Hydrogenic Matrix Elements with Different Effective Charges: Non-Relativistic and Relativistic Cases
This work explores the evaluation of hydrogenic matrix elements for non-relativistic and relativistic cases under the Screened Hydrogenic Model (SHM). It focuses on scenarios where the initial and final states have different effective charges <inline-formula><math xmlns="http://www.w3....
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-06-01
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| Series: | Atoms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2218-2004/13/7/60 |
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| Summary: | This work explores the evaluation of hydrogenic matrix elements for non-relativistic and relativistic cases under the Screened Hydrogenic Model (SHM). It focuses on scenarios where the initial and final states have different effective charges <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Z</mi><mn>1</mn></msub><mo>≠</mo><msub><mi>Z</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula>, deriving closed-form solutions for particular cases <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>n</mi><mn>1</mn></msub><mo>=</mo><msub><mi>n</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Z</mi><mn>1</mn></msub><mo>=</mo><msub><mi>Z</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula>. In addition, analytical expressions for radial matrix elements <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>⟨</mo><msup><mi>n</mi><mo>′</mo></msup><msup><mi>l</mi><mo>′</mo></msup><mo>|</mo><msup><mi>r</mi><mi>β</mi></msup><mo>|</mo><mi>n</mi><mi>l</mi><mo>⟩</mo></mrow></semantics></math></inline-formula> and their relativistic counterparts are presented. These are applicable for discrete–discrete transitions and allow simplifications for specific configurations using Laplace transforms. The study discusses generalizations of SHM for calculating cross-sections in hot and dense plasmas, employing the Plane Wave Born Approximation (PWBA). It also addresses the transition from LS to jj coupling for matrix elements, providing rules for such transformations. |
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| ISSN: | 2218-2004 |