SEMI-CLASSICAL CALCULATIONS OF ENERGY LEVELS AND WAVE FUNCTIONS OF HAMILTONIAN SYSTEMS WITH ONE AND SEVERAL DEGREES OF FREEDOM BASED ON THE METHOD OF CLASSICAL AND QUANTUM NORMAL FORMS
- Authors: Belyaeva I.N.1, Korsunov N.I.1, Chekanov N.A.1, Chekanov A.N.2
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Affiliations:
- Belgorod National Research University
- Putilin Belgorod Law Institute of Ministry of the Interior of Russia
- Issue: No 15 (2023)
- Pages: 255-263
- Section: Theory of nanosystems
- URL: https://journal-vniispk.ru/2226-4442/article/view/378455
- DOI: https://doi.org/10.26456/pcascnn/2023.15.255
- EDN: https://elibrary.ru/ZJROOB
- ID: 378455
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Abstract
The paper presents two schemes for the sequential construction of the classical normal form and its quantum analogue for some classes of classical Hamiltonian systems. For quantum normal forms, a method for solving their eigenvalue problem is indicated. Based on these normal forms, a semi-classical method for solving Schrodinger equations for classical Hamiltonian systems under their quantum consideration is proposed. With this proposed method, some quantum problems were solved and it was found that this method gives a very accurate prediction for energy levels. However, this accuracy in the field of the existence of classical chaos is deteriorating. The same semiclassical method solved the quantum problem for a flat hydrogen atom in a homogeneous magnetic field. The proposed method allows carrying out all calculations using modern computer systems of analytical calculations.
About the authors
Irina N. Belyaeva
Belgorod National Research University
Email: ibelyaeva@bsu.edu.RUS
Belgorod, Russia
Nikolay I. Korsunov
Belgorod National Research UniversityBelgorod, Russia
Nikolay A. Chekanov
Belgorod National Research UniversityBelgorod, Russia
Aleksandr N. Chekanov
Putilin Belgorod Law Institute of Ministry of the Interior of RussiaBelgorod, Russia
References
- Арнольд, В.И. Дополнительные главы теории обыкновенных уравнений / В.И. Арнольд. - М.: Наука, 1978. - 304 с.
- Дирак, П.А.М. К созданию квантовой теории поля: основные статьи 1925-1958 годов / П.А.М. Дирак; пер. с анг. и фр. под ред. Б.В. Медведева. - М.: Наука, 1990. - 368 с.
- Джакалья, Г.Е.О. Методы теории возмущений для нелинейных систем / Г.Е.О. Джакалья; пер. с анг. под ред. А.П. Маркеева. - М.: Наука, 1979. - 320 с.
- Гребеников, Е.А. Метод усреднения в прикладных задачах / Е.А. Гребеников. - М.: Наука, 1986.- 256 с.
- Маркеев, А.П. Точки либрации в небесной механике и космодинамике / А.П. Маркеев. - М.: Наука, 1978. - 312 с.
- Биркгофф, Дж.Д. Динамические системы / Дж.Д. Биркгофф. - Москва-Ижевск: НИЦ "Регулярная и хаотическая динамика", 2002. - 406 с.
- Богачев, В.Е. MAPLE программа вычисления нормальной формы Биркгофа-Густавсона и независимых интегралов движения для гамильтоновой системы с произвольным числом степеней свободы / В.Е. Богачев, Н.А. Чеканов. Зарегистрировано в Отраслевом фонде алгоритмов и программ.- М.: ВНТИЦ, 2011. - №2011616109.
- Basios, V. GITA: A REDUCE program for the normalization of polynomial Hamiltonian / V. Basios, N.A. Chekanov, B.L. Markovski, V.A. Rostovtsev, S.I. Vinitsky // Computer Physics Communications. - 1995. - V. 90. I. 2-3.- P. 355-368. doi: 10.1016/0010-4655(95)00080-Y.
- Лихтенберг, А. Регулярная и стохастическая динамика / А. Лихтенберг, М. Либерман. - М.: Мир, 1984. -528 с.
- Gustavson, F.G. On constructing formal integrals of a Hamiltonian systems near an equilibrium point / F.G. Gustavson // The Astronomical Journal. - 1966. - V.71. - № 8. - P. 670-686. doi: 10.1086/110172.
- Swimm, R.T. Semiclassical calculations of vibrational energy levels for nonseparable systems using Birkhoff-Gustavson normal form / R.T. Swimm, J.B. Delos // The Journal of Chemical Physics. - 1979. - V. 71. - I. 4. - P. 1706-1717. doi: 10.1063/1.438521.
- Banerjee, K. General anharmonic oscillators / K. Banerjee // Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences. - 1978. - V. 364. - I. 1717. - P. 265-275. doi: 10.1098/rspa.1978.0200.
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