fredag 19 december 2014

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


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

onsdag 17 december 2014

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 P-state $\Phi_2(x)$ of eigenvalue $E_2$ oriented in parallel with an electrical field $E=(E_1,0,0)$ with $f = E_1$.


tisdag 16 december 2014

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. 

onsdag 10 december 2014

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.

måndag 8 december 2014

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.  
   


torsdag 4 december 2014

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.

  

onsdag 3 december 2014

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.  

tisdag 2 december 2014

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.