The Radiating Atom 5: Summary

A summary of the experience gathered in the recent posts on radiating atoms is as follows:

1. Schrödinger's equation in standard multi-dimensional form is uncomputable and unphysical. 

Schrödinger's wave equation in multi-dimensional linear form commonly viewed as the basis of quantum mechanics, is uncomputable and hence unphysical. To insist that atom physics is well described by a model which is uncomputable lacks scientific rational, since a model without output cannot be compared with observation. Instead a computable model as a nonlinear system of one-electron wave equations in the spirit of Hartree, should be sought.

2. Schrödinger's equation for a non-radiating atom has a fictional time-dependence.

Schrödinger's equation in standard time-dependent form

• $ih\frac{\partial\psi}{\partial t} + H\psi =0$

with $H$ a Hamiltonian and $t$ time, supposedly describes the dynamics of an atom which is not interacting with any exterior electromagnetic field, that is, is not absorbing or emitting radiation. But such an atom cannot be observed and thus the model cannot be compared to reality. This is reflected by the fact that the charge density $\vert\psi\vert^2$ of the ground state or an excited state as a pure eigen-state of the form

• $\psi (x,t)=\exp(iE/h)\Psi(x)$ 

with $\Psi =\Psi (x)$ an eigenfunction of the Hamiltonian  $H\Psi =E\Psi$ with corresponding real eigenvalue $E$, is not changing with time. Thus the time-dependence in Schrödinger's standard form is fictional in the sense that it cannot be observed. What can be observed is the difference between eigenvalues, as shown in the next section.

3. A radiating atom can be modeled as a forced resonator with small damping.

The standard Schrödinger equation in above complex form can alternatively be formulated in real form as a second order wave equation for a resonator build from $H^2$:

• $\frac{\partial^2\phi}{\partial t^2}+H^2\phi =0$,

which can naturally be extended to include exterior forcing and radiative damping, as shown in Computational Physics of Black Body Radiation. In this setting the frequency $\nu$ of observable absorption/emission of radiation resulting from interference between two pure eigen-states with eigenvalues $E_2>E_1$, satisfies  $h\nu =E_2 - E_1$, while the forcing may have different frequency matching the resonance frequencies $E_2/h$ and $E_1/h$ and not (necessarily) $\nu =E_2/h -E_1/h$.

As above the eigen-states are determined from eigenfunctions $\Psi$ of the Hamiltonian $H$ as stationary values of the energy as the sum of kinetic and potential energies under normalization of $\Psi$. The damping term to be added to the second order wave equation can take the form $\gamma\dot\phi$ with $\gamma >0$ a damping coefficient and corresponding dissipation rate $\gamma\dot\phi^2$ balancing outgoing radiation.

The extended wave equation for a radiating atom may thus take the form

• $\frac{\partial^2\phi}{\partial t^2}+H^2\phi +\gamma\dot\phi =f$,

expressing a balance between forcing $f=f(x,t)$ and the sum of an out-of-balance atomic resonator reaction $\frac{\partial^2\phi}{\partial t^2}+H^2\phi$ and dissipation reaction $\gamma\dot\phi$.  What can here be observed is the radiation generated by a time dependent charge density $\phi^2 (t)$, and not the internal dynamics described by the wave equation, which remains hidden to inspection.

4. Conclusion 

Schrödinger's equation in standard multi-dimensional complex form is not a useful model as a basis of atom physics, because 

• The model is ad hoc and is not derived from basic physics principles.
• Multi-dimensionality makes the model uncomputable. 
• Multi-dimensionality defies physical interpretation of wave functions as solutions.
• The complex form is mystical and lacks physics rationale. 
• Introducing kinetic energy by connecting momentum to $ih\frac{\partial}{\partial x}$ represents a deep formal mysticism.     

5. Towards a more useful wave equation.

It may well be possible to construct a more useful more physical less mysterious model as a system of one-electron second order wave equations expressing a balance of attractive/repulsive Coulomb forces, Abraham-Lorentz radiation forces and forces from regularization of wave solutions.  The first step in such a process is to bring the deficiencies of Schrödinger's standard equation from obscurity and mysticism into scientific light.

Here is a reference into such work: Damping Effect of Electromagnetic Radiation and Time-Dependent Schrödinger Equation by Ji Luo.

6. Reflections on the second-order Schrödinger equation

The second order wave equation $\frac{\partial^2\phi}{\partial t^2}+H^2\phi =0$ was formulated in the 4th of Schrödinger's 1926 articles, but was then dismissed on the ground that a time dependent potential from exterior forcing would give a complicated equation. However, it may well be possible to introduce forcing instead as a time-dependent right hand side $f(x,t)$ in a non-homogeneous wave equation

• $\frac{\partial^2\phi}{\partial t^2}+H^2\phi =f$ 

including the classical ingredients of acceleration $\frac{\partial^2\phi}{\partial t^2}$ connected to kinetic energy $(\frac{\partial\phi}{\partial t})^2$, and with $H=\Delta + V$ connected to a form of "elastic" energy $\vert\nabla\phi\vert^2$ (and thus not kinetic energy) and potential energy $V\phi^2$. This model would bring quantum mechanics into a setting of classical continuum mechanics, which could remove the mysteries of standard quantum mechanics as something fundamentally different from classical continuum mechanics.

Feynman's statement that nobody understands (standard) quantum mechanics, should not be viewed as a joke but as serious criticism: A theory which cannot be understood by any human being is not a scientific theory.