torsdag 24 april 2014

Quantum Mechanics as Gift from God More Intelligent than Man

  • Quantum mechanics is, with relativity, the essence of the big conceptual revolution of the physics of the 20th century. 
  • Now, do we really understand quantum mechanics? 
  • It is probably safe to say that we understand its machinery pretty well; in other words, we know how to use its formalism to make predictions in an extremely large number of situations, even in cases that may be very intricate. 
  • Heinrich Hertz, who played such a crucial role in the understanding of electromagnetic waves in the 19th century (Hertzian waves), remarked that, sometimes, the equations in physics are “more intelligent than the person who invented them” [182]. 
  • The remark certainly applies to the equations of quantum mechanics, in particular to the Schrödinger equation, or to the superposition principle: they contain probably much more substance that any of their inventors thought, for instance in terms of unexpected types of correlations, entanglement, etc. 
  • It is astonishing to see that, in all known cases, the equations have always predicted exactly the correct results, even when they looked completely counter-intuitive. 
  • Conceptually, the situation is less clear. 
  • Nevertheless, among all intellectual constructions of the human mind, quantum mechanics may be the most successful of all theories since, despite all efforts of physicists to find its limits of validity (as they do for all physical theories), and many sorts of speculation, no one for the moment has yet been able to obtain clear evidence that they even exist. Future will tell us if this is the case; surprises are always possible!
Laloe illuminates the fact that modern physicists (and nobody else) do not understand the modern physics of quantum mechanics, and do not even pretend to do so,  as a conceptual revolution away from classical physics based on understanding. The argument is that the linear Schrödinger equation must be more intelligent than Schrödinger, since Schrödinger admittedly could not understand it and nobody else has ever claimed to understand it either. 

If the difference between science and religion is that science is all about understanding, while religion leaves understanding to divinity, modern physics appears to be more religion than science.

But it is hard to understand that an equation that cannot be solved, always predicts exactly the correct results! It is more easy to believe that any observation made can be claimed to fit exactly with the equation, since checking is impossible. It would be more convincing if observation was somewhat different from theory.

No, We Don't Understand Quantum Mechanics, But There Is Hope.

                                                            Yes, QM is a strange world.

The Preface of book Do We Really Understand Quantum Mechanics by Franck Laloe supplemented by an article with the same title, tells the truth about quantum mechanics:
  • In many ways, quantum mechanics QM is a surprising theory... because it creates a big contrast between its triumphs and difficulties. 
  • On the one hand, among all theories, quantum mechanics is probably one of the most successful achievements of science.  The applications of quantum mechanics are everywhere in our twentyfirst century environment, with all sorts of devices that would have been unthinkable 50 years ago. 
  • On the other hand, conceptually this theory remains relatively fragile because of its delicate interpretation – fortunately, this fragility has little consequence for  its efficiency. 
  • The reason why difficulties persist is certainly not that physicists have tried to ignore them or put them under the rug!
  • Actually, a large number of interpretations have been proposed over the decades, involving various methods and mathematical techniques. 
  • We have a rare situation in the history of sciences: consensus exists concerning a systematic approach to physical phenomena, involving calculation methods having an extraordinary predictive power; nevertheless, almost a century after the introduction of these methods, the same consensus is far from being reached concerning the interpretation of the theory and its
  • This is reminiscent of the colossus with feet of clay.
  • The difficulties of quantum mechanics originate from the object it uses to describe physical systems, the state vector (wave function) $\Psi$.
  • Without any doubt, the state vector is a curious object to describe reality!
The message is that QM a formidable achievement of the human intellect which is incredibly useful in practice, but like a colossus with feet of clay has a main character flaw, namely that it is a curious way to describe reality and as such not understood by physicists. 

There are two ways the handle if a physical theory is not understood because it is so curious, either the theory is dismisssed as being seriously flawed or the curiosity is chosen as a sign that the theory is correct and beyond questioning by human minds.

The reason QM is so mysterious is that the wave function $\Psi =\Psi (x_1,x_2,…,x_N)$ for an atom or molecule with $N$ electrons depends on $N$ independent three-dimensional space variables $x_1$, $x_2$,…, $x_N$, together with time, thus is a function in $3N$ space dimensions plus time and as such has no direct real physical meaning since real physics takes place in $3$ space dimensions. 

The wave function $\Psi$ is introduced as a the solution to a linear multi-dimensional linear wave equation named Schrödinger's equation of the form
  • $i\frac{\partial\Psi}{\partial t}+H\Psi = 0$,
where $H$ is a Hamiltonian operator acting on wave functions. The mysticism of QM thus originates from Schrödinger's equation and is manifested by the fact that there is no real derivation of Schrödinger's equation from basic physical laws. Instead, Schrödinger's equation is motivated as a purely formal manipulation of classical Hamiltonian mechanics without physical meaning. 

The main trouble with QM based on a linear multi-d Schrödinger equation is thus the physical interpretation of the multi-d wave function and the accepted answer to this enigma is to view 
  • $\vert\Psi (x_1,…,x_N)\vert^2$ 
as a probability distribution of a particle configuration described by the coordinates $(x_1,…,x_N)$ representing human knowledge about a physics and not physics itself. Epistemology of what we can know is thus allowed to replace ontology of what is.

The linear multi-d Schrödinger equation thus lacks connection to physical reality. Moreover, because of its many dimensions the equation cannot be solved (analytically or computationally), and the beautiful net result is that QM is based on an equation without physical meaning which cannot be solved. No wonder that physicists still after 100 years of hard struggle do not really understand QM. 

But since Schrödinger's linear multi-d equation lacks physical meaning (and neither can be solved) there is no compelling reason to view it as the foundation of atomistic physics. 

It appears to be more constructive to consider instead systems of non-linear Schrödinger equations in $N$ three-dimensional wave functions $\psi_1(x),…,\psi_N(x)$ with $x$ a 3d space coordinate,  in the spirit of of Hartree models, as physically meaningful computable models of potentially great practical usefulness. 

Sums of such wave functions then play a basic role and have physical meaning, to be compared the standard setting with $\Psi (x_1,…,x_N)$ in the form of Slater determinants as sums of muli-d products $\psi (x_1)\psi (x_2)…\psi (x_N)$ of complicated unphysical nature. 


tisdag 22 april 2014

Omodern Matematikundervisning Utan Ansvariga Matematiker

Matematikinstitutionerna vid KTH och Chalmers skickar varje år en ny larmrapport om ytterligare försämrade matematikkunskaper hos nyantagna teknologer och nu var det dags igen:
Med larmrapporten friskriver sig högskolematematikerna från sitt ansvar att se till att landets matematikutbildning är modern och funktionell, genom att skylla på skolmatematiken:  
  • De högskolelärare som SvD pratar med är överens om att studenternas svaga grundkunskaper gjort att utbildningsnivån vid högskolorna sänkts undan för undan.
  • Visst har vi anpassat nivån, men det är inget folk vill tala högt om. 
  • För svag matteundervisning i grundskola och gymnasium, i kombination med en för generös betygsättning, ligger bakom problemen.
Men skolmatematiken är en (förenklad) variant av högskolematematiken och anledningen att skolmatematiken inte längre fungerar är att högskolematematiken är omodern och inte motsvarar datorsamhällets nya möjligheter och behov.  

När jag försöker få högskolematematikerna att bära sitt ansvar och modernisera utbildningen möts jag av oförstående och uppgivenhet och mitt öppna brev till Svenska Matematikersamfundet och Nationalkommitten för Matematik leder ingenstans. Se också mitt inlägg i kommande maj-nummer av SMS-Bulletinen.

torsdag 17 april 2014

Extremism of Modern Physics as Bluff Poker Physics

Modern physics has been driven into an increasingly extremist position with focus on extremely small or large spatial or temporal scales or extremely large energies. When problems were met on a certain (extreme) scale, the study was directed to yet more extreme scales and energies, as in a steadily increasing bet in a game of poker with little on hand to never get called. When LHC does not deliver, then the bet is raised to a new bigger more powerful LHC...

When Einstein was pressed about the meaning of his special theory of relativity, he increased the bet to general relativity and when pressed about the meaning of general relativity he jumped the bet to cosmology...

When physicists after the introduction of quantum mechanics faced questions about the electronic structure of atoms and molecules, they turned to the three orders of magnitude smaller proton and neutron forming atomic kernels, and then to the quarks forming the proton and neutron and then ultimately to string theory on scales 15 orders of magnitude smaller than the proton in an ulitmate attempt to find the origin of gravitation acting on cosmological scales. In each case the problems met on one scale were met by resort to smaller or larger scales, steadily increasing the bet and preventing a call.

Today cosmology is directed to multiversa and inflation after Big Bang as the next step after Einstein's cosmology of general relativity supposedly all originating from string theory.  But this may be the last possible bet and a call is approaching anticipated as a crisis in physics.

onsdag 16 april 2014

Crisis in Physics vs Computational Physics

The May14 issue of Scientific American asks the following questions:
These questions naturally present themselves because modern theoretical physicists have driven themselves to search for the truth on scales which are either too small (string theory) or too big (cosmology) to be assessed experimentally. But theory without experiment may well be empty theory and that may be the meaning of the crisis. Of course, advocates of string theory like Lubos, forcefully denies that there is a crisis in physics. But there are other blog voicesand leading physicists show little hope..

But modern physicists have a new tool to use and that is computational physics, which offers an experimental laboratory without the scale limits of a physical laboratory. 

Computational physics needs computable models, but both quantum mechanics and general relativity are based on models which are not computable, and so there is a lot of work to be done. The question is if modern theoretical physicists have the right training to do this work.     

måndag 14 april 2014

Wanted: Constructive Physics

                                     Wanted: Constructive version of Schrödinger's equation!

The book Constructive Physics by Y.I. Oshigov has an important message:
  • Only in the rebuilding of the gigantic construction of the modern physics in the constructive manner can open doors to the understanding of the complex processes in the sense of exact sciences.
  • The modern situation in physics looks like a crisis, and the genealogy of this crisis is the same as for the crisis in mathematics in the first third of the 20th century: this is the crisis in the axiomatic method.
  • Today we possess the more exact kit of instruments of the constructive mathematics: algorithms must replace formulas.
  • (The multidimensional wave function) harbors serious defects….it does not allow the computation of such functions already for a small number of particles, for example 10, let alone for the more complex systems.
  • This complexity barrier is principal. We should not think then that the quantum theory for many bodies gives such reliable answers to questions as it was the case in one particle case.
In short, quantum mechanics based on Schrödinger's equation for a wave function in $3N$ space dimensions for $N$ particles (electrons or kernels) must be given a new constructive form. A real challenge! My answer is given as Many-Minds Quantum Mechanics. 

onsdag 9 april 2014

Popper: Realism vs Quantum Muddle vs Statistics

Karl Popper starts out Quantum Theory and the Schism of Physics, as Vol III of Postscript to Logic of Scientific Discovery, with the following declaration:
  • Realism is the message of this book. 
  • It is linked with objectivity…with rationalism, with the reality of the human mind, of human creativity, and human suffering.
In Preface 1982: On a Realistic and Commonsense Interpretation of Quantum Mechanics, Popper gives his verdict:
  • Today, physics is in a crisis….This crisis is roughly as old as the Copenhagen interpretation of quantum mechanics.
  • In my view, the crisis is, essentially, due to two things: (a) the intrusion of subjectivism into physics; and (b) the victory of the idea that quantum theory has reached a complete and final truth.
  • Subjectivism in physics can be traced to several great mistakes. One is the positivism or idealism of Mach. Another is the subjectivist interpretation of the calculus of probability.
  • The central issue here is realism. That is to say, the reality of the physical world we live in: the fact that this world exists independently of ourselves; that it existed before life existed,…and that it will continue to exist long after we have all been swept away.
  • The subjectivist dogma was too deeply entrenched within the ruling interpretation of quantum mechanics, the so-called Copenhagen interpretation… this is how the great quantum muddle started….and the whole terminology, introduced in the early period of the theory, conspired to make the muddle worse and worse.
  • Another source of the crisis in physics is the persistence of the belief that quantum mechanics is final and complete.
  • Philosophers and physicists have been all too prone under the direct influence of Machian positivism, to take up idealist positions…
  • One of the things that this volume of the Postscript tries to to is to review many of the past arguments for idealism - which many current physicists still simply take for granted - and to show their error.  
But Popper, one of the greatest philosophers of science of the 20th century, talked to deaf ears and the crisis in physics is deepening every year…

Another thing is that Popper deepened the crisis be dwelling deeper into the statistical interpretation of the wave function of Born as the basis of the Copenhagen interpretation. Popper thus identified the crisis but then himself got drowned by the muddle...

Quantum Theory: Flight from Realism

The book Quantum Theory and the Flight from Realism by Christopher Norris is introduced by:
  • Norris examines the premises of orthodox quantum theory as formulated most influentially by Bohr and Heisenberg….as requiring a drastic revision of principles which had hitherto defined the very nature of scientific method, casual explanation and rational enquiry.
  • Putting the case for a realist approach which adheres to well-tried scientific principles of casual reasoning and interference to the best explanation, Norris clarifies the debate…
Norris continues:  
  • In this book I examine various aspects of the near century-lonh debate concerning the conceptual foundation of quantum mechanics (QM) and the problems it has posed for physicists and philosophers from Einstein to the present. They include the issue of wave-particle dualism; the uncertainty attaching to measurements of particle location or momentum, the (supposedly) observer-induced "collapse of the wave-function"; and the evidence of remote superluminal interaction between widely separated particles.
  • It is important to grasp exactly how the problems arose and exactly why - on what scientific or philosophical grounds - any alternative (realist) contrual should have been so often and routinely ruled out as a matter of orthodox QM wisdom. 
This is an important book with the important mission of bringing realism back to physics after a century  of anti-realist confusion ultimately corrupting all of science and with the adoption of climate alarmism by the American Physical Society as the tragic anti-realist irrational expression. 

tisdag 8 april 2014

Essence of Quantum Mechanics: Energy vs Frequency in Wave Models

In Schrödinger's Equation: Smoothed Particle Dynamics we observed that Schrödinger's equation for Hydrogen atom with one electron (normalized to unit mass and charge) reads
  • $i\bar h\dot\psi + H\psi =0$,
  • $H\psi =\frac{\bar h^2}{2}\Delta\psi +\frac{1}{\vert x\vert}\psi$,
where $\psi (x,t)$ the complex-valued wave function depending on coordinates of space $x$ and time $t$ with the dot denoting differentiation with respect to time, $H$ is the Hamiltonian operator and $\bar h$ Planck's (reduced) constant.

In terms of the real part $\phi$ and imaginary part $\chi$ of $\psi =\phi +i\chi$, Schrödinger's equation takes the system form
  1. $\bar h\dot\phi +H\chi =0$,
  2. $\bar h\dot\chi - H\phi =0$.
If $\phi_E(x)$ is an eigenfunction of the Hamiltonian satisfying $H\phi_E =E\phi_E$ with $E$ the corresponding eigenvalue, then the solution can be represented as
  • $\phi (x,t)=\cos(\omega t)\phi_E(x)$,     $\chi (x,t)=\sin(\omega t)\phi_E(x)$, 
with $\bar h\omega =E$, which expresses a periodic exchange between the two real-valued wave functions $\phi$ and $\chi$ mediated by the Hamiltonian $H$. The parallel to a harmonic oscillator (with $H$ the identity) is obvious.

We see that the effect of the time derivative term is to connect energy $E$ to (angular) frequency $\omega$ by
  • $\bar h\omega = E$, 
  • or $h\nu =E$, 
where $h=2\pi\bar h$ and $\nu =\frac{\omega}{2\pi}$ is frequency in Hertz, where $h$ acts as scale factor.

Schrödinger's equation thus sets up a connection between frequency $\nu$, which can be observed as atomic emission lines, and a model of internal atomic energy $E$ as the sum of kinetic and potential energies of eigenfunctions of the Hamiltonian with the connection $\bar h\omega =h\nu = E$. Observations of atomic emission then show to fit with energy levels of the model, which gives support to the functionality of the model. 

The basic connection $\nu \sim E$ can also be seen in Planck's radiation law (with simplified high-frequency cut-off)
  • $R(\nu ,T)=\gamma T\nu^2$ for $\frac{h\nu}{kT} < 1$,
where $R(\nu ,T)$ is normalized radiance as energy per unit time, with $\gamma =\frac{2k}{c^2}$, $T$ is temperature and $k$ is Boltzmann's constant, which gives an energy per cycle scaling with $\nu$ and a high frequency cut-off $h\nu$ scaling with atomic energy $kT$.

The connection $h\nu =E$ also occurs in the law of photoelectricity
  • $h\nu = P + K$,
where $P$ is the release energy and $K=eU$ is the kinetic energy of a released electron with $e$ the electron charge and $U$ the stopping potential. 

The atomic connection $h\nu =E$ between frequency and energy thus has both theoretical and experimental support,  but it does not say that energy is "quantized" into discrete packets of energy $h\nu$ carried by particles named photons of frequency $\nu$. 

The relation $h\nu =E$ is compatible with wave models of both emission from atoms and radiation from clusters of atoms and if so by Ockham's razor particle models have no role to play.

Atomic emission and radiation is a resonance phenomenon much like the resonance in a musical instrument, both connecting frequency to matter.

Text books state that 
  1. Blackbody radiation and the photoelectric effect cannot be explained by wave models.
  2. Hence discrete quanta and particles must exist. 
  3. Hence there is particle-wave duality.    
I give on Computational Blackbody Radiation evidence that 1 is incorrect, and therefore also 2 and 3. Without particles a lot of the mysticism of quantum mechanics can be eliminated and progress made. 

måndag 7 april 2014

The Strange Story of The Quantum: Physics as Mysticism

The Strange Story of The Quantum by Banesh Hoffman bears witness to the general public about modern physics as mysticism:  
  • This book is designed to serve as a guide to those who would explore the theories by which the scientist seeks to comprehend the mysterious world of the atom.
  • The story of the quantum is the story of a confused and groping search for knowledge…enlivened by coincidences such as one would expect to find only in fiction.
  • It is a story about turbulent revolution…and of the tempesteous emergence of a much chastened regime - Quantum Mechanics.
  • The magnificent rise of the quantum to a dominant position in modern science and philosophy is a story of drama and high adventure often well-nigh incredible. It is a chaotic tale…apparent chaos…nonsensical…intricate jagsaw…major discovery of the human mind.
  • Planck called his bundle or quota a QUANTUM of energy…This business of bundles of energy was unpardonable heresy, frightening to even the bravest physicist. Prandtl was by no means happy... But all was to no avail….to Max Planck had fallen the immortal honor of discovering them.
  • Einstein insisted...that each quantum of energy  somehow must behave like a particle: a particle of light; what we call a photon…But how could a particle theory possibly hope to duplicate the indisputable triumphs of the wave theory? To go back to anything like the particle theory would be tantamount to admitting that the elaborately confirmed theory of electromagnetic phenomena was fundamentally false. Yet Einstein...was actually proposing such a step.
  • It is difficult to decide where science ends and mysticism begins….In talking of the meaning of quantum mechanics, physicists indulge in more or less mysticism according to their individual tastes.
  • Perhaps it is this which makes it seem so paradoxical.
  • Perhaps there is after all some innate logic in quantum theory.
  • The message of the quantum suddenly becomes clear: space and time are not fundamental.
  • Out of it someday will spring a new and far more potent theory…what will then survive of our present ideas no one can say…
  • Already we have seen waves an particles and causality and space and time all undermined.
  • Let us hasten to bring the curtain down in a rush lest something really serious should happen...
Hoffman's book was first published in 1947. Since then the mysticism of modern physics has only become deeper...

söndag 6 april 2014

Schrödinger's Equation: Smoothed Particle Dynamics

Eigenfunctions of the Hamiltonian for the Hydrogen atom with eigenvalues representing the sum of kinetic and potential energies, with Schrödinger's equation as a smoothed version of the particle dynamics of a harmonic oscillator.  

This is continuation of the previous post How to Make Schrödinger's Equation Physically Meaningful + Computable. Consider the basic case of the Hydrogen atom with one electron (normalized to unit mass and charge):
  • $ih\dot\psi + H\psi =0$,
  • $H\psi =\frac{h^2}{2}\Delta\psi +\frac{1}{\vert x\vert}\psi$,
where $\psi (x,t)$ the complex-valued wave function depending on coordinates of space $x$ and time $t$ with the dot denoting differentiation with respect to time, $H$ is the Hamiltonian operator and $h$ Planck's constant.

In terms of the real part $\phi$ and imaginary part $\chi$ of $\psi =\phi +i\chi$, Schrödinger's equation takes the system form
  1. $h\dot\phi +H\chi =0$,
  2. $h\dot\chi - H\phi =0$.
If $\phi_E(x)$ is an eigenfunction of the Hamiltonian satisfying $H\phi_E =E\phi_E$ with $E$ the corresponding eigenvalue, then the solution can be represented as
  • $\phi (x,t)=\cos(\omega t)\phi_E(x)$,     $\chi (x,t)=\sin(\omega t)\phi_E(x)$, 
with $h\omega =E$, which expresses a periodic exchange between the two real-valued wave functions $\phi$ and $\chi$ mediated by the Hamiltonian $H$.

We can see 1- 2 as an analog of the equation for a harmonic oscillator $\ddot u+\omega^2u=0$ written in system form (with $h=1$)
  • $\dot\phi  + \omega\chi =0$
  • $\dot\chi  - \omega \phi = 0$,
where $\phi =\dot u$ and $\chi =\omega u$, with solution
  • $\phi (x,t)=\cos(\omega t)$,     $\chi (x,t)=\sin(\omega t)$.  
Here the velocity $\phi =\dot u$ connects to kinetic energy $\phi^2 =\dot u^2$ and $\chi =\omega u$ to potential energy $\chi^2 =\omega^2u^2$ and the dynamics of the harmonic oscillation consists of periodic transfer back and forth between kinetic and potential energy with their sum being constant.

Returning now to the Hydrogen atom, we obtain multiplying 1 by $\phi$ and 2 by $\chi$ and integrating in space the following the energy balance
  • $h\frac{d}{2dt}\int\phi^2\, dx + \int \phi H\chi \, dx =0$
  • $h\frac{d}{2dt}\int\chi^2\, dx - \int \chi H\phi\, dx =0$,    
  • $ \int \phi H\chi \, dx = \int \chi H\phi\, dx =\frac{h^2}{2}\int\nabla\phi\cdot\nabla\chi\, dx +\int\frac{\phi\chi}{\vert x\vert}\, dx$,
which shows upon summation (by the symmetry of $H$) that
  • $\frac{d}{2dt}\int\phi^2\, dx =\frac{d}{2dt}\int\chi^2\, dx =0$, 
which allows normalization to  
  • $\int\phi^2\, dx = \int\chi^2\, dx = \frac{1}{2}$,
  • $\int\vert\psi\vert^2\, dx = 1$, for all time. 
Further, multiplying 1 by $\dot\chi$ and 2 by $\dot\phi$ and subtracting the resulting equations shows that
  • $\int (\phi H\phi + \chi H\chi)\, dx$ is constant in time. 
We can now summarize as follows:

A. We see that the solution pair $(\phi ,\chi )$ of 1 - 2 as the real and imaginary part of Schrödinger's wave function $\phi$, represents a periodic exchange mediated by the Hamiltonian $H$ with balancing associated total energies 
  • $\int \phi H\phi (x,t)\, dx = \frac{h^2}{2}\int\vert\nabla\phi (x,t)\vert^2dx +\int\frac{\phi^2(x,t)}{\vert x\vert}\, dx$,
  • $\int \chi H\chi (x,t)\, dx = \frac{h^2}{2}\int\vert\nabla\chi (x,t)\vert^2dx +\int\frac{\chi^2(x,t)}{\vert x\vert}\, dx$    
as the sum of kinetic and potential energies.

B. We see that Schrödinger's equation for the Hydrogen atom can be viewed as a smoothed version of a harmonic oscillator with the smoothing effectuated by the Laplacian and with $h$ acting as a smoothing parameter.

C. We see that the system form 1- 2 combines the spatial eigenfunction $\phi_E$ with a periodic time dependence without introducing energy beyond the kinetic and potential energies defined by the Hamiltonian, thus associating these energies to frequency as the essence of quantum mechanics.

D. We see that quantum mechanics and Schrödinger's equation can be given an interpretation which closely connects to classical mechanics, as smoothed particle mechanics, which avoids the common mystifications of particle-wave duality, complementarity, wave function collapse and statistics forced by insistence to use a multidimensional wave function defying a direct physical meaning.

Extension to several electrons can be naturally be made following the idea of smoothed particle dynamics. For details see Many-Minds Quantum Mechanics.

fredag 4 april 2014

Comparing Blackbody Radiation Spectrum to Atomic Emission Spectrum

Planck's constant $h$ appears with different roles in a blackbody radiation spectrum and an atomic emission spectrum.  Blackbody radiation can be described as a near-resonance phenomenon in a forced harmonic oscillator with small damping in a mathematical model of the form
  • $\ddot u (t) +\omega^2u(t) -\gamma\dddot u = f(t)\approx \sin(\omega t)$, 
where $u(t)$ is displacement as function of time $t$, $\omega$ is angular velocity, $\gamma$ is a small damping constant, $f(t)$ is forcing in near-resonance with $\omega$ and the dot signifies time differentiation. Here the oscillator described by $\ddot u (t) +\omega^2u(t)$ carries energy as background temperature and the dissipative term $-\gamma\dddot u$ gives off radiation balancing forcing $f(t)$.

The dynamics of near-resonance is quite subtle as explained in detail on Computational Blackbody Radiation showing that Planck's constant enters as a parameter in a high-frequency cut-off reflecting Wien's displacement law.   

Atomic emission can be described as an eigenvalue problem for Schrödinger's equation of the form
  • $ih\dot\psi = E\psi$,
where $E$ is a real eigenvalue of an atomic Hamiltonian, with solution
  • $\psi (t) =\exp(i\omega t) =\cos(\omega t)+i\sin(\omega t)$, 
which can be seen as a periodic exchange of two forms of energy represented by the real part $\cos(\omega t)$ and the complex part $\sin(\omega t)$ reflecting incoming-outgoing radiation. Atomic emission is thus a direct resonance phenomenon without background temperature. Planck's constant serves to convert angular velocity (angular momentum) $\omega$ to atomic energy $E$ as $\bar h\omega$ with $\bar h=\frac{h}{2\pi}$ with $E$ the sum of kinetic and potential energy. 

We conclude:
  1. Blackbody radiation is a near-resonance phenomenon of molecules or collections of atoms modeled as a forced harmonic oscillator with small damping. Collections of atoms vibrate without electron configurations changing energy.   
  2. Atomic radiation is a direct resonance phenomenon which can be modeled by a harmonic oscillator. Electrons oscillate between two energy levels representing eigenstates of an atom.
In both cases $h$ enters combined with frequency $\nu$ in the form $h\nu =\bar h\omega$ as quantity of energy serving in a threshold condition in blackbody radiation, and as an energy eigenvalue in atomic emission.

The value of $h$ as setting a conversion scale between light energy and electronic energy can be determined by the photoelectric effect and can then be used by definition in blackbody radiation and Schrödinger's equation. 

torsdag 3 april 2014

Water Dam Analog of Photoelectric Effect

                               Open sluice gates in the Three Gorges Dam in the Yangtze River.

Einstein was awarded the 1921 Nobel Prize in Physics for his "discovery of the law of the photoelectric effect", connecting frequency $\nu$ of light shining on a metallic surface with measured potential $U$:
  • $h\nu = h\nu_0 + e\, U$ or $h(\nu -\nu_0) = e\, U$,
where $h$ is Planck's constant with dimension $eVs = electronvolt\,\times second$,  $\nu_0$ is the smallest frequency releasing electrons and $U$ in Volts $V$ is the stopping potential bringing the current to zero for $\nu >\nu_0$ and $e$ is the charge of an electron. Observing $U$ for different $\nu$ in a macroscopic experiment shows a linear relationship between $\nu -\nu_0$ and $U$ with $h$ as scale factor with reference value 
  • $h = 4.135667516(91)\times 10^{-15}\, eVs$,
with Millikan's value from 1916 within $0.5\%$.

Determining $h$ this way makes Einstein's law of photoelectricity into an energy conversion standard attributing $h\nu$ electronvolts to the frequency $\nu$, without any implication concerning the microscopic nature of the photoelectric effect.

The award motivation "discovery of the law of the photoelectric effect" reflected that Einstein's derivation did not convince the committee as expressed by member Gullstrand: 
  • When it was formulated it was only a tentatively poorly developed hunch, based on qualitative and partially correct observations. It would look peculiar if a prize was awarded to this particular work. 
To give perspective let us as an analog of the law of the photoelectric effect consider a water dam with sluice gates which automatically open when the level of water is $\nu_0$.  The sluice gates will then remain locked as long as the water level $\nu <\nu_0$.  Lock the sluice gates and let the dam fill to some water level $\nu >\nu_0$ and then unlock the sluices. The sluices will then open and water will flow through under transformation of potential energy into kinetic energy. Assuming the work to open the sluices corresponds to a level loss of $\nu_0$, a net level of $\nu -\nu_0$ potential energy will then be transformed into kinetic energy by the water flow through the sluices. 

The dam can be seen as an illustration of the photoelectric effect with the water level corresponding to frequency $\nu$ and the gravitational constant corresponding to $h$ and the width of the dam corresponding to the amplitude of incoming light. If $\nu <\nu_0$ then nothing will happen, if $\nu >\nu_0$ then the kinetic energy will scale with $h\nu$ and the total flow will scale with the width of the dam.

Notice that noting in this model requires the water to flow in discrete lumps or quanta. The only discrete effect is the threshold $\nu_0$ for opening the sluices.

onsdag 2 april 2014

Universal Quantum of Action: Standard Without Universality

In recent posts on we have seen that Plank's constant $h$ in physics text books being presented as a universal quantum of action as a smallest "packet of action" as a fundamental constant of fundamental significance in the "quantized" world we happen to be part of, in fact is nothing but a conversion standard between two measures of energy, in terms of frequency $\nu$ in periods per second and electronvolt (eV), determined by Einstein's law of photoelectricity
  • $h(\nu - \nu_0) = e\, U$,
where $\nu_0$ is the smallest frequency releasing electrons from a metallic surface upon exposure of light, $U$ in Volts $V$ is the stopping potential bringing the current to zero for $\nu >\nu_0$ and $e$ is the charge of an electron. Observing $U$ for different $\nu$ shows a linear relationship between $\nu -\nu_0$ and $U$ with $h$ as the scale factor measured in $eVs$ $electronvolts\times second$ as $energy \times time$ as action. The reference value obtained this way is 
  • $h = 4.135667516(91)\times 10^{-15}\, eVs$,
with Millikan's value from 1916 within $0.5\%$. Determining $h$ this way makes Einstein's law of photoelectricity simply into a conversion standard (that is, a definition) of energy attributing $h\nu$ electronvolts to the frequency $\nu$. Another way of finding the conversion from frequency to electronvolt is using a Josephson junction.

We now turn to Schrödinger's equation
  • $i\bar h\frac{\partial\psi}{\partial t}+H\psi=0$,
where $\bar h=\frac{h}{2\pi}$ is Planck's reduced constant as conversion from periods $\nu$ per second to angular velocity $\omega$ with $h\nu =\bar h\omega$, and $H$ is a Hamiltonian of space dependence. An eigenvalue $E$ of the Hamiltonian represents energy with $\psi_E$ a corresponding space dependent eigenfunction satisfying $H\psi_E =E\psi_E$ and $\exp(i\omega t)\psi_E$ a corresponding solution of Schrödinger's equation with 
  • $h\nu = \bar h\omega =  E$, 
expressing energy in terms of frequency. We see that the appearance of $\bar h$ with the time derivative in Schrödinger's equation accounts for the energy conversion and is completely normal and without mystery. 

Next, we consider the space dependent Hamiltonian in the basic case of the Hydrogen atom:
  • $H\psi =  \frac{\bar h^2}{2m}\Delta\psi +  \frac{e^2}{r}\psi$  
where $\psi =\psi (x)$ with $x$ a space coordinate, $r =\vert x\vert$, and $m$ is the mass of the electron. Normalising by changing scale in space $x=a\bar x$ and time $t=b\bar t$, we obtain the Hamiltonian in normalized atomic units in the form
  • $\bar H = \bar\Delta + \frac{2}{\bar r}$ with smallest eigenvalue $1$, 
  • $a=\frac{\bar h^2}{me^2}$ as $Bohr\, radius$,
  • $b=\frac{\bar h2a}{e^2}$ as $Bohr\, time$ with $\omega =\frac{1}{b}$ angular velocity
  • $E =\frac{e^2}{2a}$ as $Rydberg\, energy$.
We now observe that
  • $E\, b = \bar h$,
  • $E = \bar h\, \omega$, 
which shows that the also the space dependent part of Schrödinger's equation is calibrated to the energy conversion standard. 

Finally, Planck's constant also appears in Planck's radiation law and then in the high-frequency cut-off factor
  • $\frac{\alpha}{\exp(\alpha )-1}$
  • $\alpha = \frac{h\nu}{kT}$,
where $k$ is Boltzmann's constant and $T$ temperature. We see that again $h\nu$ appears as an atomic energy measure with a value that is not very precisely determined in its role in the cut-off factor.
The value of $h$ from photoelectricity can then serve also in Planck's law.

We conclude that Planck's constant $h$ is a conversion standard between two energy measures and as such has no meaning as a universal quantum of action or as integral multiples $nh\nu$ with $n=1,2,3,..$ of special significance other than by connection to eigenfunctions and eigenvalues.   

Ultimately, what is measured are atomic emission spectra in terms of frequencies and wave lengths which through Planck's constant can be translated to energies expressed in electronvolts (or Joule). Nothing of the internal atomic structure (in terms of $e$ and $m$) enters into this discussion.

Planck introduced $h$ in a statistical argument in 1900 long before atoms were known, Einstein picked up $h\nu$ in his 1905 article on photoelectricity, before atoms were known, and Schrödinger put $h$ into his equation in 1926 to describe atoms. This line of events supports the idea that Planck's constant $h$ is a convention without any universal significance.

Understanding the real role of Planck's constant may help to give Schrödinger's equation a physical interpretation which is free from mysteries of "quantization" and statistics. Versions of Schrödinger's equation based on an idea of smoothed particle mechanics then naturally present themselves, with $h$ acting as a smoothing parameter.

PS Notice that the fine structure constant $\alpha = \frac{e^2}{\bar hc}=\frac{1}{137}$ can be expressed as $\alpha =\frac{2}{c}\frac{a}{b}$ which shows that $\alpha$ relates $Bohr\, speed\, =\frac{a}{b}$ to the speed of light $c$. This relation is viewed to be fundamental, but why is hidden in mystery.   

tisdag 1 april 2014

Royal Swedish Academy of Sciences: CO2 Warming Can Prevent New Ice Age

The Royal Swedish Academy has issued a New Statement on the Scientific Basis of Climate Change giving up its former support of the CO2 global warming alarmism of IPCC and returning to the standpoint of the legendary foremost leading member of the Academy Svante Arrhenius, Nobel Prize in Chemistry in 1903, who in Worlds in the Making (1908) suggested that the human emission of CO2 would be strong enough to prevent the world from entering a new ice age, and that a warmer earth would be needed to feed the rapidly increasing population, of particular importance for the Swedish people under immediate threat of being covered under 1000 m solid ice:
  • Although the sea, by absorbing carbonic acid, acts as a regulator of huge capacity, which takes up about five-sixths of the produced carbonic acid, we yet recognize that the slight percentage of carbonic acid in the atmosphere may by the advances of industry be changed to a noticeable degree in the course of a few centuries. (p54)
  • Since, now, warm ages have alternated with glacial periods, even after man appeared on the earth, we have to ask ourselves: Is it probable that we shall in the coming geological ages be visited by a new ice period that will drive us from our temperate countries into the hotter climates of Africa? 
  • There does not appear to be much ground for such an apprehension. The enormous combustion of coal by our industrial establishments suffices to increase the percentage of carbon dioxide in the air to a perceptible degree. (p61)
  • By the influence of the increasing percentage of carbonic acid in the atmosphere, we may hope to enjoy ages with more equable and better climates, especially as regards the colder regions of the earth, ages when the earth will bring forth much more abundant crops than at present, for the benefit of rapidly propagating mankind. (p63)
A major revision of Swedish and European climate politics is expected to follow from the U-turn in the  scientific view of the Academy. The Swedish King says he is ready to act, and turn on the heat in his many huge poorly insulated royal castles.

New Theory of Flight Presented to the World

Simulation movie of airflow around a jumbojet in landing configuration at large angle of attack.

The revised version of New Theory of Flight has now been submitted to Journal of Mathematical Fluid Mechanics for expected swift publication.

This article together with my former students Johan Hoffman and Johan Jansson represents the summit of my scientific career as a combination of mathematical analysis and computation. The article asks for a major revision of text book aerodynamics and opens new roads to aerodynamic design. And it is not a joke…Finally, The Secret of Flight can be revealed to humanity.

Once the article has appeared in JMFM the new theory will be launched in a press release to media. Stay tuned….

Here is the Summary of article:
  • The new theory shows that the miracle of flight is made possible by the combined effects of (i) incompressibility, (ii) slip boundary condition and (iii) 3d rotational slip separation, creating a flow around a wing which can be described as (iv) potential flow modified by 3d rotational separation. 
  • The basic novelty of the theory is expressed in (iii) as a fundamental 3d flow phenomenon only recently discovered by advanced computation and analyzed mathematically, and thus is not present in the classical theory. 
  • Finally, (iv) can be viewed as a realization in our computer age of Euler’s original dream to in his equations capture an unified theory of fluid flow. 
  • The crucial conditions of (ii) a slip boundary condition and (iii) 3d rotational slip separation show to be safely satisfied by incompressible flow if the Reynolds number is larger than 106. For lower Reynolds numbers the new theory suggests analysis and design with focus on maintaining (ii) and (iii).