# 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^{1}, N.I. Korsunov^{1}, N.A. Chekanov^{1}, A.N. Chekanov^{2}

^{1} Belgorod National Research University^{2} Belgorod Law Institute of Ministry of the Interior of Russia

**DOI:** 10.26456/pcascnn/2023.15.255

* Short communication*

**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

**Reference: **

**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): **
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