Hydrogen atom as a nonlinear oscillator under circularly polarized light: epicyclical electron orbits

arXiv:2410.00056v2 Announce Type: replace Abstract: We used Clifford algebra $Cl_{2,0}$ to find the 2D orbit of a hydrogen electron under a Coulomb force and a perturbing circularly polarized electric field of light at angular frequency~$\omega$, which is turned on at time $t = 0$ via a unit step switch. Using a coordinate system co-rotating with the electron's unperturbed circular orbit at angular frequency $\omega_0$, we derived a complex differential equation that is similar to but different from that of the Lorentz oscillator equation for light--atom interaction. We solved the homogeneous and particular differential equation and showed that the position of the electron is a linear combination of five exponential Fourier terms or orbital wave functions with frequencies $0$, $\omega_0$, $2\omega_0$, $(2\omega_0 - \omega)$, and $\omega$, whose coefficients depend on the light-to-atom frequency ratio $\omega/\omega_0$ and light-to-atom force magnitude ratio $A/r_0$. We showed that the electron orbits are approximately Keplerian at light-to-atom frequency ratio $\alpha = \omega/\omega_0 = \{0, 1, 2\}$, with the orbits becoming discontinuous and divergent as $\alpha \rightarrow 1^{\pm}$, but continuous and non-divergent at $\alpha = \{0, 2\}$. These Keplerian orbits are approximated by the sum of the zeroth, first, and second harmonics of the electron's unperturbed orbital wave function $\hat\psi_0 = e^{\hat\imath\omega_0 t}$, corresponding to the eccentric, deferent, and epicycle in the Copernican construction of planetary orbits.

Abstract:We used Clifford algebra $Cl_{2,0}$ to find the 2D orbit of a hydrogen electron under a Coulomb force and a perturbing circularly polarized electric field of light at angular frequency~$\omega$, which is turned on at time $t = 0$ via a unit step switch. Using a coordinate system co-rotating with the electron's unperturbed circular orbit at angular frequency $\omega_0$, we derived a complex differential equation that is similar to but different from that of the Lorentz oscillator equation for light--atom interaction. We solved the homogeneous and particular differential equation and showed that the position of the electron is a linear combination of five exponential Fourier terms or orbital wave functions with frequencies $0$, $\omega_0$, $2\omega_0$, $(2\omega_0 - \omega)$, and $\omega$, whose coefficients depend on the light-to-atom frequency ratio $\omega/\omega_0$ and light-to-atom force magnitude ratio $A/r_0$. We showed that the electron orbits are approximately Keplerian at light-to-atom frequency ratio $\alpha = \omega/\omega_0 = \{0, 1, 2\}$, with the orbits becoming discontinuous and divergent as $\alpha \rightarrow 1^{\pm}$, but continuous and non-divergent at $\alpha = \{0, 2\}$. These Keplerian orbits are approximated by the sum of the zeroth, first, and second harmonics of the electron's unperturbed orbital wave function $\hat\psi_0 = e^{\hat\imath\omega_0 t}$, corresponding to the eccentric, deferent, and epicycle in the Copernican construction of planetary orbits.

The full revised paper has 42 pages, 184 equations, 9 figures, and 2 tables. The sign error in the 2024 ArXiv version was corrected, leading to new results and conclusions

Quantum Physics (quant-ph); Mathematical Physics (math-ph); Atomic Physics (physics.atom-ph)

arXiv:2410.00056 [quant-ph]

(or arXiv:2410.00056v2 [quant-ph] for this version)

Sugon, Q., Jr.; Bennett, C.D.G.; McNamara, D.J. Hydrogen Atom as a Nonlinear Oscillator Under Circularly Polarized Light: Epicyclical Electron Orbits. Hydrogen 2026, 7, 92

https://doi.org/10.3390/hydrogen7030092 DOI(s) linking to related resources

From: Quirino Sugon Jr. [view email] [v1] Sun, 29 Sep 2024 03:06:34 UTC (50 KB) [v2] Fri, 31 Jul 2026 09:30:49 UTC (22 KB)

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Hydrogen atom as a nonlinear oscillator under circularly polarized light: epicyclical electron orbits

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