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
I.N. Belyaeva1, N.I. Korsunov1, N.A. Chekanov1, A.N. Chekanov2
1 Belgorod National Research University
2 Belgorod Law Institute of Ministry of the Interior of Russia
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.
Keywords: classical normal form, quantum analog of normal form, Weyl-McCoy rule, energy levels, eigenfunctions, mathematical modeling
- Irina N. Belyaeva – Ph. D, Docent, Department of Computer Science, Natural Sciences and Teaching Methods, Belgorod National Research University
- Nikolay I. Korsunov – Dr. Sc., Professor, Department of Mathematical and Software Information Systems, Belgorod National Research University
- Nikolay A. Chekanov – Dr. Sc., Professor, Department of Applied Mathematics and Computer Modeling, Belgorod National Research University
- Aleksandr N. Chekanov – Senior Lecturer, Department of Security at Transport Facilities Putilin, Belgorod Law Institute of Ministry of the Interior of Russia
Belyaeva, I.N. 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 / I.N. Belyaeva, N.I. Korsunov, N.A. Chekanov, A.N. Chekanov // Physical and chemical aspects of the study of clusters, nanostructures and nanomaterials. — 2023. — I. 15. — P. 255-263. DOI: 10.26456/pcascnn/2023.15.255. (In Russian).
Full article (in Russian): download PDF file
1. Arnold V.I. Dopolnitel'nye glavy teorii obyknovennykh uravnenij [Additional chapters in the theory ofordinary equations], Moscow, Nauka Publ., 1978, 304 p. (In Russian).
2. Dirak P.A.M. K sozdaniyu kvantovoj teorii polya: osnovnye stat'i 1925-1958 godov [Toward the creation of a quantum field theory: main papers 1925-1958], trans. and ed. by B.V. Medvedev, Moscow, Nauka Publ., 1990, 368 p. (In Russian).
3. Giacaglia G.E.O. Perturbation Methods in Non-Linear Systems, New York, Springer, 1972, IX, 369 p. DOI: 10.1007/978-1-4612-6400-2.
4. Grebenikov E.A. Metod usredneniya v prikladnykh zadachakh [Averaging method in applied problems], Moscow, Nauka Publ., 1986, 256 p. (In Russian).
5. Markeev A.P. Tochki libratsii v nebesnoj mekhanike i kosmodinamike [Libration points in celestial mechanics and cosmodynamics], Moscow, Nauka Publ., 1978, 312 p. (In Russian).
6. Birkhoff G.D. Dynamical systems, Providence, Rhode Island, American Mathematical Society, 1927, 305 p.
7. Bogachev V.E., Chekanov N.A. MAPLE programma vychisleniya normalnoj formy Birkgofa-Gustavsona i nezavisimyh integralov dvizheniya dlya gamiltonovoj sistemy s proizvolnym chislom stepenej svobody [MAPLE program for calculating the Birkhoff-Gustavson normal form and independent integrals of motion for a Hamiltonian system with an arbitrary number of degrees of freedom], Moscow, The All-Russian Scientific and Technical Information Center, 2011, no. 2011616109. (In Russian).
8. Basios V., Chekanov N.A., Markovski B.L., Rostovtsev V.A., Vinitsky S.I. GITA: A REDUCE program for the normalization of polynomial Hamiltonian, Computer Physics Communications, 1995, V. 90, issue 2-3, pp. 355-368. DOI: 10.1016/0010-4655(95)00080-Y.
9. Lichtenberg A.J., Lieberman M.A. Regular and chaotic dynamics, New York, Heidelberg, Berlin, SpringerVerlag, 499 p.
10. Gustavson F.G. On constructing formal integrals of a Hamiltonian systems near an equilibrium point, The Astronomical Journal, 1966, V.71, no. 8, pp. 670-686. DOI: 10.1086/110172.
11. Swimm R.T., Delos J.B. Semiclassical calculations of vibrational energy levels for nonseparable systems using Birkhoff-Gustavson normal form, The Journal of Chemical Physics, 1979, vol. 71, issue 4, pp. 1706-1717. DOI: 10.1063/1.438521.
12. Banerjee K. General anharmonic oscillators, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 1978, vol. 364, issue 1717, pp. 265-275. DOI 10.1098/rspa.1978.0200.