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arXiv:1308.0221

A model with two quantum particles similar to the hydrogen atom

Ruslan Abdulovich Sharipov

Published Aug 1, 2013• 0 citations
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Abstract

The hydrogen atom with the Coulomb interaction is one of the exactly solvable non-relativistic quantum models. Unlike many other exactly solvable models it describes a real physical object providing the formulas for energy levels and stationary state wave functions of a real hydrogen atom. In this paper we modify the model replacing the Coulomb interaction by the interaction of the proton and the electron with the classical electromagnetic field serving as an intermediary transmitting the electromagnetic interaction of these two charged quantum particles.

DOI
10.48550/arxiv.1308.0221

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The hydrogen atom with the Coulomb interaction is one of the exactly solvable non-relativistic quantum models. Unlike many other exactly solvable models it describes a real physical object providing the formulas for energy levels and stationary state wave functions of a real hydrogen atom. In this paper we modify the model replacing the Coulomb interaction by the interaction of the proton and the electron with the classical electromagnetic field serving as an intermediary transmitting the electromagnetic interaction of these two charged quantum particles.
In this work we are presenting a modified Coulomb potential function to describe the interaction between two micro-scopic electric charges. In particular, concerning the interaction between the proton and the electron in the hydrogen atom. The modified potential function is the product of the classical Coulomb potential and an oscillatory function dependent on a quantized phase factor. The oscillatory function picks up only selected points along the Coulomb potential, creating potential wells and barriers around the nucleus of the atom. The new potential reveals us new features of the hydrogen atom. Searching for a manner to determine the phase factor, we are using the concept of the de Broglie particle wavelike behavior and the quantum analogue of the virial theorem for describing the bound motion of a particle in a central force field. This procedure is a kind of feedback action, where we are making use of well established concepts of the quantum mechanics aiming to determine the phase factor of the new interaction potential.
It is shown that a solution of the classical Kepler problem with a binomial potential explains the stationary orbits of the hydrogen atom, its discrete spectrum, and trajectories of scattered electrons with energies from several electron-volts to relativistic values. The Schrodinger equation with the binomial potential takes the form of a classical wave equation. Hence, the binomial potential is a more appropriate model for understanding atomic phenomena than the model based on the Coulomb potential.
The hydrogen atom has historically played a foundational role in the development of quantum mechanics, where its discrete energy spectrum is conventionally derived from solutions of the Schrödinger wave equation. In this work we present an alternative formulation in which the hydrogen spectrum emerges without invoking the Schrödinger equation. We consider a semi-quantum framework in which the electron is treated as a classical particle governed by Poisson-bracket dynamics while interacting with a quantized electromagnetic field described using second quantization. The electron moves in the Coulomb potential generated by the proton and simultaneously couples to quantized electromagnetic modes through minimal coupling. The Coulomb system possesses a hidden dynamical symmetry characterized by the Runge–Lenz vector, which enlarges the rotational symmetry to an group for bound states. Within this framework, interactions between the classical particle and the quantized field induce an effective commutation structure in the particle’s phase space. Once this structure emerges, the algebra of the conserved quantities associated with the Runge–Lenz vector becomes identical to the operator algebra used in Pauli’s symmetry-based derivation of the hydrogen spectrum. Consequently, the discrete hydrogen energy levels arise naturally from the combined effects of Coulomb symmetry and particle–field interaction, offering a physically transparent interpretation of atomic quantization and suggesting that wave–particle duality may arise dynamically from interactions with quantized electromagnetic fields.