torsdag 26 januari 2017

Why Atomic Emission at Beat Frequencies Only?

An atom can emit radiation of frequency $\nu =E_2-E_1$ (with Planck's constant $h$ normalized to unity and allowing to replace energy by frequency) and $E_2>E_1$ are two frequencies as eigenvalues $E$ of a Hamiltonian $H$ with corresponding eigenfunction $\psi (x)$ depending on a space coordinate $x$ satisfying $H\psi =E\psi$ and corresponding wave function $\Psi (x,t)=\exp(iEt)\psi (x)$ satisfying Schrödingers wave equation
• $i\frac{\partial\Psi}{\partial t}+H\Psi =0$
and $t$ is a time variable.

Why is the emission spectrum generated by differences $E_2-E_1$ of frequencies of the Hamiltonian as "beat frequencies" and not the frequencies $E_2$ and $E_1$ themselves? Why does an atom interact/resonate with an electromagnetic field of beat frequency $E_2-E_1$, but not $E_2$ or $E_1$?

In particular, why is the ground state of smallest frequency stable by refusing electromagnetic resonance?

This was the question confronting Bohr 1913 when trying to build a model of the atom in terms of classical mechanics terms. Bohr's answer was that "for some reason" only certain "electron orbits" with certain frequencies "are allowed" and that "for some reason" these electron orbits cannot resonate with an electromagnetic field, and then suggested that observed resonances at beat frequencies came from "electrons jumping between energy levels".  This was not convincing and prepared the revolution into quantum mechanics in 1926.

Real Quantum Mechanics realQM gives the following answer: The charge density $\vert\Psi (t,x)\vert^2=\psi^2(x)$ of a wave function $\Psi (x,t)=\exp(iEt)\psi (x)$ with $\psi (x)$ satisfying $H\psi =E\psi$, does not vary with time and as such does not radiate.

On the other hand the difference $\Psi =\Psi_2-\Psi_1$ between two wave functions $\Psi_1(x,t)=\exp(iE_1t)\psi_1(x)$ and $\Psi_2(x,t)=\exp(iE_2t)\psi_2(x)$ with $H\psi_1=E_1$ and
$H\psi_2=E_2\psi_2$, is a solution to Schrödinger's equation and can be written
• $\Psi (x,t)=\exp(iE_1t)(\exp(i(E_2-E_1)t)\psi_2(x)-\psi_1(x))$
with corresponding charge density
• $\vert\Psi (t,x)\vert^2 = \vert\exp(i(E_2-E_1)t)\psi_2(x)-\psi_1(x)\vert^2$
with a visible time variation in space scaling with $(E_2-E_1)$ and associated radiation of frequency $E_2-E_1$ as a beat frequency.

Superposition of two eigenstates thus may radiate because the corresponding charge density varies in space with time, while pure eigenstates have charge densities which do not vary with time and thus do not radiate.

In realQM electrons are thought of as "clouds of charge" of density $\vert\Psi\vert^2$ with physical presence, which is not changing with time in pure eigenstates and thus does not radiate, while superpositions of eigenstates do vary with time and thus may radiate, because a charge oscillating at a certain frequency generates a electric field oscillating at the same frequency.

In standard quantum mechanics stdQM $\vert\Psi\vert^2$ is instead interpreted as probability of configuration of electrons as particles, which lacks physical meaning and as such does not appear to  allow an explanation of the non-radiation/resonance of pure eigenstates and radiation/resonance at beat frequencies. In stdQM electrons are nowhere and everywhere at the same time, and it is declared that speaking of electron (or charge) motion is nonsensical and then atom radiation remains as inexplicable as to Bohr in 1913.

So the revolution of classical mechanics into quantum mechanics driven by Bohr's question and unsuccessful answer, does not seem to present any real answer. Or does it?

PS I have already written about The Radiating Atom in a sequence of posts 1-11 with in particular 3: Resolution of Schrödinger's Enigma connecting to this post.