tisdag 25 november 2014

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
  • $\Psi (x,t) = \vert\Phi (x,t)\vert^2 =  \Phi (x,t)\overline\Phi (x,t)$,
  • $\Phi (x,t) = c_1\psi_1(x,t)+c_2\psi_1(x,t)$
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. 

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