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

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.

• The paper proposes a canonical regularization of the stationary Coulomb problem that introduces Langer-like inverse-square corrections in radial, orbital, and axial sectors.
• It reformulates the amplitude-phase dynamics via the de Broglie–Bohm framework, leading to separable Sturm–Liouville branches and shifted quantization conditions.
• The analysis results in an intrinsic orbital energy splitting, replicating an Aufbau/Madelung-like ordering without resorting to multi-electron effects.

Canonical Regularization of the Stationary Coulomb Problem: Intrinsic Spectral Ordering

Introduction

This paper addresses the stationary hydrogen atom problem from the standpoint of the de Broglie–Bohm representation, applying canonical regularization to amplitude–phase equations. Unlike the conventional approach—which yields degenerate energy levels across orbital angular momentum quantum numbers—the author proposes a systematic regularization procedure for the component amplitudes, resulting in separable Sturm–Liouville branches. Notably, the regularized equations incorporate Langer-like inverse-square corrections in the radial, orbital, and axial sectors. This fundamentally alters the quantization conditions and induces an intrinsic orbital energy splitting. The resulting energy spectrum thereby exhibits an Aufbau/Madelung-like ordering, which is a well-known empirical rule for electron shell occupation in many-electron atoms.

Mathematical Formulation and Regularization

The methodology begins with the stationary Schrödinger equation in amplitude–phase form, $\Psi=R\exp(iS/\hbar)$, separating the quantum dynamics into amplitude and phase equations under the de Broglie–Bohm formalism. The regularization is based on imposing canonical amplitude–current constraints, $p_i q_i = \frac{\hbar}{2}$ for each coordinate sector. This is realized within spherical coordinates, resulting in modified Hamilton–Jacobi equations with additional inverse-square terms.

The radial, orbital, and axial amplitude equations are each regularized:

• Radial sector: The modified radial equation includes a term $\frac{1}{4r^2}$, leading to a non-integral parameter in the associated Laguerre functions. Energy quantization is thereby shifted, yielding a discrete spectrum where the energy depends on both the radial quantum number $n_r$ and the orbital label $\lambda$.
• Orbital sector: The angular equation, regularized with a half-integer offset in its order, produces associated Legendre functions of non-integral degree and order. The separation labels $\lambda$ and $\mu$ are introduced, and their correspondence to conventional quantum numbers ($l$, $m$) is established via a minimum labeling scheme.
• Axial sector: The regularized azimuthal equation admits solutions in terms of Bessel functions rather than the traditional complex exponentials. The functional form includes non-integral parameters tied to $\mu$.

These modifications preserve separability and solvability through special function branches, but fundamentally lift the degeneracy between states of different $p_i q_i = \frac{\hbar}{2}$0 for fixed principal quantum number.

Energy Spectrum and Aufbau Ordering

The energy of bound states after canonical regularization is given as:

$p_i q_i = \frac{\hbar}{2}$1

For $p_i q_i = \frac{\hbar}{2}$2, this recovers the standard degenerate Rydberg sequence. For $p_i q_i = \frac{\hbar}{2}$3, the energy levels are shifted downward (i.e., further from the nucleus) in a manner that organizes states according to the empirical Aufbau principle. The sequence of energy levels follows:

$p_i q_i = \frac{\hbar}{2}$4

This ordering directly mirrors the Madelung rule for subshell filling observed in atomic physics, although here the splitting is generated analytically within a one-center Coulomb potential, absent electron–electron interactions, relativistic corrections, or screening effects.

The spectrum retains the full degeneracy only in the $p_i q_i = \frac{\hbar}{2}$5 sector; for $p_i q_i = \frac{\hbar}{2}$6, the Coulomb degeneracy is lifted. The construction allows analytical solution for the eigenstates via the regularized Sturm–Liouville approach, employing generalized Laguerre, Legendre, and Bessel functions.

Angular Amplitudes: Extensions Beyond Spherical Harmonics

The angular components are expressed as generalized special functions with non-integral parameters: associated Legendre functions $p_i q_i = \frac{\hbar}{2}$7 and Bessel functions $p_i q_i = \frac{\hbar}{2}$8. Spherical harmonic closure ($p_i q_i = \frac{\hbar}{2}$9 admissibility) is not imposed. This permits a broader set of ($\frac{1}{4r^2}$0, $\frac{1}{4r^2}$1) branches, restricted only for comparison to conventional hydrogenic labeling.

The author develops a chart-based representation to cover the full angular domain, maintaining correct phase and amplitude continuity. The parity operators $\frac{1}{4r^2}$2, $\frac{1}{4r^2}$3 are introduced for systematic assignment of these branches.

The mathematical admissibility of the amplitude functions is established by reference to the confluent hypergeometric and generalized special function theory; all branches are well-defined and reducible to conventional cases for integral labels.

Implications and Future Directions

The key implication is that intrinsic orbital splitting and Aufbau/Madelung-like ordering can arise analytically from regularization of the stationary one-center Coulomb problem, without recourse to many-body effects or empirical rules. This challenges the standard notion that such ordering only emerges in interacting many-electron systems. The analytic mechanism is grounded in the canonical constraints on quantum amplitude–phase flow and associated inverse-square corrections familiar from WKB and semiclassical quantization.

From a theoretical standpoint, this approach may serve as a model for understanding how spectral hierarchy and orbital ordering can be generated by quantum regularization mechanisms alone. It also points to the potential utility of de Broglie–Bohm analysis and canonical transformations in the study of atomic structure and quantum spectral theory.

Practically, while this formulation is not intended to replace full many-electron atom calculations, it provides a systematic analytical framework that may be relevant for effective model Hamiltonians, quantum defect theory, and the study of orbital filling effects in low-dimensional or strongly regularized quantum systems. Extensions to other central potentials, multi-center systems, and non-separable configurations are possible future avenues.

Conclusion

The paper presents a rigorous analysis of the stationary hydrogen atom in the de Broglie–Bohm representation, augmented by canonical regularization. The resulting Sturm–Liouville equations yield analytically solvable special function branches with shifted quantization conditions, leading to a non-degenerate energy spectrum that replicates the empirical Aufbau/Madelung ordering. This demonstrates that intrinsic orbital splitting and spectral hierarchy can arise purely from the mathematics of amplitude-phase regularization, without invoking multi-electron interactions. The findings establish the regularized one-center Coulomb problem as a valuable analytical model for studying the origins of spectral ordering in quantum systems (2606.17359).