The Radiating Atom

In the analysis on Computational Blackbody Radiation I used the following model of a harmonic oscillator of frequency $\omega$ with small damping $\gamma >0$ subject to near resonant forcing $f(t)$:

• $\ddot u+\omega^2u-\gamma\dddot u=f(t)$

with the following characteristic energy balance between outgoing and incoming energy:

• $\gamma\int\ddot u^2dt =\int f^2dt$

with integration over a time period and the dot signifying differentiation with respect to time $t$. 

An extension to Schrödingers equation written as a system of real-valued wave functions $\phi$ and $\psi$ may take the form

• $\dot\phi +H\psi -\gamma\dddot \psi = f(t)$            (1)
• $-\dot\psi +H\phi -\gamma\dddot \phi = g(t)$          (2)

where $H$ is a Hamiltonian, $f(t)$ and $g(t)$ represent near-resonant forcing, and $\gamma =\gamma (\dot \rho )\ge 0$ with $\gamma (0)=0$ and $\rho =\phi^2 +\psi^2$ is charge density.

This model carries the characteristics displayed of the model $\ddot\phi+H^2\phi =0$ as the 2nd order in time model obtained after eliminating $\psi$ in the case $\gamma =0$ as displayed in a previous post. 

In particular, multiplication of (1) by $\phi$ and (2) by $-\psi$ and addition gives conservation of charge if $f(t)\phi -g(t)\psi =0$ as a natural phase shift condition. 

Further, multiplication of (1) by $\dot\psi$ and (2) by $\dot\phi$ and addition gives a balance of total energy as inner energy plus radiated energy 

• $\int (\phi H\phi +\psi H\psi)dt +\gamma\int (\ddot\phi^2 +\ddot\psi^2)dt$

in terms of work of forcing.