torsdag 27 november 2014

The Radiating Atom 3: Resolution of Schrödinger's Enigma

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

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

Let us then first rewrite Schrödinger's equation (with $H$ the Hamiltonian)
  • $ih\dot{\Psi} + H\Psi =0$, 
where $\Psi = \psi + i\phi$ with $\psi (x,t)$ and $\phi (x,t)$ real-valued functions of space $x$ and time $t$ with the dot representing time differentiation, into the system (with h=1)
  • $\dot\psi +H\phi =0$,
  • $-\dot \phi + H\psi  =0$,     
which has the form of a harmonic oscillator and can be written as a scalar second order in time equation
  • $\ddot\psi+H^2\psi =0$ and/or $\ddot\phi+H^2\phi =0$. 
We see that the quantum mechanical model of an atom has the form of the wave equation studied in Mathematical Physics of Blackbody Radiation.  The analysis therein of the extended equation with near-resonant forcing and small radiative damping/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. 

2 kommentarer:

  1. How could this explain radiation in the form of fluorescense? Seems kind of hard. I think there are other reasons that Schrödinger left the deterministic model than Bohr.

  2. I think I wrote fluorescent earlier, while I of course meant to write phosphorescent. Just a clarification.