Algebraic Solution of Gaunt Coefficients via the Angular Momentum Ladder Operators

Abstract

In this study, Gaunt coefficients, frequently encountered in quantum mechanical calculations of atomic and molecular structures, have been algebraically derived. Firstly, the Gaunt coefficient, equal to the integral over the solid angle of the product of three spherical harmonics, is written in terms of angular momentum ladder operators. Subsequently, raising or lowering operators are applied to spherical harmonics, and the obtained integrals are solved using the recurrence and orthogonality relations of spherical harmonics. As a result, algebraic expressions for Gaunt coefficients are obtained in terms of quantum numbers.

Keywords

• Gaunt coefficients
• ladder operators
• spherical harmonics

Açısal Momentum Merdiven İşlemcileri ile Gaunt Katsayılarının Cebirsel Çözümü

Abstract

Bu çalışmada, atomik ve moleküler yapıların kuantum mekaniksel hesaplamalarında sıklıkla karşılaşılan Gaunt katsayıları cebirsel olarak türetilmiştir. İlk olarak, üç küresel harmoniğin çarpımının katı açı üzerinden integraline eşit olan Gaunt katsayısı, açısal momentum merdiven işlemcileri cinsinden yazılır. Daha sonra, yükseltme veya alçaltma işlemcileri küresel harmoniklere uygulanır ve elde edilen integralleri çözmek için küresel harmoniklerin tekrarlama ve diklik bağıntıları kullanılır. Sonuç olarak, Gaunt katsayıları için cebirsel ifadeler, kuantum sayıları cinsinden elde edilir.

Keywords

• Gaunt katsayıları
• merdiven işlemcileri
• küresel harmonikler

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