Magnetic quantum number distribution from $ q $-binomial identities

Jean-Christophe Pain; Jean-Christophe Pain

doi:10.3934/math.20261383

AIMS Mathematics

2026, Volume 11, Issue 10: 35698-35717. doi: 10.3934/math.20261383

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Research article

Magnetic quantum number distribution from $ q $-binomial identities

Jean-Christophe Pain ^{1,2
,
,}

1.
CEA, DAM, DIF, F-91297 Arpajon, France
2.
Université Paris-Saclay, CEA, Laboratoire Matière en Conditions Extrêmes, F-91680 Bruyères-le-Châtel, France

Received: 26 June 2026 Revised: 29 September 2026 Accepted: 19 October 2026 Published: 30 October 2026
MSC : 05A10, 05A30, 11B65, 81V35

The distribution $ \mathscr{P}(M) $ of the total atomic magnetic quantum number $ M $ is a fundamental quantity for the computation of atomic structure and spectra. It enables one to deduce the distribution $ \mathscr{Q}(J) $ of the total angular momentum $ J $, which determines the sizes of the Hamiltonian blocks. Various techniques have been developed to obtain $ \mathscr{P}(M) $, based on recurrence relations, group theory, or $ q $-binomial coefficients (Gaussian polynomials). In this work, we derived new recurrence relations for $ \mathscr{P}(M) $ based on identities that have not yet been exploited for this purpose, including a $ q $-analogue of the so-called "Star of David" theorem. We showed that classical combinatorial identities, when extended to Gaussian polynomials and combined with generating-function differentiation, systematically yielded new recurrence relations for the magnetic quantum number distribution. While these new relations may not uniformly surpass existing algorithms in raw computational efficiency, they offer significant theoretical novelty by connecting advanced $ q $-combinatorial theorems to atomic physics. Furthermore, because the recurrence relations involved a distinct stencil over the index set (and thus different subsets of $ M $-values), they provided valuable cross-validation tools to test the consistency and reliability of large-scale calculations.
- magnetic quantum number distribution,
- angular momentum,
- $ q $-binomial coefficients,
- Gaussian polynomials,
- Star of David theorem,
- combinatorial identities,
- recurrence relations,
- generating functions,
- atomic structure,
- shell model,
- fermionic systems,
- $ q $-analogues
Citation: Jean-Christophe Pain. Magnetic quantum number distribution from $ q $-binomial identities[J]. AIMS Mathematics, 2026, 11(10): 35698-35717. doi: 10.3934/math.20261383

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Abstract

The distribution $ \mathscr{P}(M) $ of the total atomic magnetic quantum number $ M $ is a fundamental quantity for the computation of atomic structure and spectra. It enables one to deduce the distribution $ \mathscr{Q}(J) $ of the total angular momentum $ J $, which determines the sizes of the Hamiltonian blocks. Various techniques have been developed to obtain $ \mathscr{P}(M) $, based on recurrence relations, group theory, or $ q $-binomial coefficients (Gaussian polynomials). In this work, we derived new recurrence relations for $ \mathscr{P}(M) $ based on identities that have not yet been exploited for this purpose, including a $ q $-analogue of the so-called "Star of David" theorem. We showed that classical combinatorial identities, when extended to Gaussian polynomials and combined with generating-function differentiation, systematically yielded new recurrence relations for the magnetic quantum number distribution. While these new relations may not uniformly surpass existing algorithms in raw computational efficiency, they offer significant theoretical novelty by connecting advanced $ q $-combinatorial theorems to atomic physics. Furthermore, because the recurrence relations involved a distinct stencil over the index set (and thus different subsets of $ M $-values), they provided valuable cross-validation tools to test the consistency and reliability of large-scale calculations.

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