Statistics of Quantum Numbers for Non-Equivalent Fermions in Single-<i>j</i> Shells
This work addresses closed-form expressions for the distributions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mo>(</mo><mi>M</mi><mo>)</mo><...
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| author | Jean-Christophe Pain |
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| author_sort | Jean-Christophe Pain |
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| description | This work addresses closed-form expressions for the distributions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mo>(</mo><mi>M</mi><mo>)</mo></mrow></semantics></math></inline-formula> of the magnetic quantum numbers <i>M</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Q</mi><mo>(</mo><mi>J</mi><mo>)</mo></mrow></semantics></math></inline-formula> of total angular momentum <i>J</i> for non-equivalent fermions in single-<i>j</i> orbits. Such quantities play an important role in both nuclear and atomic physics, through the shell models. Using irreducible representations of the rotation group, different kinds of formulas are presented, involving multinomial coefficients, generalized Pascal triangle coefficients, or hypergeometric functions. Special cases are discussed, and the connections between <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mo>(</mo><mi>M</mi><mo>)</mo></mrow></semantics></math></inline-formula> (and therefore <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Q</mi><mo>(</mo><mi>J</mi><mo>)</mo></mrow></semantics></math></inline-formula>) and mathematical functions such as elementary symmetric, cyclotomic, and Jacobi polynomials are outlined. |
| format | Article |
| id | doaj-art-4878a79a224742eaaf0d2c81ef938566 |
| institution | OA Journals |
| issn | 2218-2004 |
| language | English |
| publishDate | 2025-03-01 |
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| spelling | doaj-art-4878a79a224742eaaf0d2c81ef9385662025-08-20T02:24:43ZengMDPI AGAtoms2218-20042025-03-011342510.3390/atoms13040025Statistics of Quantum Numbers for Non-Equivalent Fermions in Single-<i>j</i> ShellsJean-Christophe Pain0CEA, DAM, DIF, F-91297 Arpajon, FranceThis work addresses closed-form expressions for the distributions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mo>(</mo><mi>M</mi><mo>)</mo></mrow></semantics></math></inline-formula> of the magnetic quantum numbers <i>M</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Q</mi><mo>(</mo><mi>J</mi><mo>)</mo></mrow></semantics></math></inline-formula> of total angular momentum <i>J</i> for non-equivalent fermions in single-<i>j</i> orbits. Such quantities play an important role in both nuclear and atomic physics, through the shell models. Using irreducible representations of the rotation group, different kinds of formulas are presented, involving multinomial coefficients, generalized Pascal triangle coefficients, or hypergeometric functions. Special cases are discussed, and the connections between <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mo>(</mo><mi>M</mi><mo>)</mo></mrow></semantics></math></inline-formula> (and therefore <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Q</mi><mo>(</mo><mi>J</mi><mo>)</mo></mrow></semantics></math></inline-formula>) and mathematical functions such as elementary symmetric, cyclotomic, and Jacobi polynomials are outlined.https://www.mdpi.com/2218-2004/13/4/25quantum mechanicsangular momentumelectron configurationsPauli exclusion principleenergy levelscomplex spectra |
| spellingShingle | Jean-Christophe Pain Statistics of Quantum Numbers for Non-Equivalent Fermions in Single-<i>j</i> Shells Atoms quantum mechanics angular momentum electron configurations Pauli exclusion principle energy levels complex spectra |
| title | Statistics of Quantum Numbers for Non-Equivalent Fermions in Single-<i>j</i> Shells |
| title_full | Statistics of Quantum Numbers for Non-Equivalent Fermions in Single-<i>j</i> Shells |
| title_fullStr | Statistics of Quantum Numbers for Non-Equivalent Fermions in Single-<i>j</i> Shells |
| title_full_unstemmed | Statistics of Quantum Numbers for Non-Equivalent Fermions in Single-<i>j</i> Shells |
| title_short | Statistics of Quantum Numbers for Non-Equivalent Fermions in Single-<i>j</i> Shells |
| title_sort | statistics of quantum numbers for non equivalent fermions in single i j i shells |
| topic | quantum mechanics angular momentum electron configurations Pauli exclusion principle energy levels complex spectra |
| url | https://www.mdpi.com/2218-2004/13/4/25 |
| work_keys_str_mv | AT jeanchristophepain statisticsofquantumnumbersfornonequivalentfermionsinsingleijishells |