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.