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arXiv:2606.17359

Canonical regularization of the stationary Coulomb problem and an Aufbau-like spectral ordering

Anand Aruna Kumar

quant-phmath-phSubmitted Jun 15, 2026• Announced Jun 17, 2026• 0 citations• Revised Jun 21, 2026 (v2)
v1 · Jun 15, 2026v2 · Jun 21, 2026
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Abstract

The stationary hydrogen atom has Coulomb degeneracy across orbital levels, whereas the Aufbau/Madelung ordering is an empirical, many-electron rule established in atomic physics. We examine the hydrogen atom through a regularized de Broglie--Bohm representation, in which stationary amplitude current constraints generate separable Sturm--Liouville branches. In this formulation, the radial, orbital, and magnetic sectors acquire canonical Langer-like inverse square corrections. The modified boundary value problems allow analytical solutions and produce a hydrogen-like spectrum with regularized radial and angular indices. Consequently, radial Coulomb quantization acquires an orbital dependent shift, lifting the Coulomb degeneracy and producing a spectral ordering that follows the Aufbau/Madelung sequence. On this basis, we construct the ordering of the regularized de Broglie--Bohm states and show that the spectral structure retains the standard degenerate Rydberg sequence in the l=0 sector. The separated amplitudes are represented by generalized special function branches, including the associated Laguerre, Legendre, and Bessel functions with non-integral parameters arising from regularized separation. Therefore, the treatment is intended as an analytical examination of spectral ordering in a regularized one center Coulomb problem rather than as a replacement for the many electron atomic structure theory. Keywords: de Broglie--Bohm representation; Coulomb spectrum; canonical regularization; Langer correction; Sturm--Liouville equations; Aufbau principle; Madelung ordering; associated Legendre functions; associated Laguerre functions; Bessel functions.

Comments
11 pages, 3 figures, revised and expanded sections 3 and 4

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The stationary hydrogen atom has Coulomb degeneracy across orbital levels, whereas the Aufbau/Madelung ordering is an empirical, many-electron rule established in atomic physics. We examine the hydrogen atom through a regularized de Broglie--Bohm representation, in which stationary amplitude current constraints generate separable Sturm--Liouville branches. In this formulation, the radial, orbital, and magnetic sectors acquire canonical Langer-like inverse square corrections. The modified boundary value problems allow analytical solutions and produce a hydrogen-like spectrum with regularized radial and angular indices. Consequently, radial Coulomb quantization acquires an orbital dependent shift, lifting the Coulomb degeneracy and producing a spectral ordering that follows the Aufbau/Madelung sequence.
On this basis, we construct the ordering of the regularized de Broglie--Bohm states and show that the spectral structure retains the standard degenerate Rydberg sequence in the l=0 sector. The separated amplitudes are represented by generalized special function branches, including the associated Laguerre, Legendre, and Bessel functions with non-integral parameters arising from regularized separation. Therefore, the treatment is intended as an analytical examination of spectral ordering in a regularized one center Coulomb problem rather than as a replacement for the many electron atomic structure theory.
Keywords: de Broglie--Bohm representation; Coulomb spectrum; canonical regularization; Langer correction; Sturm--Liouville equations; Aufbau principle; Madelung ordering; associated Legendre functions; associated Laguerre functions; Bessel functions.
In recent years, it has been recognized that treatment of the spectral problem should be associated with closed orbits. This paper proves that the closed orbit obtained by Bohr using numerical algebraic equation is reasonable both mathematically and physically and that the de Broglie wavelength of the electron in the hydrogen atom is equal to the circumference of its orbit. By Bohr model and de Broglie’s matter-wave thought, some quantization conditions, electronic transition formulae, a mechanism and details of emitting photon wave train, a filling order of electrons in the periodic table, and Madelung rule have been deduced theoretically. We solve the Löwdin’s challenge problem.
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This chapter begins with the simple solution of the Schrödinger equation for hydrogen atom for which the energy value has been derived. The Bohr radius of H atom has also been computed. After this, this chapter provides the generalised solution of the Schrödinger equation for H atom/hydrogen-like species in which F equation, R equation, and T equations have been derived in a simple and lucid way. This chapter also deals with the Laguerre differential equation, Laguerre polynomials, Rodrigues formula for the Laguerre polynomials, Laguerre associated equation and its solution, associated Laguerre polynomials, Rodrigues formula for the associated Laguerre polynomials, and solution of the radial equation. It also provides normalisation of the radial wave function and complete wave function; for the hydrogen atom, hydrogenic atomic orbital; electron density of 1 s electron; mathematical background of shape of orbitals; the most probable distance of electron from nucleus of hydrogen atom, and average distance of electron from the nucleus of hydrogen atom. The end of this chapter describes the references and adequate computational problems (solved problems and question on concepts).
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We derive a regularized semiclassical radial propagator for the Coulomb potential, a case for which standard approaches run into well-known difficulties associated with a non-Cartesian radial coordinate and a potential singularity. Following Kleinert [Path Integrals in Quantum Mechanics, Statistics and Polymer Physics (World Scientific, Singapore, 1990)], we first perform a quantum-mechanical regularization of the propagator. The semiclassical limit is then obtained by stationary phase approximation of the resulting integrals. The semiclassical propagator so derived has the standard Van Vleck--Gutzwiller form for the radial Coulomb problem with a potential correction (Langer modification) term included. The regularized semiclassical propagator is applied to compute the autocorrelation function for a Gaussian Rydberg wave packet.