Araştırma Makalesi
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Algebraic Solution of Gaunt Coefficients via the Angular Momentum Ladder Operators

Yıl 2023, , 229 - 244, 28.12.2023
https://doi.org/10.33484/sinopfbd.1358148

Öz

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.

Kaynakça

  • Griffiths, D. J. (2017). Introduction to Quantum Mechanics (2nd ed.). Cambridge University Press, Cambridge.
  • McQuarrie, D. A. (2008). Quantum Chemistry (2nd ed.). University Science Books, California.
  • Condon, E. U., & Shortley, G. H. (1935). Theory of Atomic Spectra. Cambridge University Press, Cambridge.
  • Cohen-Tannoudji, C., Diu, B., & Laloe, F. (1977). Quantum Mechanics. John Wiley & Sons, New York.
  • Townsend, J. S. (2000). A Modern Approach to Quantum Mechanics. University Science Books, USA.
  • Zettili, N. (2009). Quantum Mechanics: Concepts and Applications. John Wiley & Sons, USA.
  • Weitzman, M., & Freericks, J. K. (2018). Calculating spherical harmonics without derivatives. Condensed Matter Physics, 21 (3), 1-12. https://doi.org/10.5488/CMP.21.33002
  • Edmonds, A. R. (1960). Angular Momentum in Quantum Mechanics (2nd ed.). Princeton University Press, New Jersey.
  • Rose, M. E. (1957). Elementary Theory of Angular Momentum. John Wiley & Sons, New York.
  • Zare, R. N. (1988). Angular Momentum, Understanding Spatial Aspects in Chemistry and Physics. John Wiley & Sons, New York.
  • Varshalovich, D. A., Moskalev, A. N., & Khersonskii, V. K. (1988). Quantum Theory of Angular Momentum, Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols. World Scientific Publishing Co. Pte. Ltd, Singapore.
  • Shimpuku, T. (1960). General theory and numerical tables of clebsch-gordan coefficients. Supplement of the Progress of Theoretical Physics, 13, 1-135. https://doi.org/10.1143/PTPS.13.1
  • Tarter, C. B. (1970). Coefficients connecting the stark and field-free wavefunctions for hydrogen. Journal of Mathematical Physics, 11, 3192-3195. http://dx.doi.org/10.1063/1.1665113
  • Schulten, K., & Gordon, R. G. (1976). Recursive evaluation of 3j and 6j coefficients. Computer Physics Communications, 11, 269-278. https://doi.org/10.1016/0010-4655(76)90058-8
  • Lai, S. T., & Chiu, Y. N. (1990). Exact computation of the 3-j and 6-j symbols. Computer Physics Communications, 61, 350-360. https://doi.org/10.1016/0010-4655(90)90049-7
  • Guseinov, I. I., Özmen, A., Atav, Ü., & Yüksel, H. (1995). Computation of clebsch-gordan and gaunt coefficients using binomial coefficients. Journal of Computational Physics, 122, 343-347. https://doi.org/10.1006/jcph.1995.1220
  • Wei, L. (1999). Unified approach for exact calculation of angular momentum coupling and recoupling coefficients. Computer Physics Communications, 120, 222-230. https://doi.org/10.1016/S0010-4655(99)00232-5
  • Pain, J. -C. (2020). Some properties of Wigner 3 j coefficients: non-trivial zeros and connections to hypergeometric functions. The Europan Physics Journal A, 56:296, 1-13. https://doi.org/10.1140/epja/s10050-020-00303-9
  • Akdemir, S., Özay, S., & Öztekin E. (2023). Asymptotic behavior of clebsch-gordan coefficients. Journal of Mathematical Chemistry, https://doi.org//10.1007/s10910-023-01544-x
  • Gaunt, J. A. (1929). The triplets of Helium. Philosophical Transactions of the Royal Society of London Series A 228, 151-196. https://royalsocietypublishing.org/doi/10.1098/rsta.1929.0004
  • Weniger, E. J., & Steinborn, E. O. (1982). Programs for the coupling of spherical harmonics. Computer Physics Communications, 25, 149-157. https://doi.org/10.1016/0010-4655(82)90031-5
  • Xu, Y. L. (1998). Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories. Journal of Computational Physics, 139, 137-165. https://doi.org/10.1006/jcph.1997.586
  • Yükçü, S. A., Yükçü, N., & Öztekin, E. (2019). New representations for Gaunt coefficients. Chemical Physics Letters, 735, 136769. https://doi.org/10.1016/j.cplett.2019.136769
  • Rasch, J., & Yu, A. C. H. (2004). Efficient storage scheme for precalculated wigner 3j, 6j and Gaunt coefficients. SIAM Journal on Scientific Computing, 25, 1416-1428. https://doi.org/10.1137/S1064827503422932
  • Özay, S., Akdemir, S., & Öztekin, E. (2023). New orthogonality relationships for the Gaunt coefficients. http://dx.doi.org/10.2139/ssrn.4529971
  • Homeier, H. H. H., & Steinborn, E. O. (1996). Some properties of the coupling coefficients of real spherical harmonics and their relation to Gaunt coefficients. Journal of Molecular Structure: THEOCHEM, 368, 31-37. https://doi.org/10.1016/S0166-1280(96)90531-X
  • Sebilleau, D. (1998). On the computation of the integrated products of three spherical harmonics. Journal of Physics A: Mathematical and General, 31, 7157-7168. https://doi.org/10.1088/0305-4470/31/34/017
  • Pinchon, D., & Hoggan, P. E. (2007). New index functions for storing Gaunt coefficients. International Journal of Quantum Chemistry, 107, 2186-2196. https://doi.org/10.1002/qua.21337
  • Dunlap, B. I. (2002). Generalized Gaunt coefficients. Physical Review A, 66, 032502. https://doi.org/10.1103/PhysRevA.66.032502
  • Akın, E. (2016). Gaunt katsayılarının binom katsayıları kullanılarak hesaplanması. Selçuk Üniversitesi Fen Fakültesi Fen Dergisi, 42 (2), 129-135.
  • Arfken, G. B., & Weber, H. J. (2005). Mathematical Methods for Physicists. Academic Press, London.

Açısal Momentum Merdiven İşlemcileri ile Gaunt Katsayılarının Cebirsel Çözümü

Yıl 2023, , 229 - 244, 28.12.2023
https://doi.org/10.33484/sinopfbd.1358148

Öz

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.

Kaynakça

  • Griffiths, D. J. (2017). Introduction to Quantum Mechanics (2nd ed.). Cambridge University Press, Cambridge.
  • McQuarrie, D. A. (2008). Quantum Chemistry (2nd ed.). University Science Books, California.
  • Condon, E. U., & Shortley, G. H. (1935). Theory of Atomic Spectra. Cambridge University Press, Cambridge.
  • Cohen-Tannoudji, C., Diu, B., & Laloe, F. (1977). Quantum Mechanics. John Wiley & Sons, New York.
  • Townsend, J. S. (2000). A Modern Approach to Quantum Mechanics. University Science Books, USA.
  • Zettili, N. (2009). Quantum Mechanics: Concepts and Applications. John Wiley & Sons, USA.
  • Weitzman, M., & Freericks, J. K. (2018). Calculating spherical harmonics without derivatives. Condensed Matter Physics, 21 (3), 1-12. https://doi.org/10.5488/CMP.21.33002
  • Edmonds, A. R. (1960). Angular Momentum in Quantum Mechanics (2nd ed.). Princeton University Press, New Jersey.
  • Rose, M. E. (1957). Elementary Theory of Angular Momentum. John Wiley & Sons, New York.
  • Zare, R. N. (1988). Angular Momentum, Understanding Spatial Aspects in Chemistry and Physics. John Wiley & Sons, New York.
  • Varshalovich, D. A., Moskalev, A. N., & Khersonskii, V. K. (1988). Quantum Theory of Angular Momentum, Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols. World Scientific Publishing Co. Pte. Ltd, Singapore.
  • Shimpuku, T. (1960). General theory and numerical tables of clebsch-gordan coefficients. Supplement of the Progress of Theoretical Physics, 13, 1-135. https://doi.org/10.1143/PTPS.13.1
  • Tarter, C. B. (1970). Coefficients connecting the stark and field-free wavefunctions for hydrogen. Journal of Mathematical Physics, 11, 3192-3195. http://dx.doi.org/10.1063/1.1665113
  • Schulten, K., & Gordon, R. G. (1976). Recursive evaluation of 3j and 6j coefficients. Computer Physics Communications, 11, 269-278. https://doi.org/10.1016/0010-4655(76)90058-8
  • Lai, S. T., & Chiu, Y. N. (1990). Exact computation of the 3-j and 6-j symbols. Computer Physics Communications, 61, 350-360. https://doi.org/10.1016/0010-4655(90)90049-7
  • Guseinov, I. I., Özmen, A., Atav, Ü., & Yüksel, H. (1995). Computation of clebsch-gordan and gaunt coefficients using binomial coefficients. Journal of Computational Physics, 122, 343-347. https://doi.org/10.1006/jcph.1995.1220
  • Wei, L. (1999). Unified approach for exact calculation of angular momentum coupling and recoupling coefficients. Computer Physics Communications, 120, 222-230. https://doi.org/10.1016/S0010-4655(99)00232-5
  • Pain, J. -C. (2020). Some properties of Wigner 3 j coefficients: non-trivial zeros and connections to hypergeometric functions. The Europan Physics Journal A, 56:296, 1-13. https://doi.org/10.1140/epja/s10050-020-00303-9
  • Akdemir, S., Özay, S., & Öztekin E. (2023). Asymptotic behavior of clebsch-gordan coefficients. Journal of Mathematical Chemistry, https://doi.org//10.1007/s10910-023-01544-x
  • Gaunt, J. A. (1929). The triplets of Helium. Philosophical Transactions of the Royal Society of London Series A 228, 151-196. https://royalsocietypublishing.org/doi/10.1098/rsta.1929.0004
  • Weniger, E. J., & Steinborn, E. O. (1982). Programs for the coupling of spherical harmonics. Computer Physics Communications, 25, 149-157. https://doi.org/10.1016/0010-4655(82)90031-5
  • Xu, Y. L. (1998). Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories. Journal of Computational Physics, 139, 137-165. https://doi.org/10.1006/jcph.1997.586
  • Yükçü, S. A., Yükçü, N., & Öztekin, E. (2019). New representations for Gaunt coefficients. Chemical Physics Letters, 735, 136769. https://doi.org/10.1016/j.cplett.2019.136769
  • Rasch, J., & Yu, A. C. H. (2004). Efficient storage scheme for precalculated wigner 3j, 6j and Gaunt coefficients. SIAM Journal on Scientific Computing, 25, 1416-1428. https://doi.org/10.1137/S1064827503422932
  • Özay, S., Akdemir, S., & Öztekin, E. (2023). New orthogonality relationships for the Gaunt coefficients. http://dx.doi.org/10.2139/ssrn.4529971
  • Homeier, H. H. H., & Steinborn, E. O. (1996). Some properties of the coupling coefficients of real spherical harmonics and their relation to Gaunt coefficients. Journal of Molecular Structure: THEOCHEM, 368, 31-37. https://doi.org/10.1016/S0166-1280(96)90531-X
  • Sebilleau, D. (1998). On the computation of the integrated products of three spherical harmonics. Journal of Physics A: Mathematical and General, 31, 7157-7168. https://doi.org/10.1088/0305-4470/31/34/017
  • Pinchon, D., & Hoggan, P. E. (2007). New index functions for storing Gaunt coefficients. International Journal of Quantum Chemistry, 107, 2186-2196. https://doi.org/10.1002/qua.21337
  • Dunlap, B. I. (2002). Generalized Gaunt coefficients. Physical Review A, 66, 032502. https://doi.org/10.1103/PhysRevA.66.032502
  • Akın, E. (2016). Gaunt katsayılarının binom katsayıları kullanılarak hesaplanması. Selçuk Üniversitesi Fen Fakültesi Fen Dergisi, 42 (2), 129-135.
  • Arfken, G. B., & Weber, H. J. (2005). Mathematical Methods for Physicists. Academic Press, London.
Toplam 31 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Klasik Fizik (Diğer)
Bölüm Araştırma Makaleleri
Yazarlar

Selda Akdemir 0000-0002-5487-8703

Yayımlanma Tarihi 28 Aralık 2023
Gönderilme Tarihi 10 Eylül 2023
Yayımlandığı Sayı Yıl 2023

Kaynak Göster

APA Akdemir, S. (2023). Algebraic Solution of Gaunt Coefficients via the Angular Momentum Ladder Operators. Sinop Üniversitesi Fen Bilimleri Dergisi, 8(2), 229-244. https://doi.org/10.33484/sinopfbd.1358148


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