Prototype RealQM for Helium Atom: Test Yourself!

Real Quantum Mechanics (RealQM) offers a new atom/ion/molecule model in the form of a system of non-overlapping electron densities as a classical continuum mechanics Schrödinger model in 3 space dimensions + time. This model describes a reality/actuality and is readily computable also for many electrons, which is not the case for standard QM (stdQM) as a multi-dimensional statistical model describing possibilities instead of actualities. See previous post for a connection to the 2023 Nobel Prize in Physics depicting electron charge densities, thus supporting RealQM, while real pictures of stdQM are missing. 

A prototype computational realisation of RealQM for the Helium atom with two electrons can be inspected here. The ground state of Helium is computed by minimising the total energy consisting of kinetic energy, attractive kernel potential energy of kernel at $x=0$ and repulsive mutual electron energy:  

• $\frac{1}{2}\int w(x)\vert\nabla u(x)\vert^2 dx -\int\frac{2u^2(x)}{\vert x\vert}dx+\int\int\frac{u^2(x)u^2(y)}{\vert x-y\vert}dxdy$

over a decomposition over a fixed grid of $u(x)=u_1(x)+u_2(x)$ into two electron density functions $u_1(x)$ and $u_2(x)$ with disjoint supports in 3d space given by characteristic functions $w_1(x)$ and $w_2(x)$ with $u_1^2(x)$ the charge density of electron 1 and $u_2^2(x)$ that of electron 2, with the boundary between $w_1$ and $w_2$ acting as a free boundary determined to achieve that u_1 and u_2 agree on the free boundary so that the electrons meet with same density.

The code consists of a couple of lines expressing gradient minimisation involving

1. Explicit relaxation of Hamiltonian in u + charge density normalisation (involves homogeneous Neuman conditions for u_1 and u_2 on free boundary enforced by the presence of $w(x)$ in the kinetic energy).
2. Explicit relaxation of Poisson problem for electron potentials.
3. Explicit level set front tracking of w to reach continuity of u over fixed grid.

which can be seen as a Bernoulli free boundary problem with homogeneous Neumann + continuity as free boundary condition.  

Run the code by clicking the arrow in the p5js code to see electrons initiated away from the kernel approaching the kernel and meeting at a free boundary with continuity (and approx homogeneous Neumann condition because of coarse resolution):   

and experiment further by modifying the code. It is fun and illuminating!

Compare with 1st excited state with 1 electron in 1st shell and 1 electron in 2nd shell (with p5js code):

Here you can test RealQM for the H2 molecule including minimisation over kernel distance:

Here you can test 2-shell atoms/ions starting with Lithium with 2 electrons in 1st shell and 1 electron in 2nd shell, continuing with always 2 electrons in 1st shell and Beryllium/ions with 2 electrons in 2nd shell.

This is a preparation for atoms/ions with more than 2 shells starting with Boron with 2+2+1 in 3 shells, Carbon 2+2+2, Nitrogen 2+2+3, Oxygen 2+2+4, Fluorine 2+3+4 and Neon 2+4+4 filling the first period.

And so on through the whole table... 

The electrons in each shell are captured by one electron density function carrying the total electron charge.

Here a test (with p5js-code to test) for Beryllium with 4 electrons in 2 shells initiated with a gap between of the 2 electrons in 2nd shell which closes under free boundary level set tracking:

See further RealQM results:

• Lithium (2++1)
• Carbon (2+4) 
• more to come

Computations in 3d without assuming spherical symmetry. Compare with earlier Atom Simulator in spherical symmetry.

Nobel Prize Physics 2023 vs RealQM

The Nobel Prize in Physics 2023 has been awarded to Pierre Agostini, Ferenc Krausz and Anne L’Huillier for

• Experimental methods that generate attosecond pulses of light for the study of electron dynamics in matter.
• The laureates’ experiments have produced pulses of light so short that they are measured in attoseconds, thus demonstrating that these pulses can be used to provide images of processes inside atoms and molecules.

This directly connects to Real Quantum Mechanics, which describes the dynamics of N interacting electrons as a 3d continuum mechanics system of N non-overlapping distributed charge densities, instead of the standard QM statistical particle model in 3N spatial dimensions. 

The Prize shows that pictures of electron charge densities can be taken, which gives support of an idea that they do indeed exist, as the essence of RealQM.  

This is to be compared with pictures of stdQM electron particles, which have not been taken, which gives support of an idea that they do not exist.

Hopefully, the 2023 Physics Prize can give an incentive to take a look at RealQM. See also this recent post  and earlier posts on RealQM.

 

Recall Wittgenstein: What you can take a picture of, you can speak of and believe to exist.

Here is an illuminating picture of two interacting electron charge densities from the presentation of the Prize by the Royal Swedish Academy of Sciences (assuming a preschool level audience?): 

 Compare with a RealQM picture of the interacting electrons of a H2O molecule

which confirms the chemist (but not physicist!) conception of H2O:

PS A closer look at the work awarded the prize, somewhat disappointingly reveals that it does not really contain images of processes inside atoms and molecules. Strange.

String Theory = Theory of Everything?

Peter Woit on Note Even Wrong recalls a podcast with Eli Frenkel about string theory:

• In particular, one thing that happened to Frenkel since last spring is that he attended Strings 2023 and gave a talk there (slides here, video here). The experience opened his eyes to just how bad some of the long-standing problems with string theory have gotten, and starting around here in the podcast he has a lot to say about them.
• It’s pretty clear that his reaction to what he saw going on at the conference was colored by his experience growing up in late Soviet-era Russia, where the failure of the system had become clear to everyone, but you weren’t supposed to say anything about this. He pins responsibility for this situation on senior leaders of the field, who have been unwilling to admit failure.

String theory was initiated in the 1970s as a grand attempt to unite all forces into a Theory of Everything ToE. My good friend Lars Brink at Chalmers was one of the early pioneers with a thesis in 1973 and subsequent work with all the senior leaders during a 50 year career until Sept 2022 when he had to give up his search for a ToE. A Memorial Meeting took place on the web Febr 18 2023 remembering a great physicist.

According to Eli Frenkel, sad to say, string theory has not delivered a credible ToE and so it is time to move on to something different building on the experience of a failed attempt. But this is not what the  leaders of string theory still alive are saying, which creates a big problem for the young generation asking for guidance and hope/inspiration for a career in physics. String theory? If not string theory, what then? What has fundamental physics de facto delivered the last 50 years? Why are the foundations of modern physics (quantum mechanics and relativity) still incompatible after 100 years? Why is there a crisis in fundamental physics if its leaders are smartest on Earth?

Eli Frenkel gives a perspective that is important to listen to if you are a young scientist in search of a mission. Old professors/theories never die, they just fade way.

What Can Quantum Mechanics Predict?

The previous post recalled the fact that standard Quantum Mechanics QM based on a $3N$ spatial dimensional Schrödinger equation for a system with $N$ electrons is effectively uncomputable for almost all atoms in the periodic table. With a resolution of $10^2$ in each spatial dimension, the total number of mesh points is daunting $10^{6N}$ which fills any thinkable computer already for $N=5$ (Boron). The wave function for all atoms beyond Boron is thus uncomputable. This means that the wave function for almost all atoms in the period table is uncomputable!

We compare with the standard message by physicists that all predictions of QM perfectly match with observation, or more precisely that there are no observations which contradict QM. 

The crucial question is then what predictions QM offers? Does QM predict the periodic table? A physicist would say that certainly this is so, in principle, while precise computations are lacking, because the full wave function is uncomputable. 

What can be computed are various low-dimensional approximations based on a variety of ad hoc assumptions. If such an ad hoc approximate solution happens to match observation, it is accepted as a demonstration that the full wave function (although unknown) also matches observation. If the ad hoc solutions does not match observation, it is discarded. With this strategy it is impossible to find a contradiction between theory and experiment! Success story!

But there is a caveat. Do you see it? Compare with How far does quantum mechanics explain the periodic table?

Real Quantum Mechanics presents a new Schrödinger equation as a non-linear system of non-overlapping electron charge densities sharing a common 3d space, thus a classical continuum model, which is computable for many electrons. A laptop computation for H2O is reported here with more to come.

Summary: QM in fact predicts very little (ab initio without ad hoc assumptions), which is consistent with the fact that there is no observation in contradiction with a QM prediction. But a theory which predicts nothing is not falsifiable and as such not a scientific theory. This is the reason behind the crisis of modern physics based on QM witnessed by so many physicists. Real QM opens a door out of the crisis. Why not give it a try? Permanent QM crisis over 100 years is not healthy to physics.  

Out of Physics Crisis: Reality instead of Possibility

The crisis of physics with nothing really new for 100 years since the arrival of Einstein's General Theory of Relativity GR theory and Quantum Mechanics QM (modulo the Atomic Bomb and the Standard Model from the 1960s), is well captured in the books Bankrupting Physics: How Today's Top Scientists are Gambling Away Their Credibility by Alexander Unzicker and Sheilla Jones and The Quantum Ten: A Story of Passion, Tragedy, Ambition by Sheilla Jones. The crisis is deepened by incompatibility of GR and QM, which has persisted with no resolution in sight. 

I argue on this blog under Quantum Contradictions  supported by Real QM that the basic reason for incompatibility is that QM concerns possibilities while GR concerns actualities or realities. This means that the main burden for the crisis is carried by QM since the prime role of physics must be to say something about the real world rather than a possible world which will/can not ever occur. It is here possible to reduce GR to Newton's theory of gravitation, since GR only concerns utterly small effects or utterly extreme situations. 

The fact that QM is concerned with possibilities is expressed by the nature of the wave function 

• $\Psi (x_1,x_2,....,x_N)$

describing the configuration of a system of $N$ particles/electrons with $x_i$ describing the possible position in 3d space (at some time) of particle $i=1,2,...,N$. The wave function $\Psi (x_1,...,x_N)$ thus depends on $3N$ spatial variables (plus time) with $\vert\Psi\vert^2(x_1,....,x_N)$ expressing the probability of the particle configuration with coordinates $(x_1,...,x_N)$, and evolves in time according to the Schrödinger equation, thus from possibility to possibility. 

The basic trouble with QM in this $3N$-dimensional form is that computation of $\Psi $ with a resolution of say $10^2$ in each spatial variable requires $10^{6N}$ mesh points which fills any thinkable computer as soon as N is not very small. The QM world of possibilities is thus so rich that it is not accessible through computation, and if it is not accessible through computation, then it cannot tell something. The step from possibility to actuality/reality is in QM taken by something called "collapse of the wave function", which is the basic mystery behind the crisis/incompatibility. 

Real Quantum Mechanics presents an alternative theory in the form of a system of $N$ electron charge densities in shared 3d space, thus basically a classical continuum mechanical system in 3d space, which is readily computable even for large $N$. 

Real QM thus offers a way out of the crisis. Give it a try, after 100 years of unsuccessful attempts to reconciliation with GR/Newton: 

• Real QM is fully compatible with GR/Newton's theory! 
• Real QM is computable and can tell something! 
• Real QM fits with observation.  

Physics Collectivism vs Mathematics Individualism

It is sometimes claimed that physics is a collective effort, while mathematics is rather a collection of individual efforts. Let us test this hypothesis by analysing the use of the first person plural pronoun "we".  

Let us start with physics noting how Andrew Strominger as a modern physicist of reputation invited by Lex Fridman to an interview entitled Black Holes, Quantum Gravity, and Theoretical Physics responds the following question by Lex:

• What is the effort of quantum gravity (to reconcile the standard model and general theory of relativity as the main open problem of modern physics)?

Andrew tells Lex:

• The one fully consistent model we have is string theory.
• We do not know if in any sense string theory describes the world.

We see that Andrew uses the pronoun "we" to send a message that modern physics is a collective effort, or more precisely a selective collective effort performed by an inner circle of leading modern physicists. For someone outside this group (which is not well defined) to say "we" in this context would be pretentious. By using an exclusive "we" Andrew signals that he belongs to the inner circle. 

There is another inclusive "we" often used in mathematical reasoning starting with a phrase like "We assume that x is a positive real number". Here "we" includes the reader or just anybody without asking for any qualification.

We know that the mathematics of Calculus was invented (in competition) by Leibniz and Newton, who both could say with confidence that "I invented Calculus", but never "We invented Calculus". 

More generally, a mathematician would rarely use "we" in the exclusive inner circle meaning, since the inner circle typically would have only one member. Mathematical theorems or theories mostly carry just one name like Brouwer's Fixed Point Theorem or Galois Theory. 

Maybe this indicates a difference between (modern) physics and mathematics, with physics being shared by physicists as some form of common reality or fiction, while mathematics is a human construct created by individuals like pieces of literature or art as individual efforts. 

But it may be tempting for both a physicist and mathematician to use "we" to signal membership in some group of prestige behind some theorem/theory. You can make your own observations...

On Wolfram's Concept of Computational Irreducibility

This is a continuation of the previous post.

Computational Irreducibility CI is used by Stephen Wolfram to describe that prediction of the state of a system requires full resolution of the evolution of the system by computation; to predict the outcome of tossing a coin by computation requires precise resolution of the motion of the coin from toss to landing. Wolfram seeks an explanation of the 2nd Law in terms of CI as an expression of Computational Impossibility. 

Wolfram's agenda (initiated when he was 12) connects to mine (since 90s), both serving to give an explanation of the 2nd Law or irreversibility or direction of time, based on computation, although fundamentally different in details. 

As a key example I consider turbulent fluid flow which is (mean-value) computable forward in time but not backward in time, thus computationally reducible forward in time but computationally irreducible backward in time, which gives time a direction. More precisely, consider an airplane wing moving through air from left to right meeting still air and leaving a wake of turbulent air. Drag and lift of the wing as force mean values are computable without resolving the flow to physical scale, thus expressing computational reducibility as shown in the book Computational Turbulent Incompressible Flow. 

However, reversing the flow with thus incoming turbulent flow from the right, will not give back the still air to the left, but instead a turbulent wake. To return to still air at the left after reversing the flow, would require solving an exponentially unstable problem without any cancellation (in associated linearised dual problem), thus suffering from CI.

The flow of air around a wing is thus an example of irreversible flow, with the irreversibility resulting from lack of cancellation in laminar flow, as carefully explained in the book. The forward problem is computationally reducible, while the backward problem is computationally irreducible = Irreversibility = 2nd Law. 

Turbulent dissipation of large scale kinetic energy into small scale heat energy can serve as a prime example of an irreversible process, which is open to study by computational solution of the Euler/Navier-Stokes equations, see post on Euler CFD and Euler's Dream. Construction of such a model from scratch in the spirit of Wolfram's New Kind of Science has not been made. More generally, processes involving some form of friction or other losses are irreversible for the same reasons as presented. 

A key fact is that these losses cannot be avoided, because forward evolution without losses requires infinite precision/infinite computational yet is necessary, while the show must go on and the show is not just heat diffusion. 

Recall that heat diffusion is easy forward but difficult backward and so is an example of irreversibility, but a trivial example. Turbulent flow is a non-trivial example. 

Wolfram's computational physics has been met with skepticism by the physics community, and so his version of the 2nd Law may not be applauded. My version stays closer to real physics in the form of real turbulent flow... 

It is remarkable the the simple code of Euler CFD (as an extended version of the 5-point scheme for the 2d Laplacian) generates the whole spectrum of turbulent flow. Remarkable.

Recall the essence of the 2nd Law: Physical systems with nearby particles moving with different velocities, typically arising in slightly viscous flow turning turbulent or moving parts mechanics under friction, develop increasing velocity gradients controlled by turbulent/friction dissipation into heat energy as small scale unordered kinetic energy. This is an irreversible process which cannot be avoided. 

In other words: Perpetuum mobile in the form of a system with moving parts is impossible: There will always be some turbulence/friction transforming motion into heat energy in an irreversible process. 

On the other hand an atom in ground state is a form of perpetuum mobile, but then in the form of a system without moving parts, as explained in Real Quantum Mechanics: Electrons in ground state do not move around the kernel, but rather turn around like someone unable to get to sleep, while the electron density is stationary. On the other hand, in a radiating atom, electron densities do shift between energy levels under energy exchange.   

Elaboration of Wolfram's Explanation of the Second Law

da Vinci's explanation of the 2nd Law = Mine.

Stephen Wolfram is now presenting work on the 2nd Law of Thermodynamics based on the notion of computation, which connects to a view I have been advocating. The role the 2nd Law is to give an explanation in mathematical terms of the observed fact that certain macroscopical physical processes are irreversible even though the underlying microscopics based on Newton's Laws appears to be reversible. 

Boltzmann tried to give an explanation in terms of statistics, which has not convinced Wolfram nor me, since we have both tried to replace statistics by computation as a process taking a system from one configuration at a certain time $t$ to a next configuration at $t+dt$ with $dt$ a small time step.  My theory is presented in the books Computational Turbulent Incompressible Flow and Computational Thermodynamics with Johan Hoffman and blog posts.

I will not here seek to summarise Wolfram's theory, which he admits is not easy to fathom, but just briefly recall the elements of my theory which is different by qualifying the general idea of computation in terms of the concepts of (i) finite precision computation and (ii) stability/well-posedness (connecting to Wolfram's computational irreducibility).

The phenomenon of turbulence of fluid motion demonstrates the role of (i) and (ii) in explaining why certain processes are not reversible. The basic feature of turbulent flow is that it contains a range of scales from large to small with large carrying kinetic energy in ordered coherent form and small carrying heat energy as unordered incoherent form. 

Fluid flow may be seen as initiated in large scale coherent form like the laminar flow in a river before a water fall in which it transforms into small scale turbulent flow by the fall developing strong velocity gradients, see picture above. In the fall coherent laminar flow develops into turbulent flow thus transforming large scale coherent kinetic energy into small scale incoherent heat energy. 

This is a stable process in the sense that the large scales and amount of heat energy will change little under finite precision of the computational process over a time. In other words, the large scale forward process is stable or well-posed. It is possible to break down large scale kinetic motion into small scale heat energy with only finite precision = Easy. You can easily break a glass by a hammer without much precision.

On the other, reversing the flow through the water fall is impossible, because it would require infinite precision to coordinate incoherent small scale turbulent flow/heat energy into large scale laminar flow /kinetic energy = Difficult. To assemble a glass smashed into pieces is difficult/impossible. 

Digital computation has finite precision. We do not have to ask physical processes to be carried out by some form of computation with infinite precision; finite precision is enough.  

Summary of Irreversibity/2nd Law: 

• Forward motion in time is possible/easy under finite precision: physics. 
• Backward motion in time is impossible/difficult under finite precision: not physics.          

PS1 Wolfram gives a (human) observer with certain perception capabilities a key role, like in the Copenhagen interpretation of quantum mechanics. But a Universe without observer is also a Universe maybe of more fundamental interest. 

PS2  There is a connection to both previous and next post describing turbulent flow as predictable chaotic and irreversible flow.   

Predictive Chaos

A common argument in the current debate about global climate is that the Earth climate system is chaotic and so cannot be computationally modeled and predicted. The argument is used to question alarmistic predictions of climate change, but also open to unrestricted alarmism in the sense that anything can happen from virtually nothing. 

Basic examples of chaotic systems are (i) coin tossing and (ii) turbulent fluid motion. In both cases pointwise prediction in space and time is impossible, but mean-values can be predicted with high precision. In coin tossing with a perfect coin the quotient of number of heads and tails quickly approach 1 as the number of tosses increases. The drag and lift as mean-values in space and time of the forces on an airplane wing subject to turbulent flow of air, can be accurately computationally predicted by solving the Navier-Stokes equations. This is the subject of the monumental treatise Computational Turbulent Incompressible Flow. See also previous posts under label chaotic system.  

How is it possible that mean-values can be accurately computed but point-values not? The reason is that turbulent flow is (like the weather) fluctuating with alternating ups and downs (like the motion of a tossed coin before landing). Therefore mean-values can be predicted by computational simulation of the fluctuations without asking for point-wise accuracy. 

We understand that chaotic motion as fluctuating turbulent motion in a certain sense is more predictive than non-fluctuating motion like that of a pen left in upright position.

While the weather as pointwise chaotic motion is not predictable over more than a week, climate as mean value of weather changes only slowly and so can be predictive, more or less. For example, we can well expect to enter into a new ice age within hundreds of years as an effect of major factors like the orbit of the Earth. 

Stock markets appear chaotic, yet can be predicted over time using models including major relevant factors. 

In any case, reference to chaotic systems as being unpredictable can be misleading and so has to be qualified.  

PS This post connects to the previous post on the unfortunate present formulation of the Navier-Stokes Clay Millennium Problem forgetting the completely foundational aspect of well-posedness in the sense of small output effects of small perturbations, which is a property of turbulent solutions of Navier-Stokes equations: Drag and lift of a wing remain the same under small perturbations of incoming flow and geometry. The present formulation appears to confuse smoothness with well-posedness.

Terence Tao on the Navier-Stokes Millennium Problem

This is a continuation a previous post on the Navier-Stokes Clay Mathematics Institute Millennium Problem stimulated by a recent article uploaded by Terence Tao: 

• Quantitative bounds for critically bounded solutions to the Navier-Stokes equations

arguing that:

• If a certain norm of the solution stays bounded (or grows at a controlled rate), then the solution stays regular.
• Taken in the contrapositive, they assert that if a solution blows up at a certain finite time T, then certain norms of the solution must also go to infinity.

The certain norm can be the $L^3$ norm over the spatial domain of the velocity $u(t)$ for $t<T$, which thus would serve as an indicator for solution of the Navier-Stokes Millennium Problem. 

I have in sequence of posts expressed criticism of the formulation of the Millennium Problem which can be condensed into: 

• The normalisation to unit viscosity means that the physics of turbulence, which appears for small viscosity and bounded flow velocities, is missing. This makes the Millennium Problem into a purely academic mathematical problem without significance to the real world of fluid flow (and so probably violating the intention of Mr Clay). 
• More precisely, a turbulent solution would not be a classical regular solution nor a solution with unbounded velocities, thus a solution outside the present formulation of the Millennium Problem.

It is thus natural to ask for a reformulation of the Navier-Stokes Problem, in particular since no advance towards a solution in the original formulation has been made since 2000. Compare with this article on Knowino. 

PS This post was triggered by a new comment to the original post by Terence from 2019. See also my next post here.