The quantization of the classical two-dimensional Hamiltonian systems

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The paper considers the class of Hamiltonian systems with two degrees of freedom. Based on the classical normal form, according to the rules of Born-Jordan and Weyl-MacCoy, its quantum analogs are constructed for which the eigenvalue problem is solved and approximate formulas for the energy spectrum are found. For particular values of the parameters of quantum normal forms using these formulas, numerical calculations of the lower energy levels were performed, and the obtained results were compared with the known data of other authors. It was found that the best and good agreement with the known results is obtained using the Weyl-MacCoy quantization rule. The procedure for normalizing the classical Hamilton function is an extremely time-consuming task, since it involves hundreds and even thousands of polynomials for the necessary transformations. Therefore, in the work, normalization is performed using the REDUCE computer algebra system. It is shown that the use of the Weyl-MacCoy and Born-Jordan correspondence rules leads to almost the same values for the energy spectrum, while their proximity increases for large quantities of quantum numbers, that is, for highly excited states. The canonical transformation is used in the work, the quantum analog of which allows us to construct eigenfunctions for the quantum normal form and thus obtain analytical formulas for the energy spectra of different Hamiltonian systems. So, it is shown that quantization of classical Hamiltonian systems, including those admitting the classical mode of motion, using the method of normal forms gives a very accurate prediction of energy levels.

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Introduction Representation of the original classical Hamilton function in normal form as the sum of homogeneous polynomials in canonically conjugate coordinates and momenta [1] allows us to carry out its quantum-mechanical description. The main provisions of the new quantum mechanics were discovered by W. Heisenberg in 1925 [2]. In the same year, the paper was published by © Belyaeva I.N., 2022 This work is licensed under a Creative Commons Attribution 4.0 International License M. Born and P. Jordan [3], in which the commutation relation for the quantummechanical operators of coordinate


About the authors

Irina N. Belyaeva

Belgorod State National Research University

Author for correspondence.

Candidate of Physical and Mathematical Sciences, Associate Professor

85, Pobedy St., Belgorod, 308015, Russian Federation


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