Hydrogen atom as a nonlinear oscillator under circularly polarized light: epicyclical electron orbits
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| Format: | Preprint |
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2024
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| _version_ | 1866913524755726336 |
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| author | Sugon Jr, Quirino Bennett, Clint Dominic G. McNamara, Daniel J. |
| author_facet | Sugon Jr, Quirino Bennett, Clint Dominic G. McNamara, Daniel J. |
| contents | In this paper, we use Clifford algebra $Cl_{2,0}$ to find the 2D orbit of Hydrogen electron under a Coulomb force and a perturbing circularly polarized electric field of light at angular frequency~$ω$, 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 $ω_0$, we derive the complex nonlinear differential equation for the perturbation which is similar to but different from the Lorentz oscillator equation: (1) the acceleration terms are similar, (2) the damping term coefficient is not real but imaginary due to Coriolis force, (3) the term similar to spring force is not positive but negative, (3) there is a complex conjugate of the perturbation term which has no Lorentz analog but which makes the equation nonlinear, and (4) the angular frequency of the forcing term is not $ω$ but $ω- ω_0$. By imposing that the position and velocity of the electron are continuous at time $t = 0$, we show that the orbit of the electron is a sum of five exponential Fourier terms with frequencies 0, $ω_0$, $2ω_0$, $(2ω_0 - ω)$, and $ω$, which correspond to the eccentric, deferent, and three epicycles in Copernican astronomy. We show that at the three resonant light frequencies $0$, $ω_0$, and $2ω_0$, the electron's orbit is divergent, but approximates a Keplerian ellipse. At other light frequencies, the orbits are nondivergent with periods that are integer multiples of $π/ω_0$ depending on the frequency ratio $ω/ω_0$. And as $ω/ω_0\rightarrow \pm\infty$, the orbit approaches the electron's unperturbed circular orbit. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_00056 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hydrogen atom as a nonlinear oscillator under circularly polarized light: epicyclical electron orbits Sugon Jr, Quirino Bennett, Clint Dominic G. McNamara, Daniel J. Quantum Physics Mathematical Physics Atomic Physics 34M04, 42A16, 70K60, 34C15 (primary) 15A67, 70F07, 78A35, 78A40 (secondary) In this paper, we use Clifford algebra $Cl_{2,0}$ to find the 2D orbit of Hydrogen electron under a Coulomb force and a perturbing circularly polarized electric field of light at angular frequency~$ω$, 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 $ω_0$, we derive the complex nonlinear differential equation for the perturbation which is similar to but different from the Lorentz oscillator equation: (1) the acceleration terms are similar, (2) the damping term coefficient is not real but imaginary due to Coriolis force, (3) the term similar to spring force is not positive but negative, (3) there is a complex conjugate of the perturbation term which has no Lorentz analog but which makes the equation nonlinear, and (4) the angular frequency of the forcing term is not $ω$ but $ω- ω_0$. By imposing that the position and velocity of the electron are continuous at time $t = 0$, we show that the orbit of the electron is a sum of five exponential Fourier terms with frequencies 0, $ω_0$, $2ω_0$, $(2ω_0 - ω)$, and $ω$, which correspond to the eccentric, deferent, and three epicycles in Copernican astronomy. We show that at the three resonant light frequencies $0$, $ω_0$, and $2ω_0$, the electron's orbit is divergent, but approximates a Keplerian ellipse. At other light frequencies, the orbits are nondivergent with periods that are integer multiples of $π/ω_0$ depending on the frequency ratio $ω/ω_0$. And as $ω/ω_0\rightarrow \pm\infty$, the orbit approaches the electron's unperturbed circular orbit. |
| title | Hydrogen atom as a nonlinear oscillator under circularly polarized light: epicyclical electron orbits |
| topic | Quantum Physics Mathematical Physics Atomic Physics 34M04, 42A16, 70K60, 34C15 (primary) 15A67, 70F07, 78A35, 78A40 (secondary) |
| url | https://arxiv.org/abs/2410.00056 |