torsdag 27 november 2014

The Radiating Atom 3: Resolution of Schrödinger's Enigma

What we observe as material bodies and forces are nothing but shapes and variations in the structure of space....A lecture course that I gave this winter (1952) on the current views of quantum mechanics has convinced me definitively that that they are inadequate from the outset, viz. from Born's probability interpretation, which I disliked from the first moment on and have ever since. So I have decided to take a firm stand  against it, pointing out its philosophical shortcomings. I have little hope of convincing many people now, the credo is too firmly established.

Encouraged by Schrödinger's view on quantum mechanics as deterministic continuous waves rather than statistics of discrete particles subject to quantum jumps, let me suggest a possible solution to the basic enigma of the mechanics of an atom capable of being observed by emission of radiation, then in line of the analysis of Mathematical Physics of Blackbody Radiation (also exposed here) starting from the two previous posts.

Let us then first rewrite Schrödinger's equation (with $H$ the Hamiltonian)
  • $ih\dot{\Psi} + H\Psi =0$, 
where $\Psi = \psi + i\phi$ with $\psi (x,t)$ and $\phi (x,t)$ real-valued functions of space $x$ and time $t$ with the dot representing time differentiation, into the system (with h=1)
  • $\dot\psi +H\phi =0$,
  • $-\dot \phi + H\psi  =0$,     
which has the form of a harmonic oscillator and can be written as a scalar second order in time equation
  • $\ddot\psi+H^2\psi =0$ and/or $\ddot\phi+H^2\phi =0$. 
We see that the quantum mechanical model of an atom has the form of the wave equation studied in Mathematical Physics of Blackbody Radiation.  The analysis therein of the extended equation with near-resonant forcing and small radiative damping
  • $\ddot\phi+H^2\phi -\gamma\dddot\phi=f$,
thus should apply, with $\gamma (\phi )$ a small damping coefficient depending on $\phi$ to be determined and $f=f(x,t)$ the forcing. Let then $\phi_1=\phi_1(x)$ and $\phi_2=\phi_2(x)$ be two eigen-functions of $H$ satisfying
  • $H\phi_1=\nu_1\phi_1$ and $H\phi_2=\nu_2\phi_2$
with eigen-values $\nu_1<\nu_2$, and thus 
  • $H^2\phi_1=\nu_1^2\phi_1$ and $H^2\phi_2=\nu_2^2\phi_2$,
with corresponding solutions of $\ddot\phi+H^2\phi=0$ as pure eigen-states 
  • $\Phi_1(x,t)=\exp(i\nu_1t)\phi_1(x)$ and $\Phi_2(x,t)=\exp(i\nu_2t)\phi_2(x)$. 
Here $\Phi_1$ may be the ground state of smallest energy $\nu_1^2$. Note here that the energy scales with $\nu_1^2$ and not $\nu_1$ as in Einstein's relation $h\nu_1 = E$ which is not a true energy relation, but instead a frequency relation. 

We observe that the charge density
  • $\vert\Phi_j(x,t)\vert^2 =\Phi_j(x,t)\overline{\Phi_j(x,t)}=\phi_j(x)^2$ for $j=1,2$,
is constant in time, which means that a pure eigen-state is not radiating, because real (observable) time-dependence is lacking. In other words,
  • $\gamma (\Phi) = 0$ if $\Phi$ is a pure eigen-state. 
On the other hand, if $\Phi = c_1\Phi_1 + c_2\Phi_2$ is a non-trivial linear combination of such pure eigen-states with both $c_1$ and $c_2$ non-zero, then the corresponding charge density $\vert\Phi\vert^2$ has a time dependence of the form $\cos((\nu_2-\nu_1)t)$ with a resonant beat frequency $\nu = \nu_2 -\nu_1 >0$ and thus is (must be) radiating under resonant forcing. Therefore
  • $\gamma (\Phi) >0$ if  $\Phi$ is a non-trivial linear combination of pure eigen-states of different frequencies. 
The analysis in Mathematical Physics of Blackbody Radiation then shows under the assumption that $\gamma >0$ is small and near-resonant forcing, that the radiated energy balances the forcing in in sustained oscillation $\phi(x,t)$ between pure eigen-states, in the sense that
  • $\int \gamma\ddot\phi^2(x,t)dxdt \approx \int f^2(x,t)dx dt$. 
It is important to notice that the energy balance holds for any small value of $\gamma >0$. The precise value of $\gamma$ is thus irrelevant. 

We are thus led to the following mathematical description of an atom capable of emitting radiation subject to forcing:
  1. Pure eigen-states do not radiate and thus correspond to harmonic oscillations. In this case $\gamma =0$.
  2. Forcing with frequency $\nu =\nu_2$ with $\nu_2>\nu_1$ with $\nu_2$ and $\nu_1$ eigenvalues of the Hamiltonian, is capable of generating an eigen-state $\Phi_2$ with energy $\nu_2^2$ starting from an eigen-state $\Phi_1$ with lower energy. Here it is important that $\gamma$ is small to allow energy to be pumped into the oscillator and not just be radiated/dissipated.
  3. Forcing with frequency $\nu_2>\nu_1$ can thus generate a non-trivial combination of pure eigen-states, which can be radiating with a beat frequency $\nu =\nu_2 -\nu_1$, which can be sustained by forcing of frequency $\nu$.
  4. Once the non-trivial linear combination is formed by forcing with frequency $\nu_2$ starting from e.g. the ground state, radiation can be sustained by forcing with the beat frequency $\nu =\nu_2-\nu_1$, which thus appears as the observable frequency of emitted radiation. 
  5. Since a pure eigen-state is not radiating it is impossible to check if a pure eigen-state other than the ground state is present, other than observing radiation with some beat frequency. If the beat is observable then somehow two pure eigen-states must have been generated.
This resolution of the enigma of the atom is I think in the line of thinking of Schrödinger (and would maybe have made him as happy as on the picture), which unfortunately was suppressed by Bohr who crushed Schrödinger's understandable wave mechanics and replaced it with a non-understandable (horrible) mixture of statistics of particles and quantum jumps.  Maybe Schrödinger as the creator of quantum mechanics is not dead after all...

PS Since the inner physics of a pure eigen-state is hidden to inspection, because it is not radiating, it may well be that a Schrödinger wave equation for an atom with $N$ electrons can be found as a (non-linear) system of $N$ electronic wave functions depending on a common 3d space coordinate and time, instead of the linear scalar equation depending on $3N$ space coordinates usually named Schrödinger's equation, which is both unphysical and uncomputable.    


onsdag 26 november 2014

The Radiating Atom 2: Those Damn Quantum Jumps

If we are going to have to put up with those damn quantum jumps, I am sorry I ever had anything to do with quantum theory.

Schrödinger formulated the Schrödinger equation as the foundation of quantum mechanics in 1926, but his equation was then hijacked by Bohr, Born and Heisenberg, who gave it a meaning as statistics of discrete energy quanta, which Schrödinger could not accept and forced him out of business.

Schrödinger returned to the  in 1952 in his article Are There Quantum Jumps? seeking to resurrect quantum mechanics as wave mechanics resonances without any need of particles and discrete energy quanta or light quanta (photons). Schrödinger's view was present in the previous post considering interference resonance in superposition (linear combination with (real say) coefficients $c_1$ and $c_2$)
  • $\psi (x,t) = c_1\psi_1(x,t)+c_2\psi_2(x,t)$
of two eigen-states $\psi_1(x,t)=\exp(i\nu_1t)\phi_1(x)$ and $\psi_2(x,t)=\exp(i\nu_2t)\phi_2(x)$ satisfying Schrödinger's equation
  • $ih\frac{\partial\psi_j}{\partial t} + H\psi_j = 0$  for $j=1,2$,
where $H\phi_1=E_1\phi_1$  and $H\phi_2=E_2\phi_2$ with $E_1=h\nu_1$ and $E_2=h\nu_2$ and $H$ is the Hamiltonian operator acting with respect to a space coordinate $x$, thus with $\phi_1$ and $\phi_2$ eigen-functions of the Hamiltonian with eigen-values $E_1$ and $E_2$ and corresponding frequencies $\nu_1$ and $\nu_2$ (with $\nu_2 > \nu_1$).

Introducing
  • $\rho (x,t) = \vert\psi (x,t)\vert^2 =  \psi (x,t)\overline{\psi (x,t)}$,
as a measure of electronic charge distribution, direct computation shows that  
  • $\rho (x,t) = c_1^2+c_2^2 + 2c_1c_2\cos((\nu_2 -\nu_1)t)$.
We see that if either $c_1=0$ or $c_2=0$, then the electronic charge distribution $\rho$ is constant in time and thus does not generate any electromagnetic radiation. An atom in a simple eigen-state such as the ground state does not radiate.

On the other hand, in real superposition with if $c_1c_2 > 0$, the electronic charge varies in time with frequency $\nu_2-\nu_1$, and thus generates electromagnetic radiation according to the Abraham-Lorentz law or Larmor formula stating that radiation power is proportional to the square of charge acceleration.

This means that an electron in true superposition of two states of different eigenstates of different frequencies, must radiate and thus needs external forcing to persist. This is what happens in emission/absorption spectrography with a hot/cold gas emitting/absorbing light of specific frequencies.

This phenomena of interference in superposition is the (sincere and true Schrödinger) rational of the Einstein-Planck's relation
  • $h\nu = E$      
with $E=h\nu_2 - h\nu_2$ by Bohr-Heisenberg-Born instead viewed as a difference in "energy" between two states, and $h\nu$ a so-called "quantum of energy" supposedly being emitted/absorbed when an electron "jumps" between two eigen-states.

Schrödinger's main point is that there is no need to introduce any concept of "energy quanta" and electron "jump" to give the relation $h\nu = E = h\nu_2 -h\nu_1$ a meaning, because its (sincere and true) meaning is that the frequency $\nu$ emitted from superposition is simply equal to the difference $\nu_2 -\nu_1$, that is a beat frequency. This is highly remarkable and gives strong support to Schrödinger's view.

But without energy quanta the quantum mechanics of Bohr-Heisenberg-Born has no meaning and that is why Schrödinger left the field in dismay.

It remains to continue from where Schrödinger ended in 1952 (or 1927). My idea is then to extend the analysis in Mathematical Physics of Blackbody Radiation (proving Planck's radiation law using finite precision wave mechanics without the statistics of energy quanta used by Planck in his proof)  to atom physics following the (Vedanta) spirit of Schrödinger.
   

tisdag 25 november 2014

The Radiating Atom 1: Schrödinger's Enigma

                                                   Are there quantum jumps?

This is a first step in my search for a wave equation for a radiating atom as an analog of the wave equation with small damping studied in Mathematical Physics of Blackbody Radiation.

Schrödinger formulated his basic equation of quantum mechanics in the last of his four legendary articles on Quantisation as a Problem of Proper Values I-IV from 1926. Central to quantum mechanics is the basic relation (with $h$ Planck's constant)
  • $\nu = (E_2 - E_1)/h$
between the frequency $\nu$ of emitted radiation, and the difference in energy $E_2 - E_1$ between two solutions $\psi_1(x,t)=\exp(i\nu_1t)\phi_1(x)$ and $\psi_2(x,t)=\exp(i\nu_2t)\phi_2(x)$ satisfying Schrödinger's equation 
  • $ih\frac{\partial\psi}{\partial t} + H\psi = 0$ 
where $H\phi_1=E_1\phi_1$  and $H\phi_2=E_2\phi_2$ with $E_1=h\nu_1$ and $E_2=h\nu_2$ and $H$ is the Hamiltonian operator acting with respect to a space coordinate $x$.

To connect to the basic relation, consider the function
  • $\Psi (x,t) = \vert\Phi (x,t)\vert^2 =  \Phi (x,t)\overline\Phi (x,t)$,
with
  • $\Phi (x,t) = c_1\psi_1(x,t)+c_2\psi_1(x,t)$
a linear combination with coeffcients $c_1$ and $c_2$.

Direct computation shows that $\Psi (x,t)$ has a time dependency of the form 
  • $\exp(i(\nu_2 -\nu_1)t)$,
and thus corresponds to a beat between two frequencies as an interference phenomenon.  

Interference between two eigen-states of energies $E_2$ and $E_2$ can thus naturally be viewed as a resonance phenomenon or beat-interference of frequency $\nu =(E_2 - E_1)/h$, which can be associated with emitted radiation from an oscillation of the modulus $\Psi (x,t)$ of the same frequency , because a pulsating charge generates a pulsating electromagnetic field.

It remains to formulate a Schrödinger equation with (small) radiation damping for an atom as an analogue of the wave equation studied in Mathematical Physics of Blackbody Radiation, an equation describing atomic oscillation between two energy levels as the origin of observable emitted radiation.

It is encouraging to note that Schrödinger in his article IV directly connects to radiation damping as an essential element of a mathematical model for an atom, a connection which is not present in the standard Schrödinger equation without radiation damping.

The mantra that presents itself is:
  • Listen to the beat of the atom!
The model should contain a damping coefficient which vanishes when $\nu$ is an eigenvalue of the Hamiltonian and is small else. This makes the beat observable, while eigenvalues and eigenfunctions of the Hamiltonian are not. 


måndag 10 november 2014

CJ70: A Posteriori Scientific Summary and A Priori Extrapolation

I am very happy to here announce the upcoming event CJ70 at Mathematical Sciences at Chalmers Nov 13 gathering former students and coworkers into a joyful a posteriori recollection of past victories, summaries of state-of-the-art and a priori extrapolations towards the 2045 Singularity resulting from computing power doubling every 18 months. My own thoughts to be expressed at this memorable event, are available here.

fredag 10 oktober 2014

Löfven till EU: Sverige Skall ha Lägst Arbetslöshet i EU 2020


Regeringskansliet meddelar triumfatoriskt att Sveriges nye statsminister Stefan Löfven gjorde en grandios entre bland övriga satschefer i EU vid EU-mötet "Growth and Employment"  i Milan genom att "share the experience that" Sverige skall ha lägst arbetslöshet i EU 2020:
  • My Government, which came into office only last Friday, has made jobs and employment its top priority. 
  • We have set ourselves an ambitious mark: to have the lowest unemployment rate in the EU by 2020.
  • To share experiences, like we are doing today, is a great way to enhance our understanding and to increase the impact of our policies.
Hur detta mål skall uppnås står skrivet i stjärnorna, men det är klart att ett sätt att nå dit är att verka för att övriga EU skall få så hög arbetslöshet som möjligt. Återstår att se om Löfven kan finna gehör för en sådan politik hos övriga EU-ledare.

onsdag 8 oktober 2014

Löfven Styr mot Nyval

Dagens riksdagsdebatt visade att Löfven styr spikrakt mot nyval. Ett nyval där SD kan komma att åter fördubbla sina röstsiffror och därmed kunna bli landets största parti med 26%, eftersom S mycket väl kan falla till 25%, eller 20. Löfven vände sig direkt till de 13% som röstat på SD (med "jag ser Dig och jag hör Dig") med upplysningen att de röstat på ett rasistiskt parti med mörkt förflutet men att Löfven lyssnar, ser och hör  alla de som röstat på SD.

Frågan är vilken partiledare som först kommer att ta avstånd från den koldioxidalarmism som är  utgångspunkten för den "energiöverenskommelse" som både S och MP påstår sig ha träffat. Det kan inte vara Löfven, som lovat svenska folket att reducera användningen av fossil energi med 90% (och kärnkraften till 0%).

Begrunda även vad Lars Bern skriver på Antropocene.

PS När Stefan Löfven utsågs till ny partiledare för S försökte jag (utan framgång) att få S att förstå att matematikundervisningen skulle kunna förnyas och fyllas med meningsfullt innehåll för många elever genom att skrota den gamla skolmatematik, som nu (utan framgång) presenteras under alla de många timmar många elever tvingas genomlida (utan utbyte), med det nya skolämnet Matematik-IT. Något nytt försök till samtal med S om denna möjlighet är knappast att tänka på.

måndag 6 oktober 2014

Nobel Prize in Medicine to Computational Adaptive Finite Element Brain Mesh


The Nobel Prize in Medicine 2014 has been awarded the discovery that both a fly, rat and human being orient in space using a computational gps-map constructed by and stored in the brain in the form of a multilevel adaptive finite element mesh. This was what I conjectured 30 years ago.

Löfvens Ordning och Reda = Kaos

Löfven har utgående från sitt mantra "Ordning och Reda" inom energi, vård-skola-omsorg, utrikespolitik, you name it,  efter en arbetsdag som statsminister bäddat för kaos inom energi, vård-skola-omsorg och utrikespolitik. Återstår så att bädda för budgetkaos.

Endast SD kan nu rädda Löfven, eftersom den till Alliansen omtalade utsträckta handen visat sig vara ett slag i ansiktet, eller ovänlig knuff i bröstet.

lördag 4 oktober 2014

Regeringsförklaringen: Kärnkraften Ersätts med Förnybar Energi


Sveriges nye statsminister Stefan Löfven deklarerar glatt i Regeringsförklaringen:
  • Klimatfrågan är vår tids ödesfråga. 
  • Klimatförändringarna är ett globalt säkerhetshot. 
  • En ny miljöbilsbonus för bilar med liten klimatpåverkan införs. Ett miljömålsråd inrättas. 
  • Kärnkraften ska ersättas med förnybar energi. 
  • Stödet till havsbaserad vindkraft och till solkraft ska stärkas.
  • Sysselsättningen i Sverige är helt beroende av att det finns en god och tillförlitlig tillgång till el till konkurrenskraftiga priser.
Se där Löfvens "ingångsvärden" i de förhandlingar över blockgränsen, som nu skall forma framtiden för vårt lilla land. Sol och vind och vår.

måndag 29 september 2014

S + MP: Obligatoriskt Gymnasium

S+ MP meddelar stolt att man kommit överens om om att utvidga skolplikten att omfatta även förskoleklass och gymnasium, inalles 13 årig obligatorisk skola för alla. Skälet upp ges vara
  • Många börjar gymnasiet, men när det blir lite motigt hoppar många unga av. 
  • Genom att göra gymnasiet obligatoriskt kan detta förhindras:
  • Alla resurser skall sättas in så fort någon är på väg att hoppa av.
Lärarfacket uppges jubla. I förslaget ingår obligatorisk matematikundervisning under 13 år, i de fyra räknesätten inklusive bråkräkning och regula de tri.  Alla resurser skall sättas in så fort någon elev härför visar bristande intresse eller färdighet. Efter 13årig skolplikt skall ingen svensk tveka inför uppgiften att räkna ut vad 1/2 + 1/3 blir, en för arbetslivet både nödvändig och tillräcklig kunskap.

S+MP förväntar sig en bred blocköverskridande överenskommelse med speciellt FP,  även om FP har diametralt motsatt uppfattning, ingen match för Stefan Löfvens med sin omvittnade förhandlingsfärdighet och 2åriga gymnasieutbildning i botten.

Den utvidgade 13åriga obligatoriska skolplikten kan ses som en kompensation för att under Alliansen värnplikten avskaffats, skatteplikten urholkats och det fria skolvalet exploderat, som en blivande svag regerings fromma förhoppning om att kunna stärka Staten och därmed finna en uppgift.

Återstår att se om S+MP kan anamma mitt förslag om reformerad skolmatematik som Matematik-IT.
Inte helt osannolikt. Obligatorisk Matematik-IT! Sveriges matematiker borde jubla!