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Communications in Analysis and Mechanics

Quantization of Hamiltonian and non-Hamiltonian systems

Sergey A. Rashkovskiy

Published Jan 1, 2023• 2 citations
Publisher

Abstract

The quantization process was always tightly connected to the Hamiltonian formulation of classical mechanics. For non-Hamiltonian systems, traditional quantization algorithms turn out to be unsuitable. Numerous attempts to quantize non-Hamiltonian systems have shown that this problem is nontrivial and requires the development of new approaches. In this paper, we present the quantization methods that do not depend upon the Hamiltonian formulation of classical mechanics. Two approaches to the quantization of mechanical systems are considered: axiomatic and hydrodynamic. It is shown that the formal application of these approaches to the classical Hamilton-Jacobi theory allows obtaining the wave equation for the corresponding quantum system in natural way. Examples are considered that show the effectiveness of the proposed approaches, both for Hamiltonian and non-Hamiltonian systems. The spinor form of the relativistic Hamilton-Jacobi theory for classical particles is considered. It is shown that it naturally leads to the Dirac equation for the corresponding quantum particle and to its non-Hamiltonian generalization, the bispinor relativistic Kostin equation.

DOI
10.3934/cam.2023014

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The quantization process was always tightly connected to the Hamiltonian formulation of classical mechanics. For non-Hamiltonian systems, traditional quantization algorithms turn out to be unsuitable. Numerous attempts to quantize non-Hamiltonian systems have shown that this problem is nontrivial and requires the development of new approaches. In this paper, we present the quantization methods that do not depend upon the Hamiltonian formulation of classical mechanics. Two approaches to the quantization of mechanical systems are considered: axiomatic and hydrodynamic. It is shown that the formal application of these approaches to the classical Hamilton-Jacobi theory allows obtaining the wave equation for the corresponding quantum system in natural way. Examples are considered that show the effectiveness of the proposed approaches, both for Hamiltonian and non-Hamiltonian systems. The spinor form of the relativistic Hamilton-Jacobi theory for classical particles is considered. It is shown that it naturally leads to the Dirac equation for the corresponding quantum particle and to its non-Hamiltonian generalization, the bispinor relativistic Kostin equation.
The quantization method based on the quantum Hamiltonian Jacobi equation, is extended to two-dimensional non-separable but integrable Hamiltonians. It is shown that each wave function for those systems corresponds to a well-defined family of classical trajectories, enveloped by a caustic. The energy eigenvalues and the values of the wave functions on the caustic are obtained by solving the 1-dim quantum Hamilton Jacobi equation of the caustic' arcs. Results are in good agreement with those obtained by usual methods.
The numerical version of the Hamilton-Jacobi quantization method, recently proposed, is applied to the one dimensional quartic oscillator. A suitable quantization condition is formulated and various energy levels and wave functions are computed. The results very well agree with those obtained by means of the Schroedinger equation, and confirm that the Quantum Hamilton Jacobi approach, which is the exact version of the semiclassical WKB scheme, is a self-contained quantization procedure, equivalent and independent from the Schoedinger's one but more general. Indeed, with respect to this latter, the Quantum Hamilton-Jacobi equation can be used to investigate the limit h->0, where the Schroedinger equation loses its significance, and explains how the fundamental quantities of the classical mechanics, the Hamilton's characteristic function and the classical momentum, are generated from the corresponding quantum ones.
Abstract A free particle may be characterized abstractly in terms of its invariance group. Both the Hamilton-Jacobi and Schrödinger equations for classical and quantum free particles are invariant under the same group, the Schrödinger group. The aim of this paper is to discuss possible restrictions on non-linear generalizations of quantum mechanics. The basic idea is to search for those non-linear generalization of quantum mechanics for which a description of the non-relativistic free particle includes a representation of the Schrödinger group, as in classical and quantum mechanics. This condition imposes strong constraints on possible models of free particles. To carry out such an investigation it is necessary to select an appropriate mathematical formalism, one which can describe physical systems of non-relativistic particles and allows for the incorporation of non-linearities in a natural way. Here the formalism of ensembles on configuration space is used, which is an approach that is capable of describing both classical and quantum systems and, in addition, allows for more general theories that describe mixed classical-quantum systems. Modifications of the equations involving derivatives of up to fourth order are considered
The Schrodinger variational approach (1926) to quantization of the natural Hamilton mechanics in 2n2n2n-dimensional phase space is revised in the modern paradigm of quantum mechanics in application to the system the Hamilton function of which is a positive-definite quadratic form of the n momenta with the coefficients depending on coordinates in the generic configuration space VnV_nVn​. The quantum Hamilonian thus obtained has a paradoxical potential term depending on choice of coordinates in VnV_nVn​, which was discovered first by B. DeWitt in 1952 in the framework of canonical quantization of the system by a particular ordering of the operators of observables of momenta and coordinates. It is shown that the Schrodinger approach in the standard paradigm of quantum mechanics determines uniquely the ordering selected by DeWitt among a continuum of other possibilties to determine the quantum Hamiltonian. Two particular classes of observables of localization in VnV_nVn​ are considered in detail. It is shown that, in general, the quantum-mechanical potential does not vanish even in the Euclidean configuration space VnV_nVn​ except the case when Cartesian coordinates are taken as the observables of localization of the system. It is noted also that, in the quasi-classical approach to the quantization considered by DeWitt in 1957, the quantum Green function (propagator) is also non-unique and depends on the choice of a line in VnV_nVn​ connecting these points. All three formalisms have the same local asymtotics of the quantum Hamiltonian if the normal Riemannian cordinates are used, at least implicitly, to localize the system in VnV_nVn​. Keywords: Hamilton function; Quantization; Quantum-Mechanical Potential; Observables of Localization; Quantum Anomaly of (non-relativistic) General Covariance.
In this initial paper in a series, we first discuss why classical motions of small particles should be treated statistically. Then we show that any attempted statistical description of any nonrelativistic classical system inevitably yields the multi-coordinate Schrödinger equation, with its usual boundary conditions, as an essential statistical equation for the system. We derive the general "canonical quantization" rule, that the Hamiltonian operator must be the classical Hamiltonian in the NNN-dimensional metric configuration space defined by the classical kinetic energy of the system, with the classical conjugate momentum NNN-vector replaced by −iℏ-i\hbar−iℏ times the vector gradient operator in that space. We obtain these results by using conservation of probability, general tensor calculus, the Madelung transform, the Ehrenfest theorem and/or the Hamilton-Jacobi equation, and comparison with results for the charged harmonic oscillator in stochastic electrodynamics. We also provide two illustrative examples and a discussion of how coordinate trajectories could be compatible with wave properties such as interference, diffraction, and tunneling.
The Hamilton-Jacobi equation of classical mechanics is approached as a model reduction of conservative particle mechanics where the velocity degrees-of-freedom are eliminated. This viewpoint allows an extension of the association of the Hamilton-Jacobi equation from conservative systems to general Newtonian particle systems involving non-conservative forces, including dissipative ones. A geometric optics approximation leads to a dissipative Schrödinger equation, with the expected limiting form when the associated classical force system involves conservative forces.