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Classical Phase Space Density for the Relativistic Hydrogen Atom

Theo M. Nieuwenhuizen

Published Jan 1, 2005• 13 citations
OpenAlex

Abstract

Abstract. Quantum mechanics is considered to arise from an underlying classical structure (“hidden variable theory”, “subquantum mechanics”), where quantum fluctuations follow from a physical noise mechanism. The stability of the hydrogen ground state can then arise from a balance between Lorentz damping and energy absorption from the noise. Since the damping is weak, the ground state phase space density should predominantly be a function of the conserved quantities, energy and angular momentum. A candidate for this phase space density is constructed for ground state of the relativistic hydrogen problem of a spinless particle. The first excited states and their spherical harmonics are also considered in this framework. The analytic expression of the ground state energy can be reproduced, provided averages of certain products are replaced by products of averages. This analysis puts forward that quantum mechanics may arise from an underlying classical level as a slow variable theory, where each new quantum operator relates to a new, well separated time interval.

Related papers

Classical Phase Space Density for the Relativistic Hydrogen Atom

Theo M. Nieuwenhuizen

Quantum mechanics is considered to arise from an underlying classical structure (“hidden variable theory”, “sub‐quantum mechanics”), where quantum fluctuations follow from a physical noise mechanism. The stability of the hydrogen ground state can then arise from a balance between Lorentz damping and energy absorption from the noise. Since the damping is weak, the ground state phase space density should predominantly be a function of the conserved quantities, energy and angular momentum.A candidate for this phase space density is constructed for ground state of the relativistic hydrogen problem of a spinless particle. The first excited states and their spherical harmonics are also considered in this framework.The analytic expression of the ground state energy can be reproduced, provided averages of certain products are replaced by products of averages. This analysis puts forward that quantum mechanics may arise from an underlying classical level as a slow variable theory, where each new quantum operator relates to a new, well separated time interval.

arXiv:quant-ph/0511144•AIP conference proceedings•Jan 1, 2006•17 citations

Wigner's phase space function and atomic structure

Jens Peder Dahl, Michael Springborg

We have constructed the Wigner function for the ground state of the hydrogen atom and analysed its variation over phase space. By means of the Weyl correspondence between operators and phase space functions we have then studied the description of angular momentum and resolved a dilemma in the comparison with early quantum mechanics. Finally we have discussed the introduction of local energy densities in coordinate space and demonstrated the validity of a local virial theorem.

Molecular Physics•Dec 10, 1982•83 citations

Classical Oscillator Model of the Ground State of Hydrogen Atom

YU Chang-feng

According to the prophesy made by quantum mechanics that the angular momentum of ground state hydrogen atom is zero,in the paper,the ground state hydrogen atom is treated as a one-dimensional classical oscillator,and by using a potential function with phase factors and repulse force,the ground state energy of the hydrogen atom as well as an analytic expression of the vibrational orbit of electron are given.

Journal of Shihezi University•Jan 1, 2004•0 citations

Spin and Statistics from a Classical Vibrational Field

Mykola Yaremenko

We extend the recently introduced purely vibrational, classical field model of the hydrogen atom in two essential directions. First, the electron scalar field is replaced by a classical Dirac spinor field, which naturally generates intrinsic spin angular momentum and, in the non-relativistic limit, yields the Pauli equation with the correct spin–orbit coupling and Darwin term; the resulting fine-structure corrections match the standard quantum mechanical formula. Second, we incorporate a classical, Lorentz-invariant zero-point electromagnetic radiation background and show that, when a measurement apparatus is modelled as a resonant detector, the time averaged detection probability for a given stationary vibrational mode is proportional to its energy fraction. The Born rule thus emerges as a statistical consequence of deterministic evolution with zero-point noise, without any fundamental quantum postulate. The combined model reproduces both the full hydrogen spectrum (including fine structure) and the probabilistic predictions of quantum mechanics from purely classical field equations.

London journal of physics.•Jul 28, 2026•0 citations

Time-Delayed Causality as a Classical Alternative to Quantum Mechanics: Hydrogen in Neutral Newtonian Mechanics

M. A. Elfouly

Abstract This study develops a strictly classical time-delayed model of the hydrogen atom, with retarded Coulomb forces and finite-difference inertia over a state-dependent delay, to reproduce key quantum features without quantum postulates. The delay law is anchored in the Abraham, Lorentz and Dirac self-time and fixed by matching the Rydberg constant, thereby determining a neutral delay scale and an effective energy frequency proportionality without inserting Planck’s constant into the equations of motion. With this single calibration, a neutral mean-power condition and a phase-locking rule between orbital motion and delay generate a discrete ladder of radii and energies reproducing the Rydberg spectrum without imposing angular-momentum quantisation. Neutral radial and azimuthal phases, constructed from the retarded trajectory, define two indices that act as classical analogues of principal and orbital quantum numbers and satisfy simple sum rules for elliptic orbits, mirroring Sommerfeld’s extension of the Bohr model. Interpreting the neutral delay as a minimal time window for inferring position and velocity yields a classical position–momentum error product of order Planck’s constant. In weakly driven settings, non-zero mean neutral power turns the delayed inertial channel into an energy reservoir, allowing deterministic level-to-level transitions between long-lived neutral orbits and preventing the catastrophic radiative collapse of the Rutherford atom.

•Dec 4, 2025•1 citation

Deriving Hydrogen Spectrum Based on a Classical Electron in a Quantized Coulomb Potential with Runge–Lenz Symmetry: An Alternative to the Standard Quantum Theory

Jau Tang

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.

•May 1, 2026•0 citations
Abstract. Quantum mechanics is considered to arise from an underlying classical structure (“hidden variable theory”, “subquantum mechanics”), where quantum fluctuations follow from a physical noise mechanism. The stability of the hydrogen ground state can then arise from a balance between Lorentz damping and energy absorption from the noise. Since the damping is weak, the ground state phase space density should predominantly be a function of the conserved quantities, energy and angular momentum. A candidate for this phase space density is constructed for ground state of the relativistic hydrogen problem of a spinless particle. The first excited states and their spherical harmonics are also considered in this framework. The analytic expression of the ground state energy can be reproduced, provided averages of certain products are replaced by products of averages. This analysis puts forward that quantum mechanics may arise from an underlying classical level as a slow variable theory, where each new quantum operator relates to a new, well separated time interval.
Quantum mechanics is considered to arise from an underlying classical structure (“hidden variable theory”, “sub‐quantum mechanics”), where quantum fluctuations follow from a physical noise mechanism. The stability of the hydrogen ground state can then arise from a balance between Lorentz damping and energy absorption from the noise. Since the damping is weak, the ground state phase space density should predominantly be a function of the conserved quantities, energy and angular momentum.A candidate for this phase space density is constructed for ground state of the relativistic hydrogen problem of a spinless particle. The first excited states and their spherical harmonics are also considered in this framework.The analytic expression of the ground state energy can be reproduced, provided averages of certain products are replaced by products of averages. This analysis puts forward that quantum mechanics may arise from an underlying classical level as a slow variable theory, where each new quantum operator relates to a new, well separated time interval.
We extend the recently introduced purely vibrational, classical field model of the hydrogen atom in two essential directions. First, the electron scalar field is replaced by a classical Dirac spinor field, which naturally generates intrinsic spin angular momentum and, in the non-relativistic limit, yields the Pauli equation with the correct spin–orbit coupling and Darwin term; the resulting fine-structure corrections match the standard quantum mechanical formula. Second, we incorporate a classical, Lorentz-invariant zero-point electromagnetic radiation background and show that, when a measurement apparatus is modelled as a resonant detector, the time averaged detection probability for a given stationary vibrational mode is proportional to its energy fraction. The Born rule thus emerges as a statistical consequence of deterministic evolution with zero-point noise, without any fundamental quantum postulate. The combined model reproduces both the full hydrogen spectrum (including fine structure) and the probabilistic predictions of quantum mechanics from purely classical field equations.
Abstract This study develops a strictly classical time-delayed model of the hydrogen atom, with retarded Coulomb forces and finite-difference inertia over a state-dependent delay, to reproduce key quantum features without quantum postulates. The delay law is anchored in the Abraham, Lorentz and Dirac self-time and fixed by matching the Rydberg constant, thereby determining a neutral delay scale and an effective energy frequency proportionality without inserting Planck’s constant into the equations of motion. With this single calibration, a neutral mean-power condition and a phase-locking rule between orbital motion and delay generate a discrete ladder of radii and energies reproducing the Rydberg spectrum without imposing angular-momentum quantisation. Neutral radial and azimuthal phases, constructed from the retarded trajectory, define two indices that act as classical analogues of principal and orbital quantum numbers and satisfy simple sum rules for elliptic orbits, mirroring Sommerfeld’s extension of the Bohr model. Interpreting the neutral delay as a minimal time window for inferring position and velocity yields a classical position–momentum error product of order Planck’s constant. In weakly driven settings, non-zero mean neutral power turns the delayed inertial channel into an energy reservoir, allowing deterministic level-to-level transitions between long-lived neutral orbits and preventing the catastrophic radiative collapse of the Rutherford atom.
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.