The Radiating Atom 8: Towards Resolution of the Riddle

Let us now collect the experience from previous posts in this series: We start recalling Schrödinger's equation for the one electron Hydrogen atom in standard form:

• $ih\dot\Psi +H\Psi =0$,            (1)  

where $\Psi =\psi +i\phi$ is a complex-valued function of space-time $(x,t)$ with real part $\psi$ and imaginary part $\phi$ as real-valued functions, and $H$ is the Hamiltonian defined by 

• $H =-\frac{h^2}{2m}\Delta + V$

where $\Delta$ is the Laplacian with respect to $x$, $V(x)=-\frac{1}{\vert x\vert}$ is the kernel potential, $m$ the electron mass, $h$ Planck's constant, and the dot signifying differentiation with respect to time $t$. The wave function $\Psi$ is normalized with

• $\int\rho (x,t)dx =1$ for all $t$
• $\rho =\vert\Psi\vert^2 =\psi^2 +\phi^2$,

where $\rho (x,t)$ is a measure of the charge intensity with total charge equal to one.  

Schrödinger's equation takes the following real-valued system form:

• $\dot\psi + H\phi =0$
• $\dot\phi -H\psi =0$,   

which upon differentiation with respect to time and recombination gives the following same second-order equation for both $\psi$ and $\phi$:

• $\ddot\psi + H^2\psi =0$, 
• $\ddot\phi + H^2\phi =0$, 

or the same equation in complex form with $\Psi =\psi +i\phi$ as a second-order Schrödinger equation:

• $\ddot\Psi + H^2\Psi =0$.       (2)

Let now $\psi_1(x)$ be the wave function of the ground state as an eigenfunction of $H$ with corresponding minimal eigenvalue $E_1$ satisfying $H\psi_1=E_1\psi_1$, that is $H_1\psi_1=0$ with $H_1=H-E_1$.

Let us then consider the following generalization of (2) into model of a radiating Hydrogen atom subject to external forcing:

• $\ddot\Psi +H_1^2\Psi -\gamma\dddot\Psi =f$,      (3)

where $-\gamma\dddot\Psi$ represents radiative damping with  $\gamma =\gamma (\Psi )$ a small non-negative radiation coefficient and corresponding radiation energy

• $R(\Psi ,t)=\int\gamma\vert\ddot\Psi (x,t)\vert^2dx$.

We see that $\Psi_1=\psi_1$ solves (3) with $f=0$. More generally, if $\psi_j$ is an eigen-function of the Hamiltonian with eigenvalue $E_j\gt E_1$, then $\Psi_j=\exp(i(E_j-E_1)t/h)\psi_j$ solves (3) with $\gamma =0$ and $f=0$ and represents a pure eigenstate of frequency in time $\nu =(E_j-E_1)/h$.

More generally, a superposition $\Psi =c_1\Psi_1+c_j\Psi_j$ of the ground state $\Psi_1$ and an excited eigen state $\Psi_j$ of frequency $\nu =(E_j-E_1)/h$ with non-zero coefficients $c_1$ and $c_j$ generates a charge

• $\rho (x,t)=\vert\Psi\vert^2=c_1^2\psi_1(x)^2+c_2^2\psi_j(x)^2+2\cos(\nu t)c_1c_j\psi_1(x)\psi_j(x)$,  

which varies in time, and thus may generate radiation.

In the spirit of Computational Physics of Black Body Radiation we are thus led to an analysis of (3) with a forcing $f$ in near-resonance and small radiative damping with eigenfrequencies $(E_j-E_1)/h$, or more generally $(E_j-E_k)/h$ with $E_j\gt E_k$, which as main result  proves the basic energy balance equation

• $\int R(\Psi ,t)dxdt \approx \int f^2(x,t)dxdt$, 

expressing that in stationary state output = input.

The following questions present themselves:

1. Which model, first order (1) or second-order (2), extends most naturally to radiation under forcing?
2. Is (3) to be viewed as a force balance with $-\gamma\dddot\psi$ as a Abraham-Lorentz radiation recoil force?
3. Which condition on $f$ guarantees that a pure eigenstate $\Psi_j$ is neither absorbing nor emitting, thus with $\gamma (\Psi_j)=0$? 

Remark 1. Note that the time dependence of an eigenstate $\Psi_j$ in superposition with an eigenstate $\Psi_k$ has frequency $(E_j-E_k)/h\gt 0$. The customary association of $\Psi_j$ 

to a frequency $E_j/h$, which can have either sign, is not needed and nor natural from physical point of view. The energy $E_j$ of an eigen-state has a physical meaning, but not $E_j/h$ as a frequency. This is a main of point of confusion in standard presentations of quantum mechanics supposedly being based on Einstein's relation $E=h\nu$ with $E$ energy and $\nu$ frequency.

Remark 2. Normalisation of wave functions under forcing and radiative damping, can be maintained by adjustment of the coefficient $\gamma (\Psi )$.

Remark 3. The energy balance in the form output = input or input = output, determines radiative equilibrium of an assembly of atoms, just as the corresponding relation in black body radiation expressed as Universality.

Remark 4. Schrödinger in the 4th and last of his 1926 articles first came up with (2) as an atomic wave equation, and then settled on (1) with the argument that a time-dependent Hamiltonian would cause problems in a transition from (1) to (2). The question is if Schrödinger gave up on (2) too easily? Maybe (2) is a better physical model than (1)?

Remark 5. Notice that (3) with an Ansatz of the form $\Psi (x,t)=c_1\Psi_1(x)+c_2\Phi (x,t)$ translates (3) into the wave equation in $\Phi$:

• $\ddot\Phi +H_1^2\Phi -\gamma\dddot\Phi =f$,

which is open to the analysis of Computational Physics of Black Body Radiation. What remains is to identify the forcing $f(x,t)$ resulting from an incoming electric or magnetic field. The basic case concerns the interaction between a $(2,1,0)$ p-state $\Phi_2(x)$ of eigenvalue $E_2$ with axis parallel to a plane-wave electrical field $E=(E_1,0,0)$ with $f = E_1$ in near-resonance with $\nu =(E_1-E_2)/h$.