$ unknowns, which already for Neon with $N=10$ is bigger that the total number of atoms in the universe.

The easy generalisation thus came with the severe side-effect of giving a computationally hopeless problem, and thus from scientific point meaningless model.

To handle the absurdity of the $3N$ dimensions rescue steps were then taken by Hartree and Fock to reduce the dimensionality by restricting wave functions to be linear combinations of products of one-electron wave functions $\psi_j(xj)$ with global support:

• $\psi_1(x1)\times\psi_2(x2)\times ....\times\psi_N(xN)$    

to be solved computationally by iterating over the one-electron wave functions. The dimensionality was further reduced by ad hoc postulating that only fully symmetric or anti-symmetric wave functions (in the variables $(x1,...,xN)$) would describe physics adding ad hoc a Pauli Exclusion Principle on the way to help the case. But the dimensionality was still large and to get results in correspondence with observations required ad hoc trial and error choice of one-electron wave functions in Hartree-Fock computations setting the standard.

We thus see an easy generalisation into many dimensions followed by a very troublesome rescue operation stepping back from the many dimensions. It would seem more rational to not give in to the temptation of easy generalisation, and in this sequence of posts we explore such a route.

PS In the second of the Four Lectures Schrödinger argues against an atom model in terms of charge density by comparing with the known Maxwell's equations for electromagnetics in terms of electromagnetic fields, which works so amazingly well, with the prospect of a model in terms of energies, which is not known to work.

New Quantum Mechanics 16: Relation to Hartree and Hartree-Fock

The standard computational form of the quantum mechanics of an atom with N electrons (Hartree or Hartree-Fock) seeks solutions to the standard multi-dimensional Schrödinger equation as linear combinations of wave functions $\psi (x1,x2,...,xN)$ depending on $N$ 3d space coordinates $x1$,...,$xN$ as a product:

• $\psi (x1,x2,...,xN)=\psi_1(x1)\times\psi_2(x2)\times ....\times\psi_N(x_N)$ 

where the $\psi_j$ are globally defined electronic wave functions depending on a single space coordinate $xj$.

The new model takes the form of a non-standard free boundary Schrödinger equation in a wave function $\psi (x)$ as a sum:

• $\psi (x)=\psi_1(x)+\psi_2(x)+....+\psi_N(x)$,

where the $\psi_j(x)$ are electronic wave functions with local support on a common partition of 3d space with common space coordinate $x$.

The difference between the new model and Hartree/Hartree-Fock is evident and profound.  A big trouble with electronic wave functions having global support is that they overlap and demand an exclusion principle and new physics of exchange energy.  The wave functions of the new model do not overlap and there is no need of any exclusion principle or exchange energy.

PS Standard quantum mechanics comes with new forms of energy such as exchange energy and correlation energy. Here correlation energy is simply the difference between experimental total energy and total energy computed with Hartree-Fock and thus is not a physical form of energy as suggested by the name, simply a computational /modeling error.

New Quantum Mechanics 15: Relation to "Atoms in Molecules"

Atoms in Molecules developed by Richard Bader is a charge density theory based on basins of attraction of atomic kernels with boundaries characterised by vanishing normal derivative of charge density.

This connects to the homogeneous Neumann boundary condition identifying separation between electrons of the new model under study in this sequence of posts.

Atoms in Molecules is focussed on the role of atomic kernels in molecules, while the new model primarily (so far) concerns electrons in atoms.

New Quantum Mechanics 14: $H^-$ Ion

Below are results for the $H^-$ ion with two electrons and a proton. The ground state energy comes out as -0.514, slightly below the energy -0.5 of $H$, which means that $H$ is slightly electro-negative and thus by acquiring an electron into $H^-$ may react with $H^+$ to form $H2$ (with ground state energy -1.17), as one possible route to formation of $H2$. Another route is covered in this post with two H atoms being attracted to form a covalent bond.

The two electron wave functions of $H^-$ occupy half-spherical domains (depicted in red and blue) and meet at a plane with a homogeneous Neumann condition satisfied on both sides.

New Quantum Mechanics 13: The Trouble with Standard QM

Standard quantum mechanics of atom is based on the eigen functions of the Schrödinger equation for a Hydrogen atom with one electron, named "orbitals" being the elements of the Aufbau or build of many-electron atoms in the form of s, p, d and f orbitals of increasing complexity, see below.

These "orbitals" have global support and has led to the firm conviction that all electrons must have global support and so have to be viewed to always be everywhere and nowhere at the same time (as a basic mystery of qm beyond conception of human minds). To handle this strange situation Pauli felt forced to introduce his exclusion principle, while strongly regretting to ever have come up with such an idea, even in his Nobel Lecture:

• Already in my original paper I stressed the circumstance that I was unable to give a logical reason for the exclusion principle or to deduce it from more general assumptions. 
• I had always the feeling and I still have it today, that this is a deficiency. 
• Of course in the beginning I hoped that the new quantum mechanics, with the help of which it was possible to deduce so many half-empirical formal rules in use at that time, will also rigorously deduce the exclusion principle. 
• Instead of it there was for electrons still an exclusion: not of particular states any longer, but of whole classes of states, namely the exclusion of all classes different from the antisymmetrical one. 
• The impression that the shadow of some incompleteness fell here on the bright light of success of the new quantum mechanics seems to me unavoidable. 

In my model electrons have local support and occupy different regions of space and thus have physical presence. Besides the model seems to fit with observations. It may be that this is the way it is.

The trouble with (modern) physics is largely the trouble with standard QM, the rest of the trouble being caused by Einstein's relativity theory. Here is recent evidence of the crisis of modern physics:
The LHC "nightmare scenario" has come true.

Here is a catalogue of "orbitals" believed to form the Aufbau of atoms:

And here is the Aufbau of the periodic table, which is filled with ad hoc rules (Pauli, Madelung, Hund,..) and exceptions from these rules:

 

New Quantum Mechanics 12: H2 Non Bonded

Here are results for two hydrogen atoms forming an H2 molecule at kernel distance R = 1.4 at minimal total energy of -1.17 and a non-bonded molecule for larger distance approaching full separation for R larger than 6-10 at a total energy of -1. The results fit quite well with table data listed below.

The computations were made (on an iPad) in cylindrical coordinates in rotational symmetry around molecule axis on a mesh of 2 x 400 along the axis and 100 in the radial direction. The electrons are separated by a plane perpendicular to the axis through the the molecule center, with a homogeneous Neumann boundary condition for each electron half space Schrödinger equation. The electronic potentials are computed by solving a Poisson equation in full space for each electron.

PS To capture energy approach to -1 as R becomes large, in particular the (delicate) $R^{-6}$ dependence of the van der Waal force, requires a (second order) perturbation analysis, which is beyond the scope of the basic model under study with $R^{-1}$ dependence of kernel and electronic potential energies.

%TABLE II. Born–Oppenheimer total, E

%Relativistic energies of the ground state of the hydrogen molecule

%L. Wolniewicz

%Citation: J. Chem. Phys. 99, 1851 (1993); 

for two hydrogen atoms separated by a distance R bohr

 R    energy

0.20 2.197803500 

0.30 0.619241793 

0.40 -0.120230242 

0.50 -0.526638671 

0.60 -0.769635353 

0.80 -1.020056603 

0.90 -1.083643180 

1.00 -1.124539664 

1.10 -1.150057316 

1.20 -1.164935195 

1.30 -1.172347104 

1.35 -1.173963683 

1.40 -1.174475671 

1.45 -1.174057029 

1.50 -1.172855038 

1.60 -1.168583333 

1.70 -1.162458688 

1.80 -1.155068699 

2.00 -1.138132919 

2.20 -1.120132079 

2.40 -1.102422568 

2.60 -1.085791199 

2.80 -1.070683196 

3.00 -1.057326233 

3.20 -1.045799627 

3.40 -1.036075361 

3.60 -1.028046276 

3.80 -1.021549766 

4.00 -1.016390228 

4.20 -1.012359938 

4.40 -1.009256497 

4.60 -1.006895204 

4.80 -1.005115986 

5.00 -1.003785643 

5.20 -1.002796804 

5.40 -1.002065047 

5.60 -1.001525243 

5.80 -1.001127874 

6.00 -1.000835702 

6.20 -1.000620961 

6.40 -1.000463077 

6.60 -1.000346878 

6.80 -1.000261213 

7.00 -1.000197911 

7.20 -1.000150992 

7.40 -1.000116086 

7.60 -1.000090001 

7.80 -1.000070408 

8.00 -1.000055603 

8.50 -1.000032170 

9.00 -1.000019780 

9.50 -1.000012855

10.00 -1.000008754 

11.00 -1.000004506 

12.00 -1.000002546

New Quantum Mechanics 10: Ionisation Energy

Below are sample computations of ground states for Li1+, C1+, Ne1+ and Na1+ showing good agreement with table data of first ionisation energies of 0.2, 0.4, 0.8 and 0.2, respectively.

Note that computation of first ionisation energy is delicate, since it represents a small fraction of total energy.

New Quantum Mechanics 9: Alkaline (Earth) Metals

The result presentation continues below with alkaline and alkaline earth metals Na (2-8-1), Mg (2-8-2), K (2-8-8-1), Ca (2-8-8-2),  Rb (2-8-18-8-1), Sr (2-8-18-8-2), Cs (2-8-18-18-8-1) and Ba (2-8-18-18-8-2):