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Вестник Российского университета дружбы народов. Серия: Математика, информатика, физика

The quantization of the classical two-dimensional Hamiltonian systems

Irina N. Belyaeva

Published Feb 25, 2022• 0 citations

Abstract

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.

DOI: 10.22363/2658-4670-2022-30-1-39-51

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Abstract One of the fundamental problems in quantum mechanics is finding the correct quantum image of a classical observable that would correspond to experimental measurements. We investigate for the appropriate quantization rule that would yield a Hamiltonian that obeys the quantum analogue of Hamilton’s equations of motion, which includes differentiation of operators with respect to another operator. To give meaning to this type of differentiation, Born and Jordan established two definitions called the differential quotients of first type and second type. In this paper we modify the definition for the differential quotient of first type and establish its consistency with the differential quotient of second type for different basis operators corresponding to different quantizations. Theorems and differentiation rules including differentiation of operators with negative powers and multiple differentiation were also investigated. We show that the Hamiltonian obtained from Weyl, simplest symmetric, and Born-Jordan quantization all satisfy the required algebra of the quantum equations of motion.

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It is well known from the work of Berezin [24] (Commun. Math. Phys. 40:153–174, 1975) in 1975 that the quantization problem of a classical mechanical system is closely related with coherent states. In particular coherent states help to understand the limiting behaviour of a quantum system when the Planck constant

\hbar

becomes negligible in macroscopic units. This problem is called the semi-classical limit problem. In this chapter we discuss properties of quantum systems when the configuration space is the Euclidean space

\mathbb R^n

. So that in the Hamiltonian formalism, the phase space is

\mathbb R^n\times \mathbb R^n

with its canonical symplectic form

\sigma

. The quantization problem has many solutions, so we choose a convient one, introduced by Weyl [259] (The Classical Groups, 1997) and Wigner [261] (Group Theory and Its Applications to Quantum Mechanics of Atomic Spectra, 1959). We study the symmetries of Weyl quantization, the operationnal calculus and applications to propagation of observables. We show that Wick quantization is a natural bridge between Weyl quantization and coherent states. Applications are given to the semi-classical limit after introducing an efficient modern tool: the semi-classical measures. We illustrate the general results proved in this chapter by explicit computations for the harmonic oscillator. More applications will be given in the following chapters, in particular concerning propagators and trace formulas for a large class of quantum systems.

A normal form transformation is carried out on one-dimensional quantum Hamiltonians that transforms them into functions of the quantum harmonic oscillator. The method works with the Weyl transform (or 'symbol') of the Hamiltonian. The Moyal star product is used to carry out the normal form transformation at the level of symbols. Diagrammatic techniques are developed for handling the expressions that result from higher order terms in the Moyal series. Once the normal form is achieved, the Bohr–Sommerfeld formula for the eigenvalues, including higher order corrections, follows easily.

We develop a quantization method, that we name decomposable Weyl quantization, which ensures that the constants of motion of a prescribed finite set of Hamiltonians are preserved by the quantization. Our method is based on a structural analogy between the notions of reduction of the classical phase space and diagonalization of selfadjoint operators. We obtain the spectral decomposition of the emerging quantum constants of motion directly from the quantization process. If a specific quantization is given, we expect that it preserves constants of motion exactly when it coincides with decomposable Weyl quantization on the algebra of constants of motion. We obtain a characterization of when such property holds in terms of the Wigner transforms involved. We also explain how our construction can be applied to spectral theory. Moreover, we discuss how our method opens up new perspectives in formal deformation quantization and geometric quantization.