The Radiating Atom 9: Hydrogen and Beyond

A plane electrical field $E_z$ acting in the $z$-direction and progressing in the $x$-direction, will interact with the $2p_z$ eigenstate of a Hydrogen atom pictured above corresponding to a charge oscillating in the $z$-direction in parallel with $E_z$.  Note that $E_z$ will not interact with the $2p_x$ and $2p_y$ eigenstates.

As a sum-up of the present series of posts on the radiating atom, we consider Schrödinger's equation for a radiating Hydrogen atom subject to forcing in the form of a second order wave equation

• $\ddot\psi +H^2\psi -\gamma\dddot\psi = f$      (1)

where $\psi (x,t)$ is a real-valued electronic wave function of a space coordinate $x=(x_1,x_2,x_3)$ and time $t$, $H$ 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, the dot signifies differentiation with respect to time $t$, $f$ is external forcing, and $\gamma =\gamma (\psi )$ is a non-negative radiation damping coefficient.

The formulation of Schrödinger's equation as a second order wave equation in terms of a real-valued wave function was considered by Schrödinger in 1926 as an alternative to the standard formulation as a 1st order complex-valued equation. In the homogeneous case with $f=0$ and  $\gamma =0$,  the two formulations are equivalent: In particular, conservation of total charge as

• $\frac{d}{dt}\int\rho (x,t)dx =0$,

with $\rho =\psi^2+(H^{-1}\dot\psi )^2$ the charge intensity, is obtained by multiplying (1) with $H^{-2}\dot\psi$ and integrating in space. In the non-homogeneous case (1) may be more natural as an expression of a force balance with $-\gamma\dddot\psi$ the Abraham-Lorentz radiation recoil force and $f$ an electrical field component, while the physical meaning of the standard formulation baffled the creators of modern physicists and followers and led into unphysical interpretations as particle statistics.  

We consider radiation of frequency $\nu =(E_2-E_1)/h$ where $E_1$ is the energy of the ground state as an eigenfunction $\Psi_1 (x)$ of $H$ with minimal eigenvalue $E_1$ and $E_2$ is a larger eigenvalue with eigenfunction $\Psi_2(x)$. We reformulate (1) in the form

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

where $H_1 = H - E_1$ and note that $H_1\Psi_1=0$ and $H_1\Psi_2=(E_2-E_1)\Psi$. 

We assume that the forcing is given as a linear combination of plane electromagnetic waves $(0,0,\cos(\omega (x_1-ct))$ of frequencies $\omega\approx\nu =(E_2-E_1)/h$ progressing in the $x_1$-direction with the speed of light $c$. We seek a solution $\psi (x,t)$ of (2) of as a linear combination of $\Psi_1$ and $\Psi_2$ of the form

• $\psi (x,t) =c_1(t)\Psi_1(x) + c_2(t)\Psi_2(x)$

with time dependent coefficients $c_1(t)$ and $c_2(t)$. Inserting this Ansatz into (2), multiplying by $\Psi_1$ and $\Psi_2$ and integrating with respect to $x$, we obtain assuming orthonormality of $\Phi_1$ and $\Psi_2$, time-periodicity and normalizing to $c=1$ and $h=1$:

• $\ddot c_1(t) -\gamma\dddot c_1(t) = f_1(t)\equiv\int f(x,t)\Psi_1(x)dx$ for all $t$,
• $\ddot c_2(t) +\nu^2c_2(t)-\gamma\dddot c_2(t) =f_2(t)\equiv\int f(x,t)\Psi_2(x)dx$ for all $t$.

By $x_1$-symmetry of $\Psi_1(x)$ it follows that $f_1(t)=0$ with the effect that $c_1(t)=c_1$ is constant. Further, if $\Psi_2(x)$ is a $(2,1,0)$ p-state oriented in the $x_3$-direction, see above figure, then $f_2(t)$ is a non-zero linear combination of $\cos(\omega t)$, and by the analysis of Mathematical Physics of Black Body Radiation  and Computational Black Body Radiation,

• $\int\gamma\ddot\psi^2(x,t)dxdt = \int\gamma\ddot c_2^2(t)dt\approx \int f_2^2(t)dt$,  (3)

which expresses that output = input as a fundamental aspect of radiation in time-periodic equilibrium as a phenomenon of near-resonance under small damping. The setting can be generalized to other eigenstates. The essence is the output = input balance, which can express both excitation into eigenstates of larger energy and radiation from such states.

The value of the radiation damping coefficient $\gamma (\psi )$ is set so that conservation of charge is maintained under forcing with the radiation balance (3). If $f=0$ and $\psi$ is a pure eigenstate, then $\gamma = 0$.

Notice that the above argument can be shifted by replacing the ground-state $\Psi_1(x)$ as a time-independent and non-radiating pure eigenstate by an eigenfunction $\Psi_j(x)$ of $H$ with larger energy, again viewed as a time-independent and non-radiating pure eigenstate. This reflects that the time-dependence of pure eigenstates is not observable and thus up to the imagination of an observer. This is not evident in the standard formulation of Schrödinger's equation.

We sum up the virtues of (1) as a semi-classical continuum wave model of a radiating atom subject to forcing, as compared to QED as a non-classical quantum particle model:

1. (1) lends itself to physical interpretation as force balance.
2. (1) lends itself to mathematical analysis. 
3. The term $\ddot\psi$ connects to kinetic energy in classical mechanics and suggests that the common terminology of quantum mechanics of connecting $\Delta\psi$ to kinetic energy, is not natural; a connection to a form of elastic energy may have better physical meaning.
4. (1) has a natural extension to a model for a many-electron atom as a system of one-electron equations, which is computable and thus potentially useful, in contrast to the standard multi-dimensional Schrödinger equation, which is uncomputable and thus potentially useless. 
5. The incoming wave is represented as forcing independent of the wave function $\psi$, which faciliates mathematical analysis and understanding, and not as in QED through a time-dependent contribution to the Hamiltonian, which opens to troublesome self-interaction.