Decemberöverenskommelsen = NyDemokrati = 1984 + 30

                    

 

Huvudpunkterna i Decemberöverenskommelsen är:

1. Den statsministerkandidat som samlar stöd från den partikonstellation som är större än alla andra tänkbara regeringskonstellationer ska släppas fram. 
2. En minoritetsregering ska kunna få igenom sin budget.

Här finns flera kluriga språkliga konstruktioner att grubbla över: Vad menas med 

• "partikonstellation"?
• "regeringskonstellation"?
• "alla tänkbara"?
• "ska kunna få igenom"?

Notera glidningen i syftning via "andra" från "partikonstellation" till "regeringskonstellation", vilket gör 1. språkligt omöjlig: Hur skall man kunna välja det äpple som är större än alla andra päron?

Varför inte bara säga att statsministerkandidaten från den största "partikonstellationen" skall "släppas fram" (ungefär som något katten släpat in)? Om det är det man vill säga? 

Och varför införa begreppet "tänkbar regeringskonstellation" när det  måste handla om "föreslagen verkligt realiserbar möjlig konstellation" och inte bara "tänkbar", och av vem i så fall? Kungen, som ju är bra på att tänka? Eller Löfven? 

Vidare: varför skrivs det  "ska kunna få igenom" och inte "ska få igenom". Med "kunna" anges att ett (outtalat) villkor föreligger: Det är ju även utan någon Decemberöverenskommelse helt möjligt att en minoritet skall "kunna" få igenom sin budget, dvs om bara majoriteten lägger ner sina röster. Varför då inte direkt skriva ner den reella innebörden nämligen att:

• Majoritetens röster räknas inte.
• Bara Minoritetens röster räknas.

Vi har här grundbulten i NyDemokrati som i Orwellsk anda med 30 års fördröjning infördes i vårt land 2014 med Decemberöverenskommelsen, utan att grundlagen behövde ändras (vilket ju alltid kan ske i efterhand, om det händelsevis skulle behövas).

PS1 Överenskommelsen innebär att RödGröna och Alliansen gjort upp om att alternera vid makten och att oppositionen därvid alltid lägger ner sina röster: Rödgröna får ha makten 2014-18 och sedan är det Alliansens tur 2018-22 osv. Ett mer demokratiskt och resurssnålt system får man nog leta efter: Varje sida får spela match under 4 år med full kraft för att sedan sitta i avbytarbåset och vila sig i 4 år.

Konstigt att de politiska partierna inte tidigare kommit överens om en sådan förnämlig effektivt fungerande kartell till allas bästa. Eftersom detta är en frivillig öppet deklarerad överenskommelse, är den helt enligt, eller snarare utanför, grundlagen och behöver således inte godkännas av folket.

PS2 Följande valutgång kan förväntas 2018 (eller tidigare): Rödgröna 33%, Allians 33%, SD 34%.
I ett sådant läge kan SD bilda regering med en minoritet om 34% som underlag medan majoriteten om 66% enligt Decemberöverenskommelsen är förbundna att lägga ner sina röster.

I riksdagsvalet i Tyskland 6/11 1932 fick NSDAP 33.1% (efter en tillbakagång från 37.3% i valet i juli samma år, men därefter en ökning till 43.9% 5/3 1933 i Weimarrepublikens sista val).

The Radiating Atom 9: Hydrogen and Beyond

A plane electrical field $E_z$ acting in the $z$-direction and progressing in the $x$-direction, will interact with the $2p_z$ eigenstate of a Hydrogen atom pictured above corresponding to a charge oscillating in the $z$-direction in parallel with $E_z$.  Note that $E_z$ will not interact with the $2p_x$ and $2p_y$ eigenstates.

As a sum-up of the present series of posts on the radiating atom, we consider Schrödinger's equation for a radiating Hydrogen atom subject to forcing in the form of a second order wave equation

• $\ddot\psi +H^2\psi -\gamma\dddot\psi = f$      (1)

where $\psi (x,t)$ is a real-valued electronic wave function of a space coordinate $x=(x_1,x_2,x_3)$ and time $t$, $H$ is the Hamiltonian defined by 

• $H =-\frac{h^2}{2m}\Delta + V$,

where $\Delta$ is the Laplacian with respect to $x$, $V(x)=-\frac{1}{\vert x\vert}$ is the kernel potential, $m$ the electron mass, $h$ Planck's constant, the dot signifies differentiation with respect to time $t$, $f$ is external forcing, and $\gamma =\gamma (\psi )$ is a non-negative radiation damping coefficient.

The formulation of Schrödinger's equation as a second order wave equation in terms of a real-valued wave function was considered by Schrödinger in 1926 as an alternative to the standard formulation as a 1st order complex-valued equation. In the homogeneous case with $f=0$ and  $\gamma =0$,  the two formulations are equivalent: In particular, conservation of total charge as

• $\frac{d}{dt}\int\rho (x,t)dx =0$,

with $\rho =\psi^2+(H^{-1}\dot\psi )^2$ the charge intensity, is obtained by multiplying (1) with $H^{-2}\dot\psi$ and integrating in space. In the non-homogeneous case (1) may be more natural as an expression of a force balance with $-\gamma\dddot\psi$ the Abraham-Lorentz radiation recoil force and $f$ an electrical field component, while the physical meaning of the standard formulation baffled the creators of modern physicists and followers and led into unphysical interpretations as particle statistics.  

We consider radiation of frequency $\nu =(E_2-E_1)/h$ where $E_1$ is the energy of the ground state as an eigenfunction $\Psi_1 (x)$ of $H$ with minimal eigenvalue $E_1$ and $E_2$ is a larger eigenvalue with eigenfunction $\Psi_2(x)$. We reformulate (1) in the form

• $\ddot\psi +H_1^2\psi -\gamma\dddot\psi = f$,      (2)

where $H_1 = H - E_1$ and note that $H_1\Psi_1=0$ and $H_1\Psi_2=(E_2-E_1)\Psi$. 

We assume that the forcing is given as a linear combination of plane electromagnetic waves $(0,0,\cos(\omega (x_1-ct))$ of frequencies $\omega\approx\nu =(E_2-E_1)/h$ progressing in the $x_1$-direction with the speed of light $c$. We seek a solution $\psi (x,t)$ of (2) of as a linear combination of $\Psi_1$ and $\Psi_2$ of the form

• $\psi (x,t) =c_1(t)\Psi_1(x) + c_2(t)\Psi_2(x)$

with time dependent coefficients $c_1(t)$ and $c_2(t)$. Inserting this Ansatz into (2), multiplying by $\Psi_1$ and $\Psi_2$ and integrating with respect to $x$, we obtain assuming orthonormality of $\Phi_1$ and $\Psi_2$, time-periodicity and normalizing to $c=1$ and $h=1$:

• $\ddot c_1(t) -\gamma\dddot c_1(t) = f_1(t)\equiv\int f(x,t)\Psi_1(x)dx$ for all $t$,
• $\ddot c_2(t) +\nu^2c_2(t)-\gamma\dddot c_2(t) =f_2(t)\equiv\int f(x,t)\Psi_2(x)dx$ for all $t$.

By $x_1$-symmetry of $\Psi_1(x)$ it follows that $f_1(t)=0$ with the effect that $c_1(t)=c_1$ is constant. Further, if $\Psi_2(x)$ is a $(2,1,0)$ p-state oriented in the $x_3$-direction, see above figure, then $f_2(t)$ is a non-zero linear combination of $\cos(\omega t)$, and by the analysis of Mathematical Physics of Black Body Radiation  and Computational Black Body Radiation,

• $\int\gamma\ddot\psi^2(x,t)dxdt = \int\gamma\ddot c_2^2(t)dt\approx \int f_2^2(t)dt$,  (3)

which expresses that output = input as a fundamental aspect of radiation in time-periodic equilibrium as a phenomenon of near-resonance under small damping. The setting can be generalized to other eigenstates. The essence is the output = input balance, which can express both excitation into eigenstates of larger energy and radiation from such states.

The value of the radiation damping coefficient $\gamma (\psi )$ is set so that conservation of charge is maintained under forcing with the radiation balance (3). If $f=0$ and $\psi$ is a pure eigenstate, then $\gamma = 0$.

Notice that the above argument can be shifted by replacing the ground-state $\Psi_1(x)$ as a time-independent and non-radiating pure eigenstate by an eigenfunction $\Psi_j(x)$ of $H$ with larger energy, again viewed as a time-independent and non-radiating pure eigenstate. This reflects that the time-dependence of pure eigenstates is not observable and thus up to the imagination of an observer. This is not evident in the standard formulation of Schrödinger's equation.

We sum up the virtues of (1) as a semi-classical continuum wave model of a radiating atom subject to forcing, as compared to QED as a non-classical quantum particle model:

1. (1) lends itself to physical interpretation as force balance.
2. (1) lends itself to mathematical analysis. 
3. The term $\ddot\psi$ connects to kinetic energy in classical mechanics and suggests that the common terminology of quantum mechanics of connecting $\Delta\psi$ to kinetic energy, is not natural; a connection to a form of elastic energy may have better physical meaning.
4. (1) has a natural extension to a model for a many-electron atom as a system of one-electron equations, which is computable and thus potentially useful, in contrast to the standard multi-dimensional Schrödinger equation, which is uncomputable and thus potentially useless. 
5. The incoming wave is represented as forcing independent of the wave function $\psi$, which faciliates mathematical analysis and understanding, and not as in QED through a time-dependent contribution to the Hamiltonian, which opens to troublesome self-interaction. 

Sverige: Humanitär Stormakt: Löfvens Jultal: Kunskap: Dynamit: Nyårsklockor: Nyval: Icke Nyval

Sveriges regering deklarerade den 13/2 2013 genom Carl Bildt att

• Sverige är en humanitär stormakt.   

Detta är idag en politisk sanning omfattad av både Allians och Rödgrön Regering, som i debatten ofta används (tex här och här) som uttryck för att svenska folket är utvalt och har förmånen att leva i den bästa av världar.

Att Sverige en gång var en stormakt (1611-1718) har vi fått lära oss i skolan, och vi läser i media att Sverige idag har en vapenexport väl värdig en stormakt. Föreställningar  om stormakt och utvaldhet är oss alltså inte främmande,  utan har tvärtom blivit en så integrerad del av "den svenska modellen" eller den "svenska mentaliteten" att de svårligen kan identifieras och än mindre ifrågasättas.

När Löfven vid EU-mötet 18/12 så höjde rösten och krävde hårdhandskar mot Putin var det förmodligen utifrån känslor stimulerade av diffusa minnen från skolan om slaget vid Poltava 1709, där Sverige som stormakt knäcktes av Ryssland, och av Palme som stormaktspolitisk spelare med Sverige som insats.

Följdriktigt utfärdade Löfven idag i sitt jultal Det Svenska Löftet: Vi skall hjälpa varandra med kunskap, trygghet och jämlikhet, och ingen människa ska lämnas bakom. Detta som en svensk form av The American Dream (frihet, broderskap och jämlikhet i franska revolutionens anda) , allt enligt Löfven, som förtydligar: Jag vet att vi kan stå emot rasistiska krafter. Sverige ska vara framåtsträvande, rikt och modernt.

Vi ser att för Löfven är kunskap = frihet, trygghet = broderskap (vi skall hjälpa varandra) och jämlikhet = jämlikhet, där det nya är frihet = kunskap. Men kunskap är ojämnt fördelad och kan svårligen kombineras med att ingen människa skall lämnas bakom, såvida inte

• maximal kunskap = minimal kunskap = gymnasiekompetens 

i enlighet med Löfvens proposition till riksdagen (som dock röstats ned).

Dock, under alla förhållanden är det klart, eftersom Sverige delar ut Nobelpriset, att:

• Sverige är även en kunskaplig stormakt. 

Att Lövfen höll sitt jultal till kunskapens lov på Stortorget, med Nobelmuseet som fond, är således helt följdriktigt. Men säger inte etablerad vishet att kunskap = makt? I så fall skulle vi enligt Löfven få kedjan

• Sverige stormakt = Sverige kunskapsnation = Sverige framåtsträvande, rikt och modernt.

I så fall ett uppmuntrande budskap till svenska folket inför den instundande julen!

PS1 Löven kommer att ringa in det nya valet vid tolv-slaget på Nyårsafton, på Skansen i par med Jan Malmsjö som givit Tennyson's Nyårsklockor en ny egen tolkning (eftersom han glömt den ursprungliga versionen enligt egen uppgift till media):

Ring, klocka, ring i bistra nyårsnatten
mot rymdens norrskenssky och markens snö;
den gamla Riksdagen lägger sig att dö...
Ring Nyval över land och vatten!

Ring in det nya valåret och ring ut det gamla
i årets första, skälvande minut.
Ring lögnens makt från SD ut,
och ring in (S)anningen till os(S) som famla.

Ring våra tankar ut ur (S)orgens häkten
och ring hugsvalelse till (S)argat parti.
Ring hatet ut emellan os(S) och (S)verige(D)emokrati
och ring försoning in till (S)amarbete hela (S)läkten

Ring ut ett dödsdömt S+MP som räknat sina dagar
och forngestaltningar av split och kiv (SD).
Ring in ett ädlare politiskt liv 
och för minoritetsregering lämpligare lagar.

Ring ut arbetslöshet och sociala nöden,
och ring den frusna (S)kolan åter varm.
Ring ut det meningslösa gatuvåldets larm,
men ring till trainee-jobben arbetsglöden (V).

Ring ut den stolthet, som blott räknar anor (SD),
förtalets lömskhet (KD), avundens försåt (FP).
Ring in det rätta på humanitära triumfens (S)tråt,
och ring till seger (S)ocial(D)emokratins framåtsträvande fanor.

Ring, klocka, ring... och (S)eklets krankhet vike;
det dagas, Partiet fram i styrka går
Ring ut, ring ut rosornas och krigens år,
ring in en tusenåra (S)amregerings rika och moderna rike!

PS2 Alliansen har idag 27/12 beslutat att förena sig med Löfven på för att ringa in det nya budgetåret genom att i kör meddela svenska folket följande Decemberöverenskommelse:  

• Ring ut, ring ut de tusen krigens år i Riksdagen.
• Ring in den tusenåra fredens rike på Skansen(till 2022).

Överenskommelsen innebär att Riksdagen flyttas till Skansen från och med årskiftet och där slås samman med Akvariets apor, halvapor och ryggradslösa djur, detta för att skapa ordning och reda enligt Anna Kinberg Batra (M). Men är det verkligen Skansendemokrati som svenska folket vill ha?

Matematik-IT: Programmering i Grundskolan: Tiden Är Inne!

SVT rapporterade igår om att (Rapport efter 22:30 min):

• Som ett av de första länderna i världen har England infört programmering i läroplanen: Alla elever mellan 5 och 16 år skall inte bara kunna använda en dator utan också förstå hur den fungerar (National curriculum in England: computing programs of study).

Detta anknyter direkt till mitt förslag om ett Matematik-IT som nytt skolämne som lär ut både analytisk och numerisk matematik och programmering i en syntes, som ersättning för det nuvarande traditionella matematikämne utformat före datorn, samt till Karin Nygårds Upprop för programmering som skolämne.

När nu väl Fridolin har lämnat över till den nya utbildningsministern är det dags för att på nytt presentera Matematik-IT som en lösning på krisen inom det traditionella matematikämnet, som kan ses som det yttersta uttrycket för skolans kris. Alliansen måste nu kunna förstå att Matematik-IT är bra för Sverige och svensk skola, och sedan handla därefter.

Det finns nu många iPad-appar som lär ut programmering för alla åldrar. Jag planerar att efter nyår testa konceptet Matematik-IT tillsammans med mitt barnbarn Konrad 9 år och hans klasskamrater i årskurs 3 genom att leda eleverna till att själva tillverka datorspel med hjälp av lämpliga appar.

The Radiating Atom 8: Towards Resolution of the Riddle

Let us now collect the experience from previous posts in this series: We start recalling Schrödinger's equation for the one electron Hydrogen atom in standard form:

• $ih\dot\Psi +H\Psi =0$,            (1)  

where $\Psi =\psi +i\phi$ is a complex-valued function of space-time $(x,t)$ with real part $\psi$ and imaginary part $\phi$ as real-valued functions, and $H$ is the Hamiltonian defined by 

• $H =-\frac{h^2}{2m}\Delta + V$

where $\Delta$ is the Laplacian with respect to $x$, $V(x)=-\frac{1}{\vert x\vert}$ is the kernel potential, $m$ the electron mass, $h$ Planck's constant, and the dot signifying differentiation with respect to time $t$. The wave function $\Psi$ is normalized with

• $\int\rho (x,t)dx =1$ for all $t$
• $\rho =\vert\Psi\vert^2 =\psi^2 +\phi^2$,

where $\rho (x,t)$ is a measure of the charge intensity with total charge equal to one.  

Schrödinger's equation takes the following real-valued system form:

• $\dot\psi + H\phi =0$
• $\dot\phi -H\psi =0$,   

which upon differentiation with respect to time and recombination gives the following same second-order equation for both $\psi$ and $\phi$:

• $\ddot\psi + H^2\psi =0$, 
• $\ddot\phi + H^2\phi =0$, 

or the same equation in complex form with $\Psi =\psi +i\phi$ as a second-order Schrödinger equation:

• $\ddot\Psi + H^2\Psi =0$.       (2)

Let now $\psi_1(x)$ be the wave function of the ground state as an eigenfunction of $H$ with corresponding minimal eigenvalue $E_1$ satisfying $H\psi_1=E_1\psi_1$, that is $H_1\psi_1=0$ with $H_1=H-E_1$.

Let us then consider the following generalization of (2) into model of a radiating Hydrogen atom subject to external forcing:

• $\ddot\Psi +H_1^2\Psi -\gamma\dddot\Psi =f$,      (3)

where $-\gamma\dddot\Psi$ represents radiative damping with  $\gamma =\gamma (\Psi )$ a small non-negative radiation coefficient and corresponding radiation energy

• $R(\Psi ,t)=\int\gamma\vert\ddot\Psi (x,t)\vert^2dx$.

We see that $\Psi_1=\psi_1$ solves (3) with $f=0$. More generally, if $\psi_j$ is an eigen-function of the Hamiltonian with eigenvalue $E_j\gt E_1$, then $\Psi_j=\exp(i(E_j-E_1)t/h)\psi_j$ solves (3) with $\gamma =0$ and $f=0$ and represents a pure eigenstate of frequency in time $\nu =(E_j-E_1)/h$.

More generally, a superposition $\Psi =c_1\Psi_1+c_j\Psi_j$ of the ground state $\Psi_1$ and an excited eigen state $\Psi_j$ of frequency $\nu =(E_j-E_1)/h$ with non-zero coefficients $c_1$ and $c_j$ generates a charge

• $\rho (x,t)=\vert\Psi\vert^2=c_1^2\psi_1(x)^2+c_2^2\psi_j(x)^2+2\cos(\nu t)c_1c_j\psi_1(x)\psi_j(x)$,  

which varies in time, and thus may generate radiation.

In the spirit of Computational Physics of Black Body Radiation we are thus led to an analysis of (3) with a forcing $f$ in near-resonance and small radiative damping with eigenfrequencies $(E_j-E_1)/h$, or more generally $(E_j-E_k)/h$ with $E_j\gt E_k$, which as main result  proves the basic energy balance equation

• $\int R(\Psi ,t)dxdt \approx \int f^2(x,t)dxdt$, 

expressing that in stationary state output = input.

The following questions present themselves:

1. Which model, first order (1) or second-order (2), extends most naturally to radiation under forcing?
2. Is (3) to be viewed as a force balance with $-\gamma\dddot\psi$ as a Abraham-Lorentz radiation recoil force?
3. Which condition on $f$ guarantees that a pure eigenstate $\Psi_j$ is neither absorbing nor emitting, thus with $\gamma (\Psi_j)=0$? 

Remark 1. Note that the time dependence of an eigenstate $\Psi_j$ in superposition with an eigenstate $\Psi_k$ has frequency $(E_j-E_k)/h\gt 0$. The customary association of $\Psi_j$ 

to a frequency $E_j/h$, which can have either sign, is not needed and nor natural from physical point of view. The energy $E_j$ of an eigen-state has a physical meaning, but not $E_j/h$ as a frequency. This is a main of point of confusion in standard presentations of quantum mechanics supposedly being based on Einstein's relation $E=h\nu$ with $E$ energy and $\nu$ frequency.

Remark 2. Normalisation of wave functions under forcing and radiative damping, can be maintained by adjustment of the coefficient $\gamma (\Psi )$.

Remark 3. The energy balance in the form output = input or input = output, determines radiative equilibrium of an assembly of atoms, just as the corresponding relation in black body radiation expressed as Universality.

Remark 4. Schrödinger in the 4th and last of his 1926 articles first came up with (2) as an atomic wave equation, and then settled on (1) with the argument that a time-dependent Hamiltonian would cause problems in a transition from (1) to (2). The question is if Schrödinger gave up on (2) too easily? Maybe (2) is a better physical model than (1)?

Remark 5. Notice that (3) with an Ansatz of the form $\Psi (x,t)=c_1\Psi_1(x)+c_2\Phi (x,t)$ translates (3) into the wave equation in $\Phi$:

• $\ddot\Phi +H_1^2\Phi -\gamma\dddot\Phi =f$,

which is open to the analysis of Computational Physics of Black Body Radiation. What remains is to identify the forcing $f(x,t)$ resulting from an incoming electric or magnetic field. The basic case concerns the interaction between a $(2,1,0)$ p-state $\Phi_2(x)$ of eigenvalue $E_2$ with axis parallel to a plane-wave electrical field $E=(E_1,0,0)$ with $f = E_1$ in near-resonance with $\nu =(E_1-E_2)/h$.

Löfven som Schrödingers Katt

Som av en händelse kopplar min senaste serie av poster om den radierande atomen till det kaos som statsminister Löfven nu målmedvetet driver vårt land emot, och som i dagens ledare i SvD av Per Gudmundson beskrivs som Schrödinger's Regeringsalternativ, som en superposition av två diametralt olika tillstånd av

• liv -- död
• S-regering -- icke S-regering
• S-budget -- Allians-budget
• MP -- icke MP
• nyval -- icke nyval
• samarbete -- icke samarbete
• förbifart -- icke förbifart
• vårdval -- icke vårdval
• kärnkraft -- icke kärnkraft
• osv -- icke osv
• ...

Nu visar min analys av den radierande atomen att superposition av två möjliga över tid bestående egen-tillstånd utan utgående strålning, varav ett kan vara grundtillståndet och det andra ett exciterat tillstånd, är förknippat med energiförlust i form av utgående strålning av en frekvens som utgör skillnanden mellan frekvenserna för de båda egen-tillstånden och alltså utgör en dissonansfrekvens. 

Energiförlusten förknippat med en sådan dissonansfrekvens beskrivs i samma SvD av LO-basen under rubriken LO: Nyval hotar jobben, och innebär att en superposition inte är bestående över tid: Dissonansen skapar frustration och dränerar systemet på konstruktiv energi.

En överslagsberäkning ger vid handen att Löfvens superposition av två oförenliga tillstånd har en energiförlust med en halveringstid på en vecka. Räknat från den 1/12 ger detta efter 4 veckor en förlust av regeringsduglighet med en faktor 16! Ingen regering kan klara en sådan förlust vilket innebär att Löfven kommer att tvingas avgå före 29/12, dagen då nyval av Löfven är utannonserat att bli utlyst. 

The Radiating Atom 7: Quantum Electro Dynamics Without Infinities?

The interaction between matter in the form of an atom and light as electro-magnetic wave is supposedly described by Quantum Electro Dynamics QED as a generalization of quantum mechanics into the "jewel of physics" according to Feynman as main creator.  However QED was from start loaded with infinities requiring  "renormalization", which made the value of the jewel as a "strange theory" questionable according to Feynman himself:

• But no matter how clever the word, it is what I call a dippy process! Having to resort to such hocus pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self consistent. ... I suspect that renormalization is not mathematically legitimate. 

Let us see what we can say from the experience of the present series of posts on The Radiating Atom leading to the following Schrödinger equation for a radiating Hydrogen atom subject to exterior forcing:

• $\dot\psi + H\phi -\gamma\dddot\phi = f$,       (1)
• $-\dot\phi + H\psi -\gamma\dddot\psi = g$,      (2)

where $\psi = \psi (x,t)$ and $\phi = \phi (x,t)$ are real-valued functions of space-time coordinates $(x,t)$ (as the real and imaginary parts of Schrödinger's complex-valued electronic wave function $\psi +i\phi$), $\dot\psi =\frac{\partial\psi}{\partial t}$,

• $H=-\frac{h^2}{2m}\Delta + V(x)$

is  the Hamiltonian with $\Delta$ the Laplacian with respect to $x$, $V(x)=-\frac{1}{\vert x\vert}$ the kernel potential, $m$ the electron mass and $h$ Planck's constant,  $-\gamma\dddot\phi$ is a Abraham-Lorentz radiation recoil force with corresponding radiation energy $\gamma\ddot\phi^2$ with $\gamma$ a small positive radiation coefficient and $f=f(x,t)$ and $g=g(x,t)$ express exterior forcing. Note that here the electron wave function is coupled to radiation and forcing through a radiative damping modeled by $(-\gamma\dddot\phi ,-\gamma\dddot\psi )$ and the right hand side $(f,g)$, and not through a time-dependent potential connecting an incoming electric field to an electronic dipole moment, which is a common alternative. An advantage of the above more phenomenological model is simpler mathematical analysis since the potential is kept independent of time.

The system (1)-(2) can be viewed as a generalized harmonic oscillator with small radiative damping subject to exterior forcing similar to the system analyzed in Mathematical Physics of Black Body Radiation. The essence of this analysis is a balance of forcing and radiation (cf. PS5 below):

• $R \equiv\int\gamma (\ddot\psi^2 +\ddot\phi^2)dxdt\approx \int (f^2 + g^2)dxdt$,

which can be viewed to express that $output \approx input$.

A radiating atom with wave function $(\psi ,\phi )$ can be viewed to interact with an electromagnetic $(E,B)$ through the charge density

• $\rho (x,t) =\psi^2(x,t) + \phi^2(x,t)$,

according to Maxwell's equations:

• $\dot B + \nabla\times E = 0$, $\nabla\cdot B =0$,
• $-\dot E + \nabla\times B = J$, $\nabla\cdot E =\rho$,

with $J$ a corresponding current. For a superposition of two pure eigen-states with eigenvalues $E_1$ and $E_2$ the charge density varies in time with frequency $\omega =(E_2 -E_1)/h$ and then as an electrical dipole generates outgoing radiation

• $P\sim\omega^4$,   

which is balanced by the radiation damping in Schrödinger's equation

• $R=\int\gamma (\ddot\psi^2 +\ddot\phi^2)dxdt\sim\omega^4$.

The above QED model combining Schrödinger's equation for an atom with Maxwell's equations for an electro-magnetic field, thus explains the physics of 

1. an electron configuration as a superposition of two pure eigen-states of different energies, 
2. which generates a time variable charge/electrical dipole, 
3. which generates an electro-magnetic field, 
4. which generates outgoing radiation,
5. under exterior forcing.

The analysis in Mathematical Physics of Black Body Radiation shows that in this system 

• $P \approx R\approx \int (f^2 + g^2)dxdt$, that is,
• outgoing radiation $\approx$ radiative damping $\approx$ exterior forcing.  

The fact that outgoing radiation $\approx$ exterior forcing makes it possible to reverse the physics (1) from an atom generating outgoing radiation as an electromagnetic field (emission) into (2) a model of the reaction of an atom subject to an incoming electro-magnetic field (absorption). This is the same reversal that can be made to use a loadspeaker as a microphone (or that an antenna reradiates about half what it absorbs allowing Swedish Television agents to detect individual watchers and check if the TV-license has been paid).

Note that the physics of (1) may be easier to explain/understand than (2), since outgoing radiation/emission can be observed, while atomic absorption of incoming electro-magnetic waves is hidden to inspection.  On the other hand if (2) is just the other side of (1), then explaining/understanding (1) may be sufficient.

The analysis thus offers an explanation of self-interaction without a catastrophy of acoustic feedback between loadspeaker and microphone, which may be at the origin of the infinities troubling Feynman's jewel of physics QED with photons being emitted and possibly directly being reabsorbed in a form of catastrophical photonic feedback.

PS1 The radiation damping $-\gamma\dddot\psi$ may alternatively take the form
$\gamma \vert\dot\rho\vert^2\dot\psi$, with again $R\sim \omega^4$ for a superposition of eigen-states, and $R=0$ for a pure eigen-state with $\dot\rho =0$. Compare PS5 below.

PS2 The basic conservation laws built into (1)-(2) with $f=g=0$ are (with PS1)

• $\frac{d}{dt}\int\rho (x,t)dx =0$   (conservation of charge), 
• $\frac{d}{dt}\int (\psi H\psi +\phi H\phi)dx = -\int(\gamma\vert\dot\rho\vert^2(\dot\psi^2+\dot\phi^2)dx$  (radiative damping of energy).

PS3 Feynman states in the above book: 

• It is very important to know that light behaves like particles, especially for those of you who have gone to school, where you were probably told something about light behaving like waves. I am telling you the way does behave - like particles. ...every instrument (photomultiplier) that has been designed to be sensitive enough to detect weak light has always ended up discovering the same thing: light is made of particles.

We read that Feynman concludes that because the output of a light detector/photo-multiplier under decreasingly weak light input, changes from a continuous signal to an intermittent signal to no signal, light must also be intermittent as if composed of a stream of isolated particles.  But this is a weak argument because it draws a general conclusion about the normal nature of light from an extreme situation where blips on a screen or sound clicks are taken as evidence that what causes the blips also must be blip-like, that is must be particles. But to draw conclusions about normality by only observing extremity or non-normality, is to stretch normal scientific methodology beyond reason. In particular, the infinities troubling QED seems to originate from particle self-interaction. With light and atom instead in the form of waves and their interaction consisting of interference of waves, self-interaction does not seem to be an issue.


PS4 The book Atoms and Light Interactions presents what its author by J. D. Dodd refers to as a semi-classical view of the interaction of electromagnetic radiation and atoms, thus as waves and not particles (which is also my view):

• It may well be that the semiclassical view falls down at some stage and is unable to predict correctly certain phenomena; my own view is that it succeeds much more widely than it is given credit for. Even if it is not justified from the point of view of many physicists, i is still useful for another reason. Even if the quantum nature of radiation (QED) is required, the underlying physics needs a firm understanding of its classical basis.  

Yes, it may well by that also atomistic physics is a form of wave mechanics and thus a form of classical continuum physics, as expressed by Zeh:

• There are no quantum jumps and nor are there any particles.

PS5 The analysis of Mathematical Physics of Black Body Radiation is more readily applicable if (1)-(2) is formulated as a second order in time wave equation of the form

• $\ddot\psi +H^2\psi + \gamma\dot\rho^2\dot\psi = F$,

with the following tentative main result as an extension of the analysis from radiative damping $-\gamma\dddot\psi$ to $\gamma\dot\rho^2\dot\psi$ (with $\gamma >0$ constant):

• $\int\gamma\dot\rho^2\dot\psi^2dxdt\approx\int F^2dxdt$.

Here $\gamma$ may have a dependence on $\psi$ to guarantee charge conservation under forcing.

Löfven-Socialdemokrati-Korporativism-Fascism-Nyfascism

Om korporativsim kan man läsa följande:

Som politisk åskådning är korporativismen nära besläktad med konservatismens organiska tendenser och förespråkar ett teknokratiskt styrelseskick under elitens ledning, vilket anses gynna de olika samhällsgrupperna mer än en demokrati av egalitär modell. Historiskt har korporativistiskt styre förespråkats av fascistiska rörelser genom deras ideologiska motstånd mot både individualism och egalitarism och istället betonas olikhet, symbios och samförstånd genom ömsesidigt beroende.

Tendenser till modifierade former av korporativism har även uppträtt i många moderna demokratiska system. Efter andra världskriget har länder som Sverige och Österrike, under långvariga socialdemokratiska maktinnehav, utvecklat korporativistiska drag med samförstånd mellan regering, fack och näringsliv (jfr. saltsjöbadsandan).

I Sverige kommer detta också till uttryck genom myndigheters och organisationers inflytande i den centrala politiska beslutsprocessen, via det statliga remissinstitutet, reglerad i regeringsformen.

När Stefan Löfven talar om nyfascism i det svenska samhället, är det denna koppling som genom en Freudiansk felsägning gör sig påmind?

PS Enligt Regeringsformen kap 3 paragraf  11 gäller:

Efter val till riksdagen får regeringen inte besluta om extra val förrän tre månader har gått från den nyvalda riksdagens första sammanträde. Regeringen får inte heller besluta om extra val under den tid då dess ledamöter, efter det att samtliga har entledigats, uppehåller sina befattningar till dess en ny regering ska tillträda.

Frågan är nu om Löfven och därmed regeringen har beslutat att extra val skall ske? I så fall vore det mot grundlagen. Å andra sidan, om regeringen inte beslutat om extra val, hur kommer det sig då att alla verkar utgå från att så har skett? Är det verkligen korrekt enligt grundlagen att regeringen nu efter bara 2 månader beslutar att beslut om nyval skall fattas när väl det föreskrivna 3 månaders förbudet mot extra val löpt ut. Kan alltså beslut fattas om att beslut om extra val skall fattas, redan dagen efter ett val? Jfr Statsministern tolkar lagen fel.

The Radiating Atom 6: Schrödinger's Equation in Real-Valued System Form

Schrödinger's equation, to start with for the electron of the Hydrogen atom, is usually written in the form

• $ih\dot\Psi = H\Psi$,

with $\Psi (x,t)$ a complex-valued function of a space-time $(x,t)$,  $\dot\Psi =\frac{\partial\psi}{\partial t}$, $H=-\frac{h^2}{2m}\Delta + V(x)$ the Hamiltonian with $\Delta$ the Laplacian with respect to $x$, $V(x)=-\frac{1}{\vert x\vert}$ the kernel potential, $m$ the electron mass and $h$ Planck's constant.  This equation can equivalently be expressed as follows in real-valued system form, with $\Psi =\phi + i\psi$ and $\phi =\phi (x,t)$ and $\psi =\psi (x,t)$ real-valued functions: 

• $\dot\psi + H\phi =0$, 
• $-\dot\phi + H\psi= 0$. 

This system can be viewed as a generalized harmonic oscillator or wave equation, which can naturally be extended to

• $\dot\psi + H\phi -\gamma\dddot\phi = f$       (1)
• $-\dot\phi + H\psi -\gamma\dddot\psi = g$     (2)

where $f(x,t)$ and $g(x,t)$ represent external electro-magnetic forcing, and $\gamma\dddot \psi$ and 

$\gamma\dddot \phi$ represents the Abraham-Lorentz recoil force from emission of radiation with $\gamma$ having a dependence on $\Phi \equiv (\psi ,\phi )$ to be specified. A system of this form as a wave equation with small damping subject to near-resonant forcing is analyzed in Mathematical Physics of Black Body Radiation.

The basic energy balance is obtained by multiplying (1) by $\dot\phi$ and (2) by $\dot\psi$, then adding and integrating in space and time, to get for $f=g=0$:

• $E(\Phi ,T)+R(\Phi ,T)= 0$ for $T>0$,
• $E(\Phi ,T)=\int (\psi (x,T)H\psi (x,T)+\phi (x,T)H\phi (x,T ))dx$
• $R(\Phi ,T)=\int_0^T\int(\gamma\ddot\psi^2(x,t)+\gamma\ddot\phi^2(x,t))dxdt$,

which expresses a balance between internal atomic energy $E(\Phi ,T)$ at time $T$ as the sum of "kinetic energy" related to the Laplacian $\Delta$ and potential energy related to V as terms in the Hamiltonian $H$, and total radiated energy until time $T$ in accordance with Larmor's formula stating that radiation scales with $\ddot q^2$, where $\ddot q=\ddot q(t)$ is the "acceleration" of a charge $q(t)$ varying in space over  time. 

Let now $\psi_1=\psi_1(x)$ and $\psi_2=\psi_2(x)$ be two eigenfunction of the Hamiltonian $H$ with corresponding eigenvalues $E_1 < E_2$ and pure eigen-states

• $\Phi_j(x,t)\equiv (\cos(E_jt/h)\psi_j(x),\sin(E_jt/h)\psi_j(x))$ for $j=1,2$, 

and corresponding charge densities

• $q_j(t)\equiv\vert \Phi_j(x,t)\vert^2\equiv(\cos^2(Et/h)+\sin^2(Et/h))\psi_j^2(x)=\psi_j^2(x)$. 

We thus find that pure eigen-states have charge densities which are constant in time and thus do not radiate.

On the other hand, the charge density $q(x,t)=\vert\Phi (x,t)\vert^2$ of a superposition $\Phi =c_1\Phi_1+c_2\Phi_2$ with $c_1$ and $c_2$ positive coefficients of the two pure eigenstates $\Phi_1$ and $\Phi_2$,  has a time dependence of the form

• $q(x,t) = a(x) + b(x)\cos((E_2-E_1)t/h)$  

with $a$ and $b$ coeffcients depending on $x$, and thus is radiating. We are thus led to a dependence of $\gamma$ on $\Phi$  of the form

• $\gamma \sim\ddot q^2$.

We conclude that (1)-(2) offers a continuum mechanical model of a radiating Hydrogen atom which can be analyzed by eigenfunction expansion as in Mathematical Physics of Black Body Radiation and thus offers an answer to the basic questions of atomic mechanics:

• Why does a pure-eigen-state not radiate and thus can persist over time as a stable atomic state?
• Why can an atom radiate under external forcing? 
• How much is an atom radiating under external forcing? 

Note that the system (1)-(2) in case with $f=g=\gamma =0$ has the equivalent form of a second order wave equation:

• $\ddot\psi + H^2\psi =0$,

a form which Schrödinger dismissed on the ground that a time dependent potential would cause complications, and probably also because the presence of the term $\ddot\psi$ appears to be asking for a physical interpretation of $\dot\psi^2$ as kinetic energy, which however was already assigned to $\vert\nabla\psi\vert^2$ connected to the Laplacian. 

On the other hand, in the real-valued system form (1)-(2), these complications do no arise, and the extension to forcing and radiation is more natural than in the standard complex form, which is commonly viewed as a complete mystery beyond human comprehension.

What remains to understand is the physical meaning of the system equations (1)-(2), which may well be possible after some imagination, which I hope to report on.  

In short (1)-(2) may be the form of Schrödinger's equation to use for extensions to multi-electron configurations. At least this is the route I am now seeking to explore.

Note that letting $h$ tend to zero, we obtain the dynamical second order system

• $\ddot\psi (t) = -V^2\psi = -\frac{\psi}{\vert x\vert^2}$

which can be interpreted as Newton's equations for a moving "particle" localized in space. Schrödinger's equation (1)-(2) can thus be viewed as regularized form of Newton's equations with regularization from the Laplacian. In this perspective there is nothing holy about the Laplacian; it is thinkable that the effective regularization in an atom is non-isotropic,  thus with different action in radial and angular variables in spherical coordinates centered at the kernel.  

An equation $\dot\psi +H\phi=\dot\psi + V(x)\phi=0$ with $h=0$ may formally be viewed as some form of force balance expressing a form of "square root of Newton's 2nd law" $\ddot\psi+V^2\psi$.

Note that in (1)-(2) $-H\phi$ connects to $\dot\psi$ and $H\psi$ to $\dot\phi$ and so the dynamics of a pure eigen-state with wave function $\Phi_j$ can be described as a "revolution/oscillation in time" of a space-dependent eigen-function of the Hamiltonian for which the charge density is constant in time without radiation,  while the charge density of a superposition of pure eigen-states varies in time and thus radiates.  With this perspective, an electron is not "moving in space" like some form of planet around the kernel, but instead has a variation in time, which gives rise to a charge density with variation in time and thus radiation, except for a pure eigen-state which does not radiate.  

   

Stefan Löfven Trotsar Parlamentarismens Princip

Efter att ha som statsminister fått sin budget nedröstad i Riksdagen bestämmer Stefan Löfven utan att höra Partiet att nyval skall utlysas. Stefan Löfven gör detta för att förhindra att talmannen undersöker om Alliansen är villig att ta över och genomföra den politik som Alliansens vinnande budgetproposition anger, vilket vore det riktiga enligt den demokratiska parlamentarismens grundläggande princip att det är Riksdagen som bestämmer.

Stefan Löfven bryter därmed denna princip, som innebär att nyval bara kan utlysas om inte en handlingsduglig regering kan formas. Ett nederlag i Riksdagen, som är lika med en misstroendeförklaring, kan inte vara tillräckligt skäl för att tillåta utlysning av nyval, eftersom om så vore fallet parlamentarismen skulle kunna  urarta till en spiral av nyval på nyval mot fullständigt kaos: Löfven skulle ju kunna fortsätta på den inslagna vägen och utlysa ännu ett nyval om han skulle förlora valet i mars 2015. Om inte Löfven avgår nu varför skulle han göra det efter ännu en förlust?

Att Stefan Löfven bryter mot parlamentarismens grundläggande princip måste bero att han med sin bakgrund som fackpamp inte förstår betydelsen av densamma. Att socialdemokratiska partiet är medlöpare i denna process visar hur långt upplösningen av detta en gång så principfasta parti nu gått.

Vi lever idag i ett Sverige där både vetenskapens och demokratins principer bryts av en regering med främsta mål att stoppa användningen av fossil energi och behålla maken oavsett vad Riksdagen bestämmer.

Hur har det kunnat bli så här tokigt? Var finns akademi och media? Alliansen?

PS1 Så sent som dagen innan Löfven röstades ned försäkrade han svenska folket att han inte skulle sitta kvar och administrera Alliansens budget, och gjorde sedan tvärtom. Tidigare minister Eskil Erlandsson anser att Löfven därmed visade sig vara ohederlig. EU-parlamentariker Gunnar Hökmark säger samma sak. Förutsättningen för att Löfven skulle utses till statsminister var att han kunde förväntas få igenom sin budget. När han nu misslyckats med detta kan han inte sitta kvar.

PS2 Tove Lifvendahl anser att Alliansen bör fälla Löven via misstroendevotum. En sådan torde inom kort krävas av SD, efter Löfvens angrepp. SD har 49 röster och det räcker med 35 för ställa krav på förtroendeomröstning.

  

The Radiating Atom 5: Summary

A summary of the experience gathered in the recent posts on radiating atoms is as follows:

1. Schrödinger's equation in standard multi-dimensional form is uncomputable and unphysical. 

Schrödinger's wave equation in multi-dimensional linear form commonly viewed as the basis of quantum mechanics, is uncomputable and hence unphysical. To insist that atom physics is well described by a model which is uncomputable lacks scientific rational, since a model without output cannot be compared with observation. Instead a computable model as a nonlinear system of one-electron wave equations in the spirit of Hartree, should be sought.

2. Schrödinger's equation for a non-radiating atom has a fictional time-dependence.

Schrödinger's equation in standard time-dependent form

• $ih\frac{\partial\psi}{\partial t} + H\psi =0$

with $H$ a Hamiltonian and $t$ time, supposedly describes the dynamics of an atom which is not interacting with any exterior electromagnetic field, that is, is not absorbing or emitting radiation. But such an atom cannot be observed and thus the model cannot be compared to reality. This is reflected by the fact that the charge density $\vert\psi\vert^2$ of the ground state or an excited state as a pure eigen-state of the form

• $\psi (x,t)=\exp(iE/h)\Psi(x)$ 

with $\Psi =\Psi (x)$ an eigenfunction of the Hamiltonian  $H\Psi =E\Psi$ with corresponding real eigenvalue $E$, is not changing with time. Thus the time-dependence in Schrödinger's standard form is fictional in the sense that it cannot be observed. What can be observed is the difference between eigenvalues, as shown in the next section.

3. A radiating atom can be modeled as a forced resonator with small damping.

The standard Schrödinger equation in above complex form can alternatively be formulated in real form as a second order wave equation for a resonator build from $H^2$:

• $\frac{\partial^2\phi}{\partial t^2}+H^2\phi =0$,

which can naturally be extended to include exterior forcing and radiative damping, as shown in Computational Physics of Black Body Radiation. In this setting the frequency $\nu$ of observable absorption/emission of radiation resulting from interference between two pure eigen-states with eigenvalues $E_2>E_1$, satisfies  $h\nu =E_2 - E_1$, while the forcing may have different frequency matching the resonance frequencies $E_2/h$ and $E_1/h$ and not (necessarily) $\nu =E_2/h -E_1/h$.

As above the eigen-states are determined from eigenfunctions $\Psi$ of the Hamiltonian $H$ as stationary values of the energy as the sum of kinetic and potential energies under normalization of $\Psi$. The damping term to be added to the second order wave equation can take the form $\gamma\dot\phi$ with $\gamma >0$ a damping coefficient and corresponding dissipation rate $\gamma\dot\phi^2$ balancing outgoing radiation.

The extended wave equation for a radiating atom may thus take the form

• $\frac{\partial^2\phi}{\partial t^2}+H^2\phi +\gamma\dot\phi =f$,

expressing a balance between forcing $f=f(x,t)$ and the sum of an out-of-balance atomic resonator reaction $\frac{\partial^2\phi}{\partial t^2}+H^2\phi$ and dissipation reaction $\gamma\dot\phi$.  What can here be observed is the radiation generated by a time dependent charge density $\phi^2 (t)$, and not the internal dynamics described by the wave equation, which remains hidden to inspection.

4. Conclusion 

Schrödinger's equation in standard multi-dimensional complex form is not a useful model as a basis of atom physics, because 

• The model is ad hoc and is not derived from basic physics principles.
• Multi-dimensionality makes the model uncomputable. 
• Multi-dimensionality defies physical interpretation of wave functions as solutions.
• The complex form is mystical and lacks physics rationale. 
• Introducing kinetic energy by connecting momentum to $ih\frac{\partial}{\partial x}$ represents a deep formal mysticism.     

5. Towards a more useful wave equation.

It may well be possible to construct a more useful more physical less mysterious model as a system of one-electron second order wave equations expressing a balance of attractive/repulsive Coulomb forces, Abraham-Lorentz radiation forces and forces from regularization of wave solutions.  The first step in such a process is to bring the deficiencies of Schrödinger's standard equation from obscurity and mysticism into scientific light.

Here is a reference into such work: Damping Effect of Electromagnetic Radiation and Time-Dependent Schrödinger Equation by Ji Luo.

6. Reflections on the second-order Schrödinger equation

The second order wave equation $\frac{\partial^2\phi}{\partial t^2}+H^2\phi =0$ was formulated in the 4th of Schrödinger's 1926 articles, but was then dismissed on the ground that a time dependent potential from exterior forcing would give a complicated equation. However, it may well be possible to introduce forcing instead as a time-dependent right hand side $f(x,t)$ in a non-homogeneous wave equation

• $\frac{\partial^2\phi}{\partial t^2}+H^2\phi =f$ 

including the classical ingredients of acceleration $\frac{\partial^2\phi}{\partial t^2}$ connected to kinetic energy $(\frac{\partial\phi}{\partial t})^2$, and with $H=\Delta + V$ connected to a form of "elastic" energy $\vert\nabla\phi\vert^2$ (and thus not kinetic energy) and potential energy $V\phi^2$. This model would bring quantum mechanics into a setting of classical continuum mechanics, which could remove the mysteries of standard quantum mechanics as something fundamentally different from classical continuum mechanics.

Feynman's statement that nobody understands (standard) quantum mechanics, should not be viewed as a joke but as serious criticism: A theory which cannot be understood by any human being is not a scientific theory.  

Prediction of Global Temperature May Well Be Possible

The recent "hiatus" of global warming, with slightly falling global temperature over now two decades under rising CO2 levels, in total contradiction to steadily rising temperature predicted by all of the complex climate models underlying the CO2 alarmism propagated by IPCC, has given support to a populistic view that "mathematical modeling of climate is impossible because the evolution of climate is chaotic". Both skeptics and alarmists have shown enthusiasm for such a scientific defaitism.

But it is not at all necessary to draw this conclusion, since chaos can sometimes be very predictable, for example as a null result of small stochastic perturbations.

For example, a simple climate model stating a balance between incoming radiation from the Sun, which is observed to be nearly constant, and outgoing radiation from the Earth system, which is observed to be nearly constant, can give the prediction that global temperature will stay nearly constant over forseeable time, say a couple of hundred years.

Such a model would be in excellent agreement with observations over the last two decades, and would also be within measurement accuracy since the start of recorded observations (with maybe half a degree Celsius nominal increase).

Climate as long-time-average of weather may thus be predictable, by the same mathematical reasons that mean-value aspects of turbulent flow like total drag and lift of an airplane are predictable (as shown in Computational Turbulent Incompressible Flow).

What may be impossible is a precise prediction of a very small effect of a small perturbation of atmospheric radiation from a change of concentration of a trace gas as CO2. But a precise prediction of something so small that it has no observable effect, is of course meaningless and thus the perceived impossibility is not real.

It is only if you like IPCC want to send an alarm of an effect of a vanishingly small cause, that you need a precise climate model supporting your case. The fact that such model is basically unthinkable is then something to hide, together with the fact that a prediction of no-change is certainly thinkable and may well be correct.

The Radiating Atom 4: Absorption vs Emission

To seek the relation between atomic absorption and emission of radiation, let us consider a near-resonantly forced harmonic oscillator with small damping as the basic model underlying the analysis presented at Computational Black Body Radiation:

• $\ddot u(t)+\nu^2u(t)+\gamma\dot u(t) = f(t)$, 

which we in mechanical terms, with the dot representing differentiation with respect to time, expresses force balance between a mass-spring oscillator with internal inertial force $\ddot u(t)$ and spring force $\nu^2u(t)$ with $u(t)$ displacement and resonance frequency $\nu$, coupled in parallel with a friction force $\gamma\dot u(t)$, which are balancing an exterior force $f(t)$. Here $\gamma >0$ is a small damping coefficient and we consider the two basic cases of 

1. Non-resonant forcing with frequency of $f(t)$ not near $\nu$: 

• $\gamma\dot u(t) \approx f(t)$ and $\ddot u(t)+\nu^2u(t)\approx 0$.  

2. Near-resonant forcing (see Computational Black Body Radiation) with frequency of $f(t)$ near $\nu$:

• $\gamma\dot u(t) \approx 0$ and $\ddot u(t)+\nu^2u(t)\approx f(t)$.
• More precisely: $\gamma \vert\dot u\vert\approx\sqrt{\gamma}\vert f\vert <<\vert f\vert$.

In case 1. the exterior force is balanced by the friction force and in case 2. by an out-of-balance harmonic oscillator. If we view $r(t) = f(t) - \gamma\dot u(t)$ as an observable net residual force, we

then have that 

1. Non-resonant forcing gives $r(t)\approx 0$: Nothing can be observed.
2. Resonant forcing $r(t) \approx f(t)$: Something can be observed.   

This gives substance to the experience that absorption and emission, as in absorption/emission spectroscopy, are related: 

• A system which can absorb radiation can also emit radiation of the same frequency, and vice versa.
• A non-resonant system does not absorb anything and nothing observable is emitted.
• Resonant absorption can be observed by some form of emission.  This does not require emission to be equal to absorption, but they come together.
• In absorption spectroscopy a cold gas is absorbing incoming radiation, which is observable as a dip in the spectrum observed after passage through the gas resulting from heating the gas.
• In emission spectroscopy of a hot gas, emission is observable but not absorption. 
• The resonance frequency connects to the difference in energy level between two electronic states since emission results from charge oscillation (connected to the Abraham-Lorentz force) of such frequencies. Hence also absorption of such these frequencies can be observable as a result of force oscillation.  

Note that both absorption and emission is a resonance phenomenon driven by forces and as such is a wave phenomenon in the spirit of Schrödinger and not a "corpuscular phenomenon", whatever that may be, as is the current wisdom rooted in Einstein's "explanation" of the photoelectric effect based on "light particles" or "photons" of energy $h\nu$ jumping stochastically back and forth seemingly without being subject to forces.

But physics is all about forces and physics without forces is non-physics.   

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/dissipation

• $\ddot\phi+H^2\phi -\gamma\dddot\phi=f$,

thus should apply, with $\gamma (\phi )$ a small (non-negative) 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 dissipated (and then radiated) energy balances the input forcing energy 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 $\nu =\nu_2 -\nu_1$. The beat frequency can be sustained by resonant forcing of frequency $\nu_2$ and the radiated energy scales with (is nearly equal to) the input energy.
4. If $\gamma (\phi )$ scales with (the modulus of) $\frac{d}{dt}\vert\phi (t)\vert^2$), then $\gamma =0$ for pure eigen-states and $\gamma  >0$ for non-trivial combinations of pure eigen-states, in correspondence with observations.  
5. Notice that the output (beat) frequency $\nu_2 - \nu_1$ is here different from the input frequency $\nu_2$. 
6. It is natural to ask if the input frequency can alternatively be the beat frequency, as in absorption spectroscopy.  In this case also heating of a cold gas is involved, which connects to the finite precision cut-off as an important feature of the analysis in Mathematical Physics of Blackbody Radiation. 

This resolution of the enigma of the atom is, I think, in the spirit of Schrödinger (and would maybe have made him as happy as on the picture if he only had been around), a spirit which unfortunately was crushed by Bohr who managed to make physicists abandon Schrödinger's understandable wave mechanics for 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...

PS1 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.

PS2 What is observable is thus the difference between energies of pure eigen-states as beat frequencies, but not energies or frequencies for such states. This is not in accordance with a basic postulate of quantum mechanics in conventional form asking eigenvalues of Hamiltonians to be observable. 

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.

   

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. 

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.

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.

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.

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.

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.

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!