Uncertainty Principle vs Real Quantum Mechanics

Heisenberg’s Uncertainty Principle UP is a cornerstone of quantum mechanics stating that there is a limit to the precision both position and velocity of a particle like an electron, can be determined. The standard hand-waving argument is that precise measurement of position changes velocity and vice versa, and so precise measurement of both to arbitrary precision is impossible. This makes quantum mechanics fundamentally different from classical mechanics, where there is no such limit to measurement precision, in principle.

The other cornerstone is Schrödinger's wave function, which does not contain UP even if it is supposed to tell everything there is to say about the system. UP is thus an add-on to standard Quantum Mechanics stdQM based on Schrödinger’s equation, and then connected to measurement.

Is there any UP in RealQM? We recall that RealQM describes an atomic system as a collection of non-overlapping extended electronic charge densities which do not have particle character. For an extended body (on macro or microscale) there is no unique point position and velocity describing the state of the body and so there is a certain fuzziness or certain uncertainty depending on size if only one point, such as the center of gravity is to be used. RealQM thus, just like classical continuum mechanics for extended elastic bodies, comes with a form of UP depending on size, but this is perfectly normal and no mystery.

In RealQM there is no built-in limit to possible measure precision. The charge densities of an atom in ground state are stationary in space and shift around under radiation and so come with the usual fuzziness of extended bodies, which on microscale of course can be very real making it very difficult to point-wise determine the electron charge density in an atom, but there is no mystery like that of Heisenberg.

No Split between Classical and Quantum Mechanics

The scales of the World without split.

À standard answer to a question about the difference between classical and quantum mechanics is that the former concerns macroscopic physics and the latter microscopic physics, and of course size makes a huge difference, right? But more precisely, what difference does size make? 

We know that an ant faces different conditions/forces than an elephant, but the same basic laws of physic apply. We know that the Moon follows the same all-size law of gravitation as ants and elephants. 

A standard physicist would now expand on the macro-micro split by informing us that macroscopic physics is deterministic while microscopic physics is non-deterministic or probabilistic, and that is huge difference: Atoms play dice while the macroscopic world is predictable or deterministic. 

You could react on this message with skepticism from a wealth of experience that a complex macroscopic world is unpredictable or non-deterministic, while an atom as an entity of utter simplicity must be expected to be predictable deterministic. So if determinism is to be what makes macro different from micro, it would be micro/simple which is deterministic and macro/complex which is non-deterministic, in direct opposition to standard physics. 

Let us now seek the reason why standard physics insists that microscopic physics is non-deterministic while macroscopic physics as meanvalues of microscopic physics is deterministic, even if that contradicts sound logic. We shall see that it all depends on the probabilistic interpretation of the wave function of quantum mechanics by Max Born, which has captured modern physics. Born proclaimed in 1926 that the square of the wave function would express the probability of the occurrence of the configuration acting as argument for the wave function as a function over configuration space, as the new buzz word which gave him the Nobel Prize in 1954 with a near 30 year delay. Let us now seek the origin of the concept of a wave function over a configuration space measuring configuration probabilities as the hall mark of quantum mechanics.  

The motion in 3d space over time $t$ of a classical mechanical system of $N$ particles is described by Newton's Laws of Motion or alternatively in terms of Lagrangian mechanics based on the Principle of Least Action for a Lagrangian

• $L(x_1,x_2,...,x_N,\dot x_1,...\dot x_N)$       (1)

where $x_i (t)$ is the position and $\dot x_i (t)\frac{dxi}{dt}$ the velocity of particle $i=1,...,N$ as function of time $t$ as a description of the evolution of the state of the system over time,  taking the form  of Newton/Lagrange's Laws of Motion. Various coordinates for 3d space may be used, such as Euclidean, spherical, cylindrical et cet.

It is here common to refer the set of all possible states $(x_1,x_2,...,x_N,\dot x_1,...\dot x_N)$ as configuration space, which thus has $6N$ dimensions. It is important to understand that all particles exist in the same 3d space and that the $x_i(t)$ with $i=1,2,,,N$ simply describe the locations of all the particles at some time $t$. The setting is a common 3d space for all particles and the $6N$ dimensional configuration space is a formality. In short (1) makes perfect sense. Out of all possible states in configuration space a specific trajectory $X(t)=(x_1(t),...,x_N(t)$ with coordinates $x_i(t)$ is determined by Lagrange's equations as a function of time $t$ only. 

We now turn to quantum mechanics with its wave function for $N$ electrons of the form

• $\Psi (x,t)=\Psi (x_1,x_2,...,x_N,t)$                            (2)

where now the $x=(x_1, ...,x_N)$ are coordinates for $S=R^3\times R^3\times ...\times R^3$ as $N$ different independent copies of 3d space $R^3$ with coordinate $x_i$. Superficially (2) resembles (1) and we may even speak of a $S$ as a configuration space for (2) with coordinates $x=(x_1,x_2,...,x_N)$.

The wave function $\Psi (x,t)$ satisfies Schrödinger's equation $i\dot\Psi =H\Psi$ with $H$ the Hamiltonian differential operator acting on the wave function $\Psi$ depending on $(x,t)$-coordinates with $x=(x_1,...,x_N)$ $3N$-dimensional. 

The hope of Schrödinger was that the generality of Lagrangian mechanics in terms of number of particles (and choice of coordinates), would allow a direct generalisation of his stunningly successful equation for the Hydrogen atom to atoms/molecules with many electrons. What could go wrong if (2) was similar to (1) both with formally a high-dimensional configuration space?

Let us compare (1) and (2) or more specifically $X(t)$ as a classical trajectory satisfying Lagrange's equations as a function depending on $t$, and $\Psi (x,t)$ satisfying Schrödinger's equations as a function of $(x,t)$. Ok, we see a clear difference in the coordinate dependence: $X(t)$ depends on a time coordinate, while $\Psi (x,t)$ depends on a time coordinate and a multidimensional spatial coordinate $x=(x1,...,xN)$. 

That is a monumental difference. Classical mechanics describes actual trajectories $X(t)$ in $R^3$ selected by Lagrange's equations from a pool of all possible trajectories, which is never covered.

On the other hand, the wave function $\Psi (x,t)$ with its multidimensional space variable $x$ freely sweeping configuration space $S$ independently of $t$, thus describes possiblities and not actuality and there are immensely many more possibilities than actualities. The wave function $\Psi (x,t)$ is thus a monster supposed to describe the probability of all possible configurations, which is such a huge undertaking that it swamps every conceivable effort by its complexity.

To reduce complexity the variation of $\Psi (x,t)$ is (ad hoc) reduced to consist of linear combinations of products of single electronic wave functions $\psi_i(x_i,t)$ each one depending on a single spatial coordinate $x_i$, typically in the form of so called Slater determinants, but even that is like seeking to emptying an ocean with a spoon as an uncomputable problem. 

On the other hand, in RealQM a different Schrödinger equations is formulated for a wave function of the form 

• $\Psi (y,t) = \sum_i\psi_i(y,t)$          

as a sum of one-electron wave functions $\psi_i(y,t)$ all depending on the same common physical 3d coordinate $y$, which is computable in the same sense as a classical deterministic continuum mechanics solid/fluid problem. 

In RealQM the physics on macro and microscope is the same, which is to expect since a split cannot be made.  

The split macro-micro of standard quantum mechanics is an artefact of the introduction of a configuration space of all possible configurations which is way too big to handle.

Another strange aspect of the standard Schrödinger equations are linear thus allowing superposition of states as something unexpected. Newton/Lagrange's equations in general are non-linear and superposition a rare effect of linearity only in very special cases, like a small amplitude vibrating string. The Schrödinger equation of RealQM is non-linear, just like the majority of similar equations in solid/fluid mechanics. 

In short, there are many serious reasons to consider the standard multi-dimensional linear Schrödinger equation as a too easily formed mathematical construct without physics. As compensation $\Psi$ is invited to belong to a mathematical Hilbert space $\mathbb{H}$ of maximal prestige and so the world can very neatly be described by a $\Psi$ in a $\mathbb{H}$, but without physics. 

One can argue that classical physics concerns real physics using concrete Calculus, while standard quantum physics rather is is a form of formal mathematics without real physics. 

Quantum Mechanics: Probability or Not?

The main mystery of quantum mechanics in its proclaimed consensus version (whatever it is), is the interpretation of the wave function in terms real physics/ontology as the 

• Probability of finding electrons at some specific locations in an atom.      (*)

When asking a physicist (or chatGPT) about the meaning of "finding an electron at some specific location in an atom", you get the answer that such a thing is impossible. Locating an electron experimentally in an atom is impossible. An electron is both everywhere and nowhere at the same time. So finding an electron in an atom is impossible and so (*) has no physical meaning. 

We must then conclude that it cannot be possible to make experimental statistics out of impossible findings and so (*) cannot be experimentally verified. It is like being prevented from tossing a specific coin in an investigation of the statistical characteristics as possible non-symmetry and bias of that specific coin, and so have to be satisfied with a message that a theoretical coin certainly is perfectly symmetric without any bias, saying nothing specific about a specific coin. Not very illuminating as concerns a real coin or atom. In short: It is impossible to give (*) a physical/ontological meaning.  

The proclaimed probabilistic nature of the wave function for an atom is in sharp contrast to very precise measurements of atomic spectra spectrum without any sign of randomness. 

If then only deterministic qualities of an atom (like spectrum) are possible to observe, and very sharply so, why are physicists still insisting on (*) while admitting that experimental verification is impossible?

The answer is simple, yet hidden to the general audience: The reason is the multi-dimensional nature of the wave function $\Psi (x1,x2,...,xN)$ depending on $N$ 3d spatial coordinates $x1, x2,...,xN$ for an atom with $N$ electrons, thus altogether $3N$ spatial coordinates, and so e g 6 spatial dimensions for Helium with two electrons. It is like each electron is given a separate 3d space to live in, and no common 3d space for all electrons. Only for Hydrogen with 1 electron does the wave function have a real physical meaning in physical 3d space as an electronic charge distribution in a classical continuum mechanical sense without any stroke of randomness. 

Schrödinger kick-started quantum mechanics in 1925 by writing down the Schrödinger equation for the Hydrogen atom guided by some heavenly inspiration, but directly stumbled on the Helium atom with two electrons. A formal mathematical extension from one to many electrons was possible with a stroke of pen without thought by introducing a multi-dimensional wave function. But Schrödinger was not happy with the result, because it lacked physicality or "Anschaulichkeit".  

For $N>1$ the multidimensional wave function $\Psi (x1,x2,...,xN)$ lives outside real 3d space and then can only be interpreted as possibility rather actuality/reality. It was Max Born who invented this probabilistic interpretation, which so upset Schrödinger that he gave up quantum mechanics saying in 1926 to Bohr:  

• If all this damned quantum jumping were really here to stay, I should be sorry, I should be sorry I ever got involved with quantum theory. 

The proclaimed probabilistic nature of atomic physics is thus a consequence of a mathematical theoretical ad hoc assumption devoid of physics making generalisation to many electrons an easy/trivial catch, but then coming with many pseudo-problems without physics and answers filling physics books. It is like postulating that all celestial motion is circular or circle-upon-circle following Aristotle, because from mathematical point of view circles are perfect, which delayed scientific progress two millennia. What progress delay has been caused by (*) over the century it has served as road block? 

It is here that RealQM in the spirit of Schrödinger comes in with a different deterministic generalisation from Hydrogen to $N>1$ with direct extension to molecules and chemistry,  100 years later. Take a look. No statistics! Instead real physics!

The Role of Differentiation and Integration in Physics

This is a further reflection on the idea of Physics as Computation in the previous post with focus on the mystery of instant action at distance (there are many posts on this topic). 

Mathematical models of physics typically take the form of differential equations such as Poisson’s equation 

• $\rho=\Delta\Phi$        (1)
• $\Delta\Phi = \rho$      (2)

connecting gravitational/electric potential $\Phi (x,t)$, depending on a space coordinate $x$ and time coordinate $t$ coordinate, to mass/charge density $\rho (x,t)$, and $\Delta$ is the Laplacian differential operator involving second order differentiation.  

In a Hen-Egg setting $\Phi$ represents Hen and $\rho$ Egg, either as local differentiation/assignment $\rho =\Delta\Phi$ as Hen-laying-Egg,  or solution of $\Delta\Phi = \rho$ by global integration/summation as Egg-generating-Hen by instant action at distance. 

How to choose between (1) and (2)? Local differentiation or global integration/instant action at distance? 

If you are a (pure) mathematician, you would without hesitation say that there is a method for symbolic differentiation and so (1) is in a sense trivial. On the other hand there is no method for symbolic solution of (2), which is the non-trivial problem of the symbolic Calculus of Leibniz/Newton. 

Mathematicians know that if symbolic solution fails, because it has no method, it is always possible to resort to numerics as a form of trivial work-horse, which case-by-case can compute solutions by number crunching. So is mathematics split into symbolic/analytical mathematics and numerical mathematics (in descending prestige) with essentially different basic elements: symbols or numbers.

As an example, symbolic differentiation is trivial while numerical differentiation is a delicate subject because a derivative $\frac{dx}{dt}$ involves the quotient of small numbers requiring precision. In general differentiation is a delicate process because precise identification is needed. So what can be trivial in symbolic mathematics can be non-trivial in numerical mathematics. 

On the other hand, symbolic integration is non-trivial while numerical integration is trivial as it is just a form of summation. 

So the world of symbolic mathematician and numerical mathematics is very different, since what is trivial or non-trivial can be opposite. 

We now turn to real physics as something real existing in the real world (ontology). What is the relation of real physics to symbolic mathematics and to numerical mathematics? 

Since symbolic mathematics works with symbols rather than numbers it has a connection to epistemology. We now ask if numerical mathematics is closer to ontology/real physics and so if we can learn anything about real physics from numerical computation.

In particular, we seek the real physics of (1) vs (2) as the Hen-Egg question posed above. We recall that numerical solution of (2) is trivial as simply global summation, while (1) is non-trivial as delicate differentiation. 

If we believe that physics is non-trivial,  then (1) represents physics: Hen lays Egg as a delicate non-trivial local operation, but not asking for instant action at distance in a global solution process. 

On the other hand, for (2) to represent physics requires instant action at distance as instant global summation. 

We are thus led to the conclusion that (1) represents real physics as a local differentiation process. The apparent instant action at distance in (2) would then represent non-real fiction.

We thus find support of an idea that computation and real physics are closely connected, while the connection of symbolic mathematics to real physics can be difficult to assess.  

The gravitational potential generates mass by local differentiation. Mass does not (have to) generate gravitational potential by global instant action at distance. 

It seems to make sense to say that physics differentiates in the sense of evaluating force differences, while physics integrates by time stepping creating motion, which opens to physics without mysterious instant action at distance. In numerics differentiation (1) can be traded with integration (2) with fictitious instant action at distance.

Another aspect is that you can see mass but not gravitational potential itself only its effect, and you may be tempted to believe that what you can see is primary and what you cannot see is secondary. But that depends on your senses of perception and so may not tell the true story from an objective physical point of view. You see a person getting smaller receding from you, but you know it is an illusion.

PS It is also possible to give up the cause-effect aspect in the potential-mass connection and like Leibniz say that potential and mass are connected in Perfect Harmony or Best of Worlds, which has been ridiculed...maybe it is time for Leibniz to come back...in any case he laid the mathematical foundations to the digital world as a world combining ontology with epistemology... 

But of course it is possible to turn the argument around as follows: Consider Newton's 2nd Law

• $\frac{dv}{dt} = f(t)$     (3)

where $v(t)$ is velocity and $f(t)$ is force. The standard view is that the force $f(t)$ is given and causes the acceleration $\frac{dv}{dt} = f(t)$ as (2). But we may also view $v(t)$ to be given and $f(t)=\frac{dv}{dt}$ simply the force required as in (1). This describes a situation where the nature of the force is unclear, while velocity/motion is very real. This is the case with the Coriolis force and of course centrifugal force. Einstein tried to get rid of gravitational force (and aether) altogether, but did not succeed…

Numerical solution of (3) is done by time stepping $dv=f(t)*dt$ updating velocity with input from force, which has direct physical meaning as motion as summation, thus with numerics in tandem along with (2).   

 

Conclusion: We may say that numerics can connect to both (1) and (2), while the role of symbolic math in physics remains to be made precise with the physical meaning of the symbolic wave function $\Psi$ of quantum mechanics, as solution to Schrödinger’s equation, after 100 years of constant brooding, still being a complete mystery in its standard so called Copenhagen interpretation. On the other hand, RealQM offers a physical meaning in classical continuum mechanics terms with the kinetic energy of the electrons appearing as a form of elastic energy preventing the electron to fall into the kernel by an elastic force balancing kernel attraction, just like the centrifugal force of motion prevents a planet to fall into its Sun. Both forces appear as necessary conditions for maintenance of certain states (Lagrange multipliers) as virtual forces without concrete physical origin: Planets move the way they do because forces balance, and electrons assemble around the kernel because forces balance. Leibniz would agree, I am sure!

Recall that the wave function $\Psi (x)$ for the ground state of the Hydrogen atom minimises the total energy E as "kinetic" energy + potential energy: 

• $E(\psi ) = \frac{1}{2}\int\vert\nabla\psi\vert^2dx- \int\frac{\psi^2 (x)}{\vert x\vert}dx$

over all real-valued functions $\psi (x)$ with $\int\psi^2dx=1$, which can alternatively be interpreted as the state of a normalised elastic cloud subject to an elastic central force with the kinetic energy appearing as "elastic" energy.  The terminology "kinetic" energy is misleading (motivated by some deep symbolism) since no motion is involved, while "elastic" energy has a concrete physical meaning as a measure of elastic compression suggesting some form of electronic charge compression for the atom.   

Physics as Computation at John Chappell's Natural Philosophy

This is an intro to a live video talk I will give on Febr 3 2024 on John Chappell's channel Natural Philosophy: Where Critical Thinking Challenges Theory (directly connecting to the slogan of this blog). If you feel that this must be crackpot science, take a look at my arguments before deciding and remember that established physics can be crackpot science.

Digital computation, with AI (or even AGI) as latest achievement, is today reshaping human conditions and it is natural to ask if also the science of physics as the inner core of existence is transformed.

Classical physics is based on mathematical models in the form of differential equations expressing balance (of forces) in some system, such as Euler’s equations for fluid mechanics and Maxwell’s equations for electro-magnetics, while modern atomic physics is based on Schrödinger’s equation. 

The equations express system forces while solutions of the equations represent evolution in time of systems under given conditions. The task of determining solutions is thus central and here digital computation opens entirely new perspectives with computational complexity or computability as key element. 

Uncomputable systems keep their information hidden to inspection, with prime example Schrödinger’s equation which in its standard multidimensional form is beyond the capacity of any thinkable digital computer. On the other hand, computing solutions to Euler’s equations resolves the enigma of turbulence, as will be shown in the talk.

It is natural to view the evolution in time of a physical system as a form of analog finite precision computation as the action of forces takes the system over small time steps from one state to the next, which can be modeled by finite precision digital computation: 

• Physics as Analog Computation as Digital Computation.

The key elements of computability are (i) finite precision and (ii) stability/wellposedness as a measure of precision required to make computational model output reliable. Forward-in-time evolution then shows to be computable because it is stable, while backward in time evolution is uncomputable because it is unstable, which can be seen to be the essence of the 2nd Law. 

Physics as Computation offers solutions to open problems of (i) turbulence and (ii) atomic physics through new computable forms of Euler's and Schrödinger’s equations, which are the subjects of the talk: 

• Secret of Flight   (Real Euler: Computational Turbulent Incompressible Flow)
• Real Quantum Mechanics

Real here directly connects to computability. A real physical system computes its own evolution forward-in-time and so is analog computable and a mimicing digital computable model can be viewed to be a real model:

• Real models are digital computable because reality is analog computable. 

The standard multidimensional Schrödinger equation is an uncomputable model without real physical meaning (only statistical). RealQM is computable and has a real physical meaning as a collection of non-overlapping interacting charge densities.

Real Euler computes real turbulent flow, and RealQM computes real atoms/molecules, which opens entirely new perspectives on physics: Physics as Computation. 

Real Euler gives an explanation of the 2nd Law (Computational Thermodynamics) as forward-in-time computability and backward-in-time uncomputability. See the book The Clock and the Arrow for a general audience.

There is a connection to Wolfram’s Computational Foundations for the Second Law of Thermodynamics in the sense that computation is central, but the essence is different: For Wolfram it is computational irreducibility, while I favor finite precision+stability.  

Instant Action at Distance in Atom Physics/Quantum Mechanics

Instant action at distance is a fundamental element of both macro-scale gravitational mechanics and micro-scale quantum mechanics in the form of Newton’s Law of gravitation and Coulomb’s Law of electrostatics. 

The idea is that the presence of a mass/charge at one point in physical space without time delay generates a force at all other points decaying with the inverse square of distance, as the fundamental force of both classical and modern physics of Newton/Einstein and Heisenberg and Feynman as the golden boys of quantum mechanics, and of course Schrödinger.  

It also formed the foundation of the now forgotten, but once great, physicist Joseph Boscovich (1711-1787) as expressed in his monumental "A Theory of Natural Philosophy reduced to one unique Law of forces that exist in Nature" stating that the World is the result of instant action at distance of attractive and repulsive forces on both small and large scales. This a nothing but a Grand Unified Theory and what remains is to fill in details about the forces and in particular to explain how instant action at distance is realised, which has remained a fundamental mystery of physics. See the book Roger Boscovich-The Founder of Modern Science, by Stoiljkovic.

One way to summarise physics is to recall that both Newton's Law and Coulomb's Law take the form of Poissons’ equation: 

• $\Delta \phi (x) = \rho (x)$                                (1)

where $\Delta$ is the Laplacian acting in 3d space with coordinates $x$, $\phi (x)$ is  gravitational/electric potential and $\rho (x)$ is mass/charge density. This is a consequence of in the equation (1) viewing $\rho (x)$ as a locally given source generating the potential $\phi (x)$ globally as a solution to Poisson's equation which can be seen as a form of instant integration/summation process sending local source information instantly around globally as instant action at distance. Forces are generated as $\nabla\phi (x)$.

Boscovich's Theory that all force is instant action at distance contradicted the classical idea that forces are transmitted by contact, adding the explanation that there is always some little distance between different material bodies including atoms maintained by ever-present repellation thus reducing physics to one unique Law. See the book Roger Boscovich- The Founder of Modern Science by Stoiljkovich. 

It is natural to consider (1) as a limit of the following time dependent heat/wave equations:

• $\epsilon\dot\phi -\Delta \phi = -\rho$,     (2)
• $\ddot\phi -\Delta\phi = -\rho$,                  (3)

where the dot indicates differentiation with respect to time $t$, and $\epsilon >0$ is small constant formally reducing (2) and (3) to (1) when tending to zero. The expanded models require some form of heat conduction or wave propagation medium/ether giving physics to action at distance with finite speed. 

On the other hand (1) could be argued to not require any medium, since force transmission is replaced by instant action at distance, but then again without explanation. 

I have argued that that there is a way out of this dilemma by shifting the conception of the meaning of the equation (1) to a view with rather the potential $\phi (x)$ as primary source from which both force $\nabla\phi (x)$ and mass $\rho (x)=\Delta\phi (x) $ are generated through the local action of differentiation by the Laplacian differential operator. 

In this view potentials are primary from which everything (force/mass/charge) is generated by local differentiation. In particular it gives a new view on the quantum mechanics of an atom, where the primary concepts are the kernel and electron potentials, and the atom with kernel and electrons is generated by the Laplacian and then required to satisfy Schrödinger's equation. 

In physics it is natural to search for sources generating effects in a cause-effect setting, but the precise mechanism of generation may be difficult to pin down, e g exactly how differentiation generates mass from gravitational potential, or how instant action at distance comes about.

This connects to Leibniz' idea of a Pre-established Harmony beyond human inspection. The gravitational potential-mass harmony expressed by (1) may be of this kind. 

You find more under Labels.

  

 

Bond and Lattice Dissociation Energies of NaH by RealQM

Let us now check if RealQM readily computes the bond dissociation energy in gaseous phase of the molecule NaH as separation into Na and H atoms, and the lattice dissociation energy in solid phase as separation into Na+ and H- ions. 

In stdQM this is viewed to be very difficult, if possible at all, which is not strange because stdQM is uncomputable (as we know). Instead, an empirical Born-Haber cycle is used. 

RealQM on the spot produces the following predictions in close agreement with reference values:

Bond dissociation energy =  E(Na) + E(H) - E(NaH)  = 0.08 Hartree  (ref 0.0765) (code) 

Lattice dissociation energy = E(Νa+) + E(H-) - E(NaH) = 0.38 Hartree (ref 0.307) (code) 

where E(X) indicates ground state energy of X, and in the code examples the distance (2*D in code) between atoms/ions is varied to capture separation while the number of iterations is kept the same for fair comparison. 

We see that the energy of a covalent bond without full electron transfer is much smaller than the lattice energy with full electron transfer from Na to H, naturally a consequence of different spatial "filling of electrons" in the sense of RealQM.  

In stdQM the mystery is deeper since the spatial presence of electrons is mysterious.

Carbon and Graphene by RealQM

A carbon atom C has up to 4 valence electrons and forms a very large variety of compounds by connecting to 1 up to 4 other atoms as CO, CO2 and CH4... 

Graphene is a 2d hexagonal pattern of carbon atoms each atom connecting to 3 other atoms, thus involving 3 valence electrons.

We now let RealQM compute the energy of C in the following shell configurations:

• 2+4                     (2 electrons in 1st shell, 4 in 2nd shell)                                (code)
• 2+2+2                (2 electrons in 1st shell, 2 in 2nd shell and 2 in 3rd shell)    (code)
• 2+3+1                (2 electrons in 1st shell, 3 in 2nd shell and 1 in 3rd shell)    (code)

and get close to the a reference ground state value of -37.7 Hartree in all three cases. We also compute ionization energies in 2+2+1 (code) and 2+3 (code) configuration in agreement with a reference value of 0.4 Hartree.

Altogether, RealQM gives a picture of C with 3 possible ground states of about the same energy available to form molecules with 1-4 bonds.

The view of standard QM stdQM is that the ground state configuration of the C atom is $1s^22s^22p^2$ with 2 electrons in spherically symmetric orbits in 1st shell, 2 electrons in spherically symmetric orbits together with 2 electrons in p-states in 2nd shell, which suggests that 1-2 valence electrons are available, but the 3 in graphene.

To get 3 valence electrons in stdQM various hybrid sp-orbital states are introduced to give the following picture of the 2d hexagonal pattern of a graphene sheet with each carbon atom connected to 3 other atoms involving 3 sp2 valence electrons and the 4th electron hoovering above and under (or around) the sheet: 

We can connect the RealQM 2+3+1 configuration to this picture with 3 electrons centered in a plane and a 4th out-of-plane electron as 3d enclosure. 

RealQM approaches the electron distribution in atoms and molecules as a real physical packing problem in 3d. RealQM is ab initio. RealQM is computable at iPad power. 

StdQM uses a formal approach based on molecular orbits which have no real physical meaning. StdQM involves a set of ad hoc rules (Pauli, Hund's rules 1-3...) and is not ab initio. StdQM fills the books on Quantum Chemistry with a message that it works, even if stdQM itself is a mystery. StdQM is uncomputable on any thinkable digital computer. 

RealQM 2nd Row of Periodic Table: Na to Argon

We now proceed to the second row of the periodic table where to speed up computation the electrons in inner shells of an atom are homogenised to a common charge density and we keep individual electrons only in the outermost shell acting as the valence shell in formation of molecules by interaction with other atoms. 

RealQM gives the following ground state energies with display of electron distribution over shells from inner to outer:  

• Ne (2+8):                   -128      (-128.5) (code)
• Ne (2+4+4):               -128      (-128.5) (code)
• Na (2+4+4+1):         -161.8    (-162.4) (code)
• Mg (2+4+4+2):         -199.4   (-200.3)  (code)
• Al  (2+4+4+2+1):     -242.4   (-242.7) (code)
• Si  (2+8+4):               -290      (-290)   (code)
• P (2+8+4+1):            -342      (-342)  (code)
• S  (2+8+4+2):          -396      (-399)   (code)       
• Cl (2+8+4+3):         - 461      (-461.4) (code)
• Ar (2+8+8):             -528       (-529)   (code) 
• K (2+8+8+1):          -601      (-602)   (code)
• Xe (2+8+18+18+8) -7458    (-7438) (code)

We see good agreement between RealQM and reference values in parenthesis. 

We see that the number of valence electrons in the outermost shell ranges from 1 to 4  (except for the noble gases Xe), which is different from stdQM with 1 to 8 valence electrons according to the "octet rule". 

We see that the 2+4+4 configuration for Neon gives about the same energy as a 2+8 configuration  suggesting that 4+4 for inner shells is the same as 8.

We note that atoms with 1-2 valence electrons are metals/metalloids and those with 3 are non-metals. This gives a very simple non-standard classification. Metals with 1-2 valence electrons have small ionization energies and those with 3 large ionization energies. 

Molecules naturally form by combining metals with non-metals such as NaCl, which will be explored in an upcoming post. 

In the above computations we have kept a full 3d resolution of all shells, with spherical charge homogenisation of inner shells to speed up. A further speed up opening to large molecules will be made by resolving inner shells in spherical symmetry and only valence shells in full 3d.  

We note that the RealQM is very simple (3 lines essentially) and and as such is essentially ab initio.  We compare with stdQM computations using Hartree-Fock or Density Functional Theory which are very complicated and thus not ab initio. 

PS It is not yet clear exactly when to stop iterations and so the number of iterations can be used to arrive exactly at reference values, if desired. Further study of stop criterion is needed. 

Helium/Neon 1st Excited RealQM

We now let RealQM compute the first excited state of Helium with one electron moving from 1st shell to 2nd shell thus giving the excited state a 1+1 configuration. We get the following results: 

• Excited state with 1st el 1st shell, 2nd el 2nd shell = -2.16 (ref -2.145) (code)
• Reference with same code but only 1 el                  = -2.00  (ref -2.000) (code)
• Reference Helium ground state                                = -2.903
• Excitation energy                                                      = 0.758  Hartree    (20.61 eV)

We see good agreement. The difference between RealQM and stdQM is that electrons do not overlap in RealQM and meet with homogeneous Neumann condition (zero flux) and charge density continuity, while in stdQM in the 1s2s configuration the outer 2s electron overlaps with the inner 1s electron. 

Spin plays no role in RealQM, while in stdQM the two electrons are assumed to have different spin (whatever the physics of spin may be). 

For Neon RealQM gives about the same excitation energy as the difference between a 2+4+4 and 2+4+3+1 configuration with 1 electron in a new outer shell (code), in accordance with observation (0.8 Hartree).

Ionization: Helium to Neon…

Ionization energies (observed) for the 2nd row of the periodic table starting with Helium and ending with Neon follows the following pattern (energy in kJ/mol with 1 Hartree = 2625 kJ/mol):

RealQM gives the following energies (with shell configuration given and list values corresponding to the above graph):

• He 2             (code)  = -2.90       (list -2.903)
• He+ 1           (code)  = -2.00      (list -2.00)
• Li 2+1          (code)  = -7.47      (list -7.48)
• Li+ 2            (code)  = -7.17      (list -7.28)
• Be 2+2          (code) = -14.6       (list -14.5)
• Be+  2+1       (code) = -14.0       (list -14.2)
• B  2+2+1       (code) =  -24.4        (list -24.5)
• B+  2+2         (code) =  -24.2      (list -24.2)
• C 2+4            (code)  = -37.7      (list -37.7)
• C 2+2+2        (code)  = -37.8       (list -37.7)
• C+ 2+2+1     (code)  = -37.3       (list -37.3)
• O 2+4+2       (code)  = -74.9       (list -74.8)
• O+ 2+4+1     (code) = -74.4        (list -74.3)
• N 2+2+3       (code) = -54.4       (list -54.4)
• N+ 2+4         (code)  = -53.7      (list -53.9)
• F 2+4+3        (code) = -99.6       (list -99.4)
• F+ 2+4+2      (code) = -98.8       (list -98.7)
• Ne 2+4+4      (code) = -128.6     (list -128.5)
• Ne+ 2+4+3    (code) = -127.5     (list -127.7)
• Na 2+4+4+1  (code) = -162     (list -162)

Fairly good agreement on 50^3 mesh with list values in the graph capturing the big ionization energies for He and Ne and steady increase from Li with kernel charge/pull, modulo the kinks Be-B and N-O. In RealQM terms the decrease Be-B can be connected to the 1 electron valence of B compared to 2 for Be, and the 2 electron valence of O compared to the 3 electron valence of N. Continue on next row starting with Na on 100^3 mesh…

Ionization: Carbon1+ and Carbon2+

Let us now test RealQM on ionization where one or more electrons are ripped off an atom at some energy expense, starting with the example of Carbon with 6 electrons. 

We have seen that a shell configuration of 2+4 (2 electrons in 1st shell and 4 electrons in 2nd shell) gives a total energy of -37.7 Hartree in correspondence with list value (code). This configuration matches CH4 (code)

On the other hand, the configuration 2+2+2 has about the same energy (code), while the electrons in the 3rd shell are less tightly bound to the kernel than the ones in the 2nd shell of the 2+4 configuration, and so would require less energy to be ripped off. 

We thus let RealQM compute the energy of Carbon1+ with one electron removed in a 2+2+1 configuration to get a total energy of -37.3 Hartree in agreement with list value (code), with thus an ionization energy of 0.4 Hartree. 

We continue with Carbon2+ with configuration 2+2 and get -36.4 with ionization energy 0.9 again in agreement with list value (code).  

Note that to compute ionization energy by subtracting total energies of atom and ion, requires three correct decimal places and so is a bit delicate, because of the required mesh size cut-off of the singular kernel potential. 

We see that both 2+4 and 2+2+2 configurations for Carbon have about the same total energy and so may both be possible, while as concerns ionization the 2+2+2 configuration in agreement with list value seems to preferred, because the ionization to 2+3 requires much bigger energy (code).

Next objective is to test if RealQM can make sense of the following pattern for 1st ionisation energies:

   

Perspective on ElectroNegativity

Let us now give more perspective on the electronegativity explored by RealQM in the previous post as the decrease of energy achieved by hypothetically adding one electron to a given atom with kernel charge Z assuming the electron configuration of the next element in the periodic table with charge Z+1. 

For example, RealQM computes a decrease of about 4 Hartree when an electron is added to Fluorine with Z=9 with electron shell configuration 2+4+3 to obtain the configuration 2+4+4 of Neon with Z=10 as the ion F-. 

In a similar way we obtain energy decrease of 0.8 Hartree for Helium- (Z=2), 1.2 for Lithium- (Z=3), 1.7 for Beryllium- (Z=4) and 2.3 for Boron- (Z=5) increasing to 4 for F- (Z=9) as the maximal electronegativity for all elements. 

RealQM gives the very small value 0.06 for H- in opposition to an accepted value of 2.  

We next ask under what conditions the ion F- will be created from F by incorporation of one electron at an energy decrease of 4 Hartree? It directly connects to the nature of the bond of  molecule HF as ionic or covalent. In an ionic bond the F atom would fully capture the electron of H with a decrease of energy of more than 3 Hartree. This is very substantial and would correspond to a dissociation energy of HF of more than 3 Hartree which is 10 times bigger than that observed.

We have earlier seen that a HF with a covalent bond has a dissociation energy in accordance with observation.

We conclude that F- appears to be hypothetical and in particular does not combine with H+ to form HF by an ionic bond. In other words, it is not clear what role electronegativity has to play if bonds are rather covalent than ionic. Any idea? Recall that direct measurement is viewed to be impossible, which gives support to a suspicion that electronegativity is more fiction than reality.

PS The accepted electronegativity of H of 2 Hartree stands out as very singular/strange:

 

 

ElectroNegativity by RealQM

Electronegativity (or rather electron affinity, see this post) of an atom measures the decrease of total energy arising from adding an electron. Pauling suggested a scale to measure electronegativity addressing the following values to the elements in the periodic table:

We see in the 2nd row electronegativity increase from 1.0 for Lithium to 4.0 for Fluorine as the maximum over all elements. 

RealQM gives the following electron affinity values measured in Hartree:

• H-    0.04  (code)
• He-   0.8    (code)
• Li-    1.2    (code)
• Ber-  1.7    (code)
• B-     2.3    (code)
• F-      5.0    (code)

We see that the the 2nd row Pauling scale matches the RealQM values in Hartrees, which makes sense to Pauling's scale. 

We note that (i) Helium is missing in the Pauling scale, and (ii) the values for H- differ fundamentally.

The reason the Pauling scale does not take up He is probably the preconceived idea of standard quantum chemistry that He as a noble gas has no incentive at all to catch an electron. RealQM tells a different story, connecting to the previous post showing that He can form a He2 molecule. 

On the other hand, RealQM gives H a very small desire to catch an electron, thus supporting the common idea that H acts as an electron donor, in particular when forming the HF molecule by combining with F with maximal electronegativity in an ionic bond.  

The Pauling value of 2.0 for H- stands out as strange and in conflict with the idea of ionic bond in HF.  

H can form H2 molecule in a covalent bond even with small electronegativity, because no entire capture of an electron is needed, only sharing. RealQM captures the difference in capturing and sharing of electrons, which standard QM does not appear to do. 

 

RealQM Predicts Helium2 Molecule

Standard physical chemistry/quantum theory says that the Helium He atom does not form molecules of two or three atoms into He2 and He3, because the two electrons "fill" the valence shell around the +2 kernel of He and so do not interact with the valence shell of any other atoms including He. In short, standard quantum theory tells that the He2 or He3 molecules do not exist. 

But this theory is contradicted by e.g. the research group of Voigtsberger reporting on observations of He3 as noted by Phys.org:

• Science has long since known that, contrary to the old school of thought, helium forms molecules of two, three or even more atoms.

Ok, so the standard quantum theory of the old school of thought says that He2 and He3 molecules do not exist, which since long is known to be wrong, but then evidently has been kept secret to not disturb the proclaimed validity of standard quantum theory. 

Let us now see what the new school of thought of Real Quantum Mechanics RealQM tells us about the total energy of He2 as two He atoms at small distance, and then compare with the energy at larger distance. 

RealQM gives He2 the energy -6.38 Hartree at a certain smaller distance, while 2He at larger distance is given the energy -5.73 Hartree, to be compared with list value for He of -2.903 or 2H of -5.806. We see that Real QM predicts the existence of a He2 molecule because it has smaller energy than 2 separated He atoms (test yourself with pjs-code):

Compare with the previous post intoxicated by the old school of thought/qunatum theory predicting that He2 does not exist, illustrating the power of theory even when contradicting observation.  Evidently reason wins: If H forms H2, then He can be expected to form He2, since He is like two H on top…

Here are results on finer mesh (try p5js-code) with the same message that He2 exists: 

    

RealQM Configuration of Neon as 2+4+4

A basic pillar of moden chemistry (supposedly with some support from standard Quantum Mechanics) is the octet rule, which says in particular that the electrons of the Neon Ne atom are arranged in two spherical shells with an inner shell around the +10 charge kernel formed by 2 electrons and an outer (valence) shell with 8 electrons, with the message that the valence shell is "full" and so Neon does not want to interact with other atoms through a covalent or ionic bond.

On the other hand, Flourine F with +9 charge kernel with 7 electrons in the outer valence shell is very keen to attach another electron to "complete the octet" and so readily forms eg an HF molecule with an H atom in a covalent/ionic bond. 

RealQM gives a different picture with Ne in 2+4+4 configuration and F as 2+4+3 with a third shell harboring 4 or 3 electrons acting as the valence shell. This means that that F combines with H to form a HF molecule, while Ne like He does not. You can interact with simulations in p5js-code here:

• Fluorine
• Neon
• HF    

For Neon in a 2+8 configuration RealQM gives a much too small energy compared to observation, which thus appears to not be attained since it requires somehow squeezing 8 electrons into one shell.

The reason H can bond to F in a 2+4+3 configuration, rather than in 2+7, appears to be that the size of an H atom better fits with the size of electrons in a valence shell of 3 electrons instead of 7.

The from observations estimated radius of an F atom is about 1 atomic unit au (50 pm). The thickness of atomic shells scales with 2/Z with Z the reduced charge reaching a shell, with thus 2/10 + 2/8 + 2/4 about 1 au for a 3-shell 2+4+4 configuration, while a 2-shell 2+8 configuration would give a too small radius compared to observation.

What can the difference be between C as 2+4, which bonds with H and O in particular, and Ne as 2+4+4, which does not want to bond with anything, when they have the same 4 electron valence  shell? It is thinkable that this is a geometric effect with the C valence shell being more compatible with those of H and O, which may be uncovered by further explorations with RealQM including the validity of the octet rule.