Multiscale Finite Element Method as Informational Structural Realism

Luciano Floridis concludes in The Philosophy of Information that the world on microscopic scales cannot be neither discrete nor continuous, since each of these standpoints is contradictory, but can maybe instead be described by Informational Structural Realism (ISR) as scale dependent structural description leaving out the true nature of the elements forming the structure. ISR can be viewed as synthesis of the discrete and continuous without internal contradiction and thus potentially as a useful world view.

This is nothing but the finite element method in multi-scale form, originally developed in structural mechanics, as scale dependent discretization of continuous differential equations, where a true physical realisation of the finite elements is not possible nor necessary.

This connects to my attempt to describe a complex world including turbulence and quantum mechanics as analog finite precision computation simulated by digital finite precision computation, where the flow of information under computation represents ISR and the true physical nature of the analog computation is unknowable, but irrelevant.

Why is then both discrete and continuous physics impossible? Because both requires infinite resolution: a discrete point particle or discontinuity has zero size and a continuum has no smallest size. Thus both discrete and continuous physics requires infinitely small resolution and thus an infinite amount of information on any scale. If you don´t think this is asking for too much, then you should reconsider your notion of the infinite.

Finite Element Quantum Mechanics 6: Basic Analysis vs Observation

Let us now inspect the basics of the atomic model considered in this sequence of posts.

Consider then a neutral atom of kernel charge $Z$ with $N=Z$ electrons occupying non-overlapping domains in space. Assume that the electrons are partitioned into a sequence of shells $S_m$ of increasing radius $r_m$ with corresponding widths $d_m$ each shell being filled by $2m^2$ electrons, for $m=1,...,M,$ with $M$ the number of shells. We consider a hypothetic atom with all shells fully filled with $2, 8, 18, 32, 50,...,$ electrons in successive shells displaying a basic aspect of the periodicity of the periodic table of elements.

Consider now the case $d_m\sim m$ with $r_m\sim m^2$, and assume $r_1=d_1\sim\frac{1}{Z}$. The electron density $\rho_m$ in $S_m$, assumed to be spherically symmetric,  then satisfies

• $\rho_mr_m^2d_m\sim m^2$

from which follows that 

• $\rho_m\sim \frac{m^3}{r_m^3}$.                            (1)

We now compute the following characteristics of this model:

1. $M^3\sim Z$, that is $M\sim Z^{\frac{1}{3}}$
2. potential energy in $S_1\sim Z^2$
3. potential energy in $S_m\sim m^2Z/r_m\sim Z/d_1\sim Z^2$
4. total potential energy and thus total energy $\sim Z^{\frac{7}{3}}$.                             (2)

We check that indeed there is room for $m^2$ electrons in shell $S_m$, because the volume of $S_m$ is $r_m^2d_m\sim m^5$, while the volume of an electron $\sim d_m^3\sim m^3$.

We observe that (2) fits with observations. We understand that the electronic density is distributed so that the potential energy and thus total energy in each full shell is basically the same, which may be viewed to be a heavenly socialistic organization of the shell structure of an atom.

Numerical computation seeking the ground state energy by relaxation in the Schrödinger model of post 5 starting from an initial density distribution according to (1), shows good correspondence with observation, supporting the basic analysis of this post. Numbers will be presented in an upcoming post.

The basic aspect of this model as a form of electron density model, is that electrons (or shells in the present spherically symmetric case) keep individuality by occupying different domains of space, which makes it possible to accurately represent electron-electron repulsion.

This feature is not present in standard density models such as Thomas-Fermi and Density Functional Theory. In these models electrons lack individuality as parts of electron clouds, which makes it difficult to represent electron-electron repulsion ab ibnitio.

Recall also that in the standard Schrödinger equations wave functions appear as multi-dimensional linear combinations of products of one-electron wave functions defined in all of space by separate spatial variables, thus with each electron "both nowhere and everywhere" without individuality, which requires a statistical interpretation of the wave function as a multi-dimensional uncomputable monster.

Another basic aspect of the presented model is continuity of electron density across inter-electron or inter-shell boundaries for the electron configuration of ground states. This allows atoms to have stable ground states as non-dissipative periodic states of minimal energy.

Notice further that the size of the atom as $r_M\sim Z^{-\frac{1}{3}}$ with decreasing size as $Z$ increases, corresponds to the observed decrease of size moving to the right in each row of the periodic table:

Gerhard 't Hooft: Improved Understanding of Quantum Mechanics Needed

Gerhard 't Hooft is one of the Nobel Laureates in Physics who is not happy with the present state of understanding of quantum mechanics and seeks to do something about it: Hooft starts out in Determinism beneath Quantum Mechanics with:

• The need for an improved understanding of what Quantum Mechanics really is, needs hardly be explained in this meeting. 
• My primary concern is that Quantum Mechanics, in its present state, appears to be mysterious.
• It should always be the scientists’ aim to take away the mystery of things. 
• It is my suspicion that there should exist a quite logical explanation for the fact that we need to describe probabilities in this world quantum mechanically. 
• This explanation presumably can be found in the fabric of the Laws of Physics at the Planck scale.
• However, if our only problem with Quantum Mechanics were our desire to demystify it, then one could bring forward that, as it stands, Quantum Mechanics works impeccably. 
• It predicts the outcome of any conceivable experiment, apart from some random ingredient. This randomness is perfect. There never has been any indication that there would be any way to predict where in its quantum probability curve an event will actually be detected. 
• Why not be at peace with this situation? 
• One answer to this is Quantum Gravity. Attempts to reconcile General Relativity with Quantum Mechanics lead to a jungle of complexity that is difficult or impossible to interpret physically. In a combined theory, we no longer see “states” that evolve with “time”, we do not know how to identify the vacuum state, and so on. 
• What we need instead is a unique theory that not only accounts for Quantum Mechanics together with General Relativity, but also explains for us how matter behaves. 
• We should find indications pointing towards the correct unifying theory underlying the Standard Model, towards explanations of the presumed occurrence of supersymmetry, as well as the mechanism(s) that break it. We suspect that deeper insights in what and why Quantum Mechanics is, should help us further to understand these issues. 

Hooft thus acknowledges that quantum mechanics is mysterious, which all prominent physicists do, but Hooft is not at peace with this situation, since after all the essence of science is understanding, although most of his colleagues seem to have accepted once and for all that quantum mechanics cannot be understood and cannot be reconciled with general relativity.
Hooft then proceeds to seek a determinism behind quantum mechanics in the form of cellular automatons (also here).

I am pursuing another route to an understandable form of quantum mechanics as analog computation with finite precision, which in a way connects to Hooft's cellular automaton's, but is expressed by Schrödinger type wave equations in a continuum mechanics framework.

In this framework the finite precision computation makes a difference between smooth (strong) solutions and non-smooth (weak) solutions of the wave equations: Smooth solutions satisfy the wave equations exactly (with infinite precision), while non-smooth solutions satisfy the equations only in a weak sense with finite precision and loss of information as a form of dissipative radiation.

This allows the ground state of an atom as a smooth solution without dissipation to be stable over time without dissipation, while an excited state as a non-smooth solution will return to the ground state under dissipative radiation.

The situation is analogous to that described in my work together with Johan Hoffman on fluid mechanics, with turbulent solutions as non-smooth dissipative solutions of formally inviscid Euler equations, which allowed us to resolve d'Alembert's paradox (J Math Fluid Mech 2008) and formulate a new theory of flight (to appear in J Math Fluid Mech 2015), among other things.

Finite Element Quantum Mechanics 5: 1d Model in Spherical Symmetry

The new Schrödinger equation I am studying in this sequence of posts takes the following form, in spherical coordinates with radial coordinate $r\ge 0$ in the case of spherical symmetry, for an atom with kernel of charge $Z$ at $r=0$ with $N\le Z$ electrons of unit charge distributed in a sequence of non-overlapping spherical shells $S_1,...,S_M$ separated by spherical surfaces of radii $0=r_0<r_1<r_2<...<r_M=\infty$, with $N_j>0$ electrons in shell $S_j$ corresponding to the interval $(r_{j-1},r_j)$ for $j=1,...,M,$ and $\sum_j N_j = N$:

Find a complex-valued differentiable function $\psi (r,t)$ depending on $r≥0$ and time $t$, satisfying for $r>0$ and all $t$,

• $i\dot\psi (r,t) + H(r,t)\psi (r,t) = 0$              (1)

where $\dot\psi = \frac{\partial\psi}{\partial t}$ and $H(r,t)$ is the Hamiltonian defined by

• $H(r,t) = -\frac{1}{2r^2}\frac{\partial}{\partial r}(r^2\frac{\partial }{\partial r})-\frac{Z}{r}+ V(r,t)$,
• $V(r,t)= 2\pi\int\vert\psi (s,t)\vert^2\min(\frac{1}{r},\frac{1}{s})R(r,s,t)s^2\,ds$,
• $R(r,s,t) = (N_j -1)/N_j$ for $r,s\in S_j$ and $R(r,s,t)=1$ else,

and 

• $4\pi\int_{S_j}\vert\psi (s,t)\vert^2s^2\, ds = N_j$ for $j=1,...,M$.                  (2)

Here $-\frac{Z}{r}$ is the kernel-electron attractive potential and $V(r,t)$ is the electron-electron repulsive potential computed using the fact that the potential $W(s)$ of a spherical uniform surface charge distribution of radius $r$ centered at $0$ of total charge $Q$, is given by $W(s)=Q\min(\frac{1}{r},\frac{1}{s})$, with a reduction for a lack of self-repulsion within each shell given by the factor $(N_j -1)/N_j$.

The $N_j$ electrons in shell $S_j$ are thus homogenised into a spherically symmetric charge distribution of total charge $N_j$.

This is a free boundary problem readily computable on a laptop, with the $r_j$ representing the free boundary separating shells of spherically symmetric charge distribution of intensity $\vert\psi (r,t)\vert^2$ and a free boundary condition asking continuity and differentiability of $\psi (r,t)$.   

Separating $\psi =\Psi +i\Phi$ into real part $\Psi$ and imaginary part $\Phi$, (1) can be solved by explicit time stepping with (sufficiently small) time step $k>0$ and given initial condition (e.g. as ground state):

• $\Psi^{n+1}=\Psi^n-kH\Phi^n$, 
• $\Phi^{n+1}=\Phi^n+kH\Psi^n$, 

for $n=0,1,2,...,$ where $\Psi^n(r)=\Psi (r,nk)$ and $\Phi^n(r)=\Phi (r,nk)$, while stationary ground states can be solved by the iteration

• $\Psi^{n+1}=\Psi^n-kH\Psi^n$, 
• $\Phi^{n+1}=\Phi^n-kH\Phi^n$, 

while maintaining (2).

A remarkable fact is that this model appears to give ground state energies as minimal eigenvalues of the Hamiltonian for both ions and atoms for any $Z$ and $N$ within a percent or so, or alternatively ground state frequencies from direct solution in time dependent form. Next I will compute excited states and transitions between excited states under exterior forcing.

Specifically, what I hope to demonstrate is that the model can explain the periods of the periodic table corresponding to the following sequence of numbers of electrons in shells of increasing radii: 2, (2, 8), (2, 8, 8), (2, 8, 18, 8), (2, 8, 18, 18, 8)... which to be true lacks convincing explanation in standard quantum mechanics (according to E. Serri among many others).

The basic idea is thus to represent the total wave function $\psi (r,t)$ as a sum of shell wave functions
with non-overlapping supports in the different in shells requiring $\psi (r,t)$ and thus $\vert\psi (r,t)\vert^2$ to be continuous across inter-shell boundaries as free boundary condition, corresponding to continuity of charge distribution as a classical equilibrium condition.

I have also with encouraging results tested this model for $N\le 10$ in full 3d geometry without spherical shell homogenisation with a wave function as a sum of electronic wave functions with non-overlapping supports separated by a free boundary determined by continuity of wave function including charge distribution.

We compare with the standard (Hartree-Fock-Slater) Ansatz of quantum mechanics with a multi-dimensional wave function $\psi (x_1,...,x_N,t)$ depending on $N$ independent 3d coordinates $x_1,...,x_N,$ as a linear combination of wave functions of the multiplicative form

• $\psi_1(x_1,t)\times\psi_2(x_2,t)\times ....\times\psi_N(x_N,t)$,  

with each electronic wave function $\psi_j(x_j,t)$ with global support (non-zero in all of 3d space). Such multi-d wave functions with global support thus depend on $3N$ independent space coordinates and as such defy both direct physical interpretation and computability, as soon as $N>1$, say. One may argue that since such multi-d wave function cannot be computed, it does not matter that they have no physical meaning, but the net output appears to be nil, despite the declared immense success of standard quantum mechanics based on this Ansatz.

Quantum Information Can Be Lost

Stephen Hawking claimed in lecture at KTH in Stockholm last week (watch the lecture here and check this announcement) that he had solved the "black hole information problem":

• The information is not stored in the interior of the black hole as one might expect, but in its boundary — the event horizon,” he said. Working with Cambridge Professor Malcolm Perry (who spoke afterward) and Harvard Professor Andrew Stromberg, Hawking formulated the idea hat information is stored in the form of what are known as super translations.

The problem arises because quantum mechanics is viewed to be reversible, because the mathematical equations supposedly describing atomic physics formally are time reversible: a solution proceeding in forward time from an initial to a final state, can also be viewed as a solution in backward time from the earlier final state to the initial state. The information encoded in the initial state can thus, according to this formal argument, be recovered and thus is never lost. On the other hand a black hole is supposed to swallow and completely destroy anything it reaches and thus it appears that a black hole violates the postulated time reversibility of quantum mechanics and non-destruction of information.

Hawking's solution to this apparent paradox, is to claim that after all a black hole does not destroy information completely but "stores it on the boundary of the event horizon". Hawking thus "solves" the paradox by maintaining non-destruction of information and giving up complete black hole destruction of information.

The question Hawking seeks to answer is the same as the fundamental problem of classical physics which triggered the development of modern physics in the late 19th century with Boltzmann's "proof" of the 2nd law of thermodynamics: Newton's equations describing thermodynamics are formally reversible, but the 2nd law of thermodynamics states that real physics is not always reversible: Information can be inevitably lost as a system evolves towards thermodynamical equilibrium and then cannot be recovered. Time has a direction forward and cannot be reversed. 

Boltzmann's "proof" was based an argument that things that do happen do that because they are "more probable" than things which do not happen. This deep insight opened the new physics of statistical mechanics from which quantum borrowed its statistical interpretation.

I have presented a different new resolution of the apparent paradox of irrreversible macrophysics based on reversible microphysics by viewing physics as analog computation with finite precision, on both macro- and microscales. A spin-off of this idea is a new resolution of d'Alemberts's paradox and a new theory of flight to be published shortly.

The basic idea here is thus to replace the formal infinite precision of both classical and quantum mechanics, which leads to paradoxes without satisfactory solution, with realistic finite precision which allows the paradoxes to be resolved in a natural way without resort to unphysical statistics. See the listed categories for lots of information about this novel idea.

The result is that reversible infinite precision quantum mechanics is fiction without physical realization, and that irreversible finite precision quantum mechanics can be real physics and in this world of real physics information is irreversibly lost all the time even in the atomic world. Hawking's resolution is not convincing.

Here is the key observation explaining the occurrence of irreversibility in formally reversible systems modeled by formally non-dissipative partial differential equations such as the Euler equations for inviscid macroscopic fluid flow and the Schrödinger equations for atomic physics:

Smooth solutions are strong solutions in the sense of satisfying the equations pointwise with vanishing residual and as such are non-dissipative and reversible.  But smooth solutions make break down into weak turbulent solutions, which are only solutions in weak approximate sense with pointwise large residuals and these solutions are dissipative and thus irreversible.

An atom can thus remain in a stable ground state over time corresponding to a smooth reversible non-dissipative solution, while an atom in an excited state may return to the ground state as a non-smooth solution under dissipation of energy in an irreversible process.      

Finite Element Quantum Mechanics 4: Spherically Symmetric Model

I have tested the new atomic model described in a previous post in setting of spherical symmetry with electrons filling a sequence of non-overlapping spherical shells around a kernel. The electrons in each shell are homogenized to spherical symmetry which reduces the model to a 1d free boundary problem with the free boundary represented by the inter-shell spherical surfaces adjusted so that the combined wave function is continuous along with derivates across the boundary. The repulsion energy is computed so as to take into account that electrons are not subject to self-repulsion, by a corresponding reduction of the repulsion within a shell.

The remarkable feature of this atomic model, in the form of a 1d free boundary problem with continuity as free boundary condition and readily computable on a lap-top, is that computed ground state energies show to be surprisingly accurate (within 1%) for all atoms including ions (I have so far tested up to atomic number 54 and am now testing excited states).

Recall that the wave function $\psi (x,t)$ solving the free boundary problem, has the form

• $\psi (x,t) =\psi_1(x,t)+\psi_2(x,t)+...+\psi_S(x,t)$         (1)

with $(x,t)$ a common space-time coordinate, where $S$ is the number of shells and $\psi_j(x,t)$ with support in shell $j$ is the wave function for the homogenized wave function for the electrons in shell $j$ with $\int\vert\psi_j(x,t)\vert^2\, dx$ equal to the number of electrons in shell $j$.

Note that the free boundary condition expresses continuity of charge distribution across inter-shell boundaries, which appears natural.

Note that the model can be used in time dependent form and then allows direct computation of vibrational frequencies, which is what can be observed. 

Altogether, the model in spherical symmetric form indicates that the model captures essential features of the dynamics of an atom, and thus can useful in particular for studies of atoms subject to exterior forcing. 

I have also tested the model without spherical homogenisation for atoms with up to 10 electrons, with  similar results. In this case the the free boundary separates diffferent electrons (and not just shells of electrons) with again continuous charge distribution across the corresponding free boundary. 

In this model electronic wave functions share a common space variable and have disjoint supports and can be given a classical direct physical interpretation as charge distribution. There is no need of any Pauli exclusion principle: Electrons simply occupy different regions of space and do not overlap, just as in a classical multi-species continuum model.

This is to be compared with standard quantum mechanics based on multidimensional wave functions $\psi (x_1,x_2,...,x_N,t)$ typically appearing as linear combinations of products of electronic wave functions

• $\psi (x_1,x_2,...,x_N,t)=\psi_1(x_1,t)\times \psi_2(x_2,t)....\times\psi_N(x_N,t)$        (2)

for an atom with $N$ electrons, each electronic wave function $\psi_j(x_j,t)$ being globally defined with its own independent space coordinate $x_j$. Such multidimensional wave functions can only be given statistical interpretation, which lacks direct physical meaning. In addition, Pauli's exclusion principle must be invoked and it should be remembered that Pauli himself did not like his principle since it was introduced ad hoc without any physical motivation, to save quantum mechanics from collapse from the very start...

More precisely, while (1) is perfectly reasonable from a classical continuum physics point of view, and as such is computable and useful, linear combination of (2) represent a monstrosity which is both uncomputable and unphysical and thus dangerous, but nevertheless is supposed to represent the greatest achievement of human intellect all times in the form of the so called modern physics of quantum mechanics.

How long will it take for reason and rationality to return to physics after the dark age of modern physics initiated in 1900 when Planck's "in a moment of despair" resorted to an ad hoc hypothesis of a smallest quantum of energy in order to avoid the "ultra-violet catastrophe" of radiation viewed to be  impossible to avoid in classical continuum physics. But with physics as finite precision computation, which I am exploring, there is no catastrophe of any sort and Planck's sacrifice of rationality serves no purpose.

PS Here are the details of the spherical symmetric model starting from the following new formulation of a Schrödinger equation for an atom with $N$ electrons organised in spherical symmetric form into $S$ shells: Find a wave function

• $\psi (x,t) = \sum_{j=1}^N\psi_j(x,t)$

as a sum of $N$ electronic complex-valued wave functions $\psi_j(x,t)$, depending on a common 3d space coordinate $x\in R^3$ and time coordinate $t$ with non-overlapping spatial supports $\Omega_1(t)$,...,$\Omega_N(t)$, filling 3d space, satisfying

• $i\dot\psi (x,t) + H\psi (x,t) = 0$ for all $(x,t)$,       (1)

where the (normalised) Hamiltonian $H$ is given by

• $H(x) = -\frac{1}{2}\Delta - \frac{N}{\vert x\vert}+\sum_{k\neq j}V_k(x)$ for $x\in\Omega_j(t)$,

where $V_k(x)$ is the potential corresponding to electron $k$ defined by 

• $V_k(x)=\int\frac{\vert\psi_k(y,t)\vert^2}{2\vert x-y\vert}dy$, for $x\in R^3$,

and the wave functions are normalised to correspond to unit charge of each electron:

• $\int_{\Omega_j}\vert\psi_j(x,t)\vert^2 =1$ for all $t$ for $j=1,..,N$.

Assume the electrons fill a sequence of shells $S_k$ for $k=1,...,S$ centered at the atom kernel with $N_k$ electrons on shell $S_k$ and 

• $\int_{S_k}\vert\psi (x,t)\vert^2 =N_k$ for all $t$ for $k=1,..,S$,
• $\sum_k^S N_k = N$.

The total wave function $\psi (x,t)$ is thus assumed to be continuously differentiable and the electronic potential of the Hamiltonian acting in $\Omega_j(t)$ is given as the attractive kernel potential together with the repulsive kernel potential resulting from the combined electronic charge distributions $\vert\psi_k\vert^2$ for $k\neq j$, with total electronic repulsion energy

• $\sum_{k\neq j}\int\frac{\vert\psi_k(x,t)\vert^2\vert\psi_k(y,t)\vert^2}{2\vert x-y\vert}dxdy=\sum_{k\neq j}V_k(x)\vert\psi_k(x)\vert^2\, dx$.

Assume now that the electronic repulsion energy is approximately determined by homogenising the $N_k$ electronic wave function $\psi_j$ in each shell $S_k$ into a spherically symmetric "electron cloud" $\Psi_k(x)$ with corresponding potential $W_k(y)$ given by

• $W_k(y)=\int_{\vert x\vert <\vert y\vert}R_k\frac{\vert\Psi_k(x)\vert ^2}{\vert y\vert}\, dx+\int_{\vert x\vert >\vert y\vert}R_k\frac{\vert\Psi_k(x)\vert ^2}{\vert x\vert}\, dx$,

and $R_k(x)=\frac{N_k-1}{N_k}$ for $x\in S_k$ is a reduction factor reflecting non self-repulsion of each electron (and $R_k=1$ else): Of the $N_k$ electrons in shell $S_k$, thus only $N_k-1$ electrons contribute to the value of potential in shell $S_k$ from the electrons in shell $S_k$. We here use the fact that the potential $W(x)$ of a uniform charge distribution on a spherical surface $\{y:\vert y\vert =r\}$ of radius $r$ of total charge $Q$, is equal to $Q/\vert x\vert$ for $\vert x\vert >r$ and $Q/r$ for $\vert x\vert <r$.

Our model then has spherical symmetry and is a 1d free boundary problem in the radius $r=\vert x\vert$ with the free boundary represented by the radii of the shells and the corresponding Hamiltonian is defined by the electronic potentials computed by spherical homogenisation in each shell. The free boundary is determined so that the combined wave function $\psi (x,t)$ is continuously differentiable across the free boundary. 

Finite Element Quantum Mechanics 3: Explaining the Periodicity of the Periodic Table

According to Eric Scerri, the periodic table is not well explained by quantum mechanics, contrary to common text book propaganda, not even the most basic aspect of the periodic table, namely its periodicity:

• Pauli’s explanation for the closing of electron shells is rightly regarded as the high point in the old quantum theory. Many chemistry textbooks take Pauli’s introduction of the fourth quantum number, later associated with spin angular momentum, as the foundation of the modern periodic table. Combining this two-valued quantum number with the ear- lier three quantum numbers and the numerical relationships between them allow one to infer that successive electron shells should contain 2, 8, 18, or $2n^2$ electrons in general, where n denotes the shell number. 
• This explanation may rightly be regarded as being deductive in the sense that it flows directly from the old quantum theory’s view of quantum numbers, Pauli’s additional postulate of a fourth quantum number, and the fact that no two electrons may share the same four quan- tum numbers (Pauli’s exclusion principle). 
• However, Pauli’s Nobel Prize-winning work did not provide a solution to the question which I shall call the “closing of the periods”—that is why the periods end, in the sense of achieving a full-shell configuration, at atomic numbers 2, 10, 18, 36, 54, and so forth. This is a separate question from the closing of the shells. For example, if the shells were to fill sequentially, Pauli’s scheme would predict that the second period should end with element number 28 or nickel, which of course it does not. Now, this feature is important in chemical education since it implies that quantum mechanics can- not strictly predict where chemical properties recur in the periodic table. It would seem that quantum mechanics does not fully explain the single most important aspect of the periodic table as far as general chemistry is concerned. 
• The discrepancy between the two sequences of numbers representing the closing of shells and the closing of periods occurs, as is well known, due to the fact that the shells are not sequentially filled. Instead, the sequence of filling fol- lows the so-called Madelung rule, whereby the lowest sum of the first two quantum numbers, n + l, is preferentially oc- cupied. As the eminent quantum chemist Löwdin (among others) has pointed out, this filling order has never been derived from quantum mechanics. 

On the other hand, in the new approach to atomic physics I am exploring, the periodicity directly connects to a basic partitioning or packing problem, namely how to subdivide the surface of a sphere in equal parts, which gives the sequence $2n^2$ by dividing first into two half spheres and then subdividing each half spherical surface in $n\times n$ pieces,  in a way similar to dividing a square surface into $n\times n$ square pieces.  With increasing shell radius an increasing number of electrons, occupying a certain surface area (scaling with the inverse of the kernel charge), can be contained in a shell. 

In this setting a "full shell" can contain 2, 8, 18, 32,.., electrons, and the observed periodicity 2, 8, 8, 18, 18, 32, 32, with each period ended by a noble gas with atomic numbers 2 (He), 10 (Neon), 18 (Argon), 36 (Krypton), 54 (Xenon),  86 (Radon), 118 (Ununoctium, unkown), with a certain repetition of shell numbers, can be seen as a direct consequence of such a full shell structure, if allowed to be repeated when the radius of a shell is not yet large enough to house a full shell of the next dignity. 

Text book quantum mechanics thus does not explain the periodicity of the periodic table, while the new approach am I pursuing may well do so in a very natural way.   Think of that.   

Finite Element Quantum Mechanics 2: Questions without Answers

Hans Primas formulates in Chemistry, Quantum Mechanics and Reductionism, the following basic questions left without answers in textbook quantum mechanics:

1. Do isolated quantal systems exist at all?
2. Is the Pauli Principle a universal and inviolable fact of nature?
3. Does quantum mechanics apply to large molecular systems?
4. Is the superposition principle universally valid?
5. Why do so many stationary states not exist?
6. Why are macroscopic bodies localised?
7. Why does quantum mechanics fail to account for chemical systematics?
8. Why can approximations be better than the exact solutions?
9. Why is the Born-Oppenheimer picture so successful?
10. Is temperature an observable? 

Despite now almost 100 years of giant efforts by giant scientific minds, no satisfactory answers to these basic questions have been delivered. There is no reason to believe that 100 more years will give any answers and the question must be posed if there is something fundamentally wrong with textbook quantum mechanics which prevents progress? 

Yes, I think so: The origin of all these questions without answers is the starting point of textbook quantum mechanics with a wave function 

• $\psi (x_1,....,x_N,t)$ depending on $3N$ space coordinates and time,
• satsifying a linear scalar wave equation in $3N$ space dimensions and time, 

for an atom with $N$ electrons as particles, with $\vert\psi (x_1,...,x_N,t)\vert^2$ interpreted as the probability that particle $j$ is at position $x_j$ at time $t$ for $j=1,...,N$.  

Such a wave function is both uncomputable (because of the many spatial dimensions) and unphysical (because an atom is not an insurance company computing probabilities, as little as an individual person paying an insurance). The fact that textbook quantum mechanics still after almost hundred years is stuck with such a hopeless scientific misconception, is nothing less than a scientific tragedy.

Hans Primas gives the following devastating verdict:

• There is no general agreement about the referent (physical meaning) of pioneer (textbook) quantum mechanics.
• Pioneer quantum mechanics has an agonising shortcoming: It cannot describe classical systems. 
• From a fundamental point of view the only adequate interpretation of quantum mechanics is an ontic (realistic) interpretation.... Bohr's epistemic interpretation expresses merely states of knowledge and misses the point of genuine scientific inquiry...If we assume that pioneer quantum mechanics is a universal theory of molecular matter, then an ontic interpretation of this theory is impossible.
• The Bohr Copenhagen (textbook) interpretation is not acceptable as a fundamental theory of matter. 

In other words, pioneer (textbook) quantum mechanics is a failed scientific project, and it is an open problem to find an ontic description of atomic physics by "genuine scientific inquiry", that is, in the spirit of the device of this blog, "by critical constructive inquiry towards understanding". 

Finite Element Quantum Mechanics 1: Listening to Bohm

David Bohm discusses in the concluding chapter of Quantum Theory the relationship between quantum and classical physics, stating the following charcteristics of classical physics:

1. The world can be analysed into distinct elements.
2. The state of each element can be described in terms of dynamical variables that are specified with arbitrarily high precision.
3. The interrelationship between parts of a system can be described with the aid of exact casual was that define the changes of the above dynamical variables with time in terms of their initial values. The behavior of the system as a whole can be regarded as the result of the interaction by its parts.

If we here replace, "arbitrarily high precision" and  "exact" with "finite precision", the description 1-3 can be viewed as a description of 

• the finite element method 
• as digital physics as digital computation with finite precision
• as mathematical simulation of real physics as analog computation with finite precision.

My long-term goal is to bring quantum mechanics into a paradigm of classical physics modified by finite precision computation, as a form of computational quantum mechanics, thus bridging the present immense gap between quantum and classical physics. This gap is described by Bohm as follows:

• The quantum properties of matter imply the indivisibility unity of all interacting systems. Thus we have contradicted 1 and 2 of the classical theory, since there exist on the quantum level neither well defined elements nor well defined dynamical variables, which describe the behaviour of these elements.

My idea is thus to slightly modify classical physics by replacing "arbitrarily high precision" with "finite precision" to encompass quantum mechanics thus opening microscopic quantum mechanics to a machinery which has been so amazingly powerful in the form of finite element methods for macroscopic continuum physics, instead of throwing everything over board and resorting to a game of roulette as in the text book version of quantum mechanics which Bohm refers to.

In particular, in this new form of computational quantum mechanics, an electron is viewed as an "element" or a "collection of elements", each element with a distinct non-overlapping spatial presence, with an interacting system of $N$ electrons described by a (complex-valued) wave function $\psi (x,t)$ depending on a 3d space coordinate $x$ and a time coordinate $t$ of the form 

• $\psi (x,t) = \psi_1(x,t) + \psi_2(x,t)+...+\psi_N(x,t)$,                             (1)

where the electronic wave functions $\psi_j(x,t)$ for $j=1,...,N$, have disjoint supports together filling 3d space, indicating the individual presence of the electrons in space and time. The system wave function $\psi (x,t)$ is required to satisfy a Schrödinger wave equation including a Laplacian 

asking the composite wave functions $\psi (x,t)$ to be continuous along with derivatives across inter element boundaries. This a is a free boundary problem in 3d space and time and as such readily computable. 

I have with satisfaction observed that a spherically symmetric shell version of such a finite element model does predict ground state energies in close comparison to observation (within a percent) for all elements in the periodic table, and I will report these results shortly.

We may compare the wave function given by (1) with the wave function of text book quantum mechanics as a linear combination of terms of the multiplicative form:

• $\psi (x_1,x_2,...x_N,t)=\psi_1(x_1,t)\times\psi_2(x_2,t)\times ...\times\psi_N(x_N,t)$,

depending on $N$ 3d space coordinates $x_1,x_2,...,x_N$ and time, where each factor $\psi_j(x_j,t)$ is part of a (statistical) description of the global particle presence of an electron labeled $j$ with $x_j$ ranging over all of 3d space. Such a wave function is uncomputable as the solution to a Schrödinger equation in $3N$ space coordinates, and thus has no scientific value. Nevertheless, this is the text book foundation of quantum mechanics.

Text book quantum mechanics is thus based on a model which is uncomputable (and thus useless from scientific point of view), but the model is not dismissed on these grounds. Instead it is claimed that the uncomputable model always is in exact agreement to all observations according to tests of this form: 

• If a computable approximate version of this model (such as Hartree-Fock with a specific suitably chosen set of electronic orbitals) happens to be in correspondence with observation (due to some unknown happy coincidence), then this is taken as evidence that the exact version is always correct. 
• If a computable approximate version happens to disagree with observation, which is often the case, then the approximate version is dismissed but the exact model is kept; after all, an approximate model which is wrong (or too approximate) should be possible to view as evidence that an exact model as being less approximate must be more (or fully) correct, right?  

PS The fact that the finite element method has been such a formidable success for macroscopic problems as systems made up of very many small parts or elements, gives good hope that this method will be at least as useful for microscopic systems viewed to be formed by fewer and possibly simpler (rather than more complex) elements. This fits into a perspective (opposite to the standard view) where microscopics comes out to be more simple than macroscopics,  because macroscopics is built from microscopics, and a DNA molecule is more complex than a carbon atom, and a human being more complex than an egg cell.