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.