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 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.
It is shown that it is possible to construct the quantum wave functions for non-separable but integrable two-dimensional Hamiltonian systems, by solving suitable Dirichlet boundary values problems inside and outside the regions spanned by particular families of classical trajectories, in one-to-one correspondence with the quantum state. The method is applied both to the Schrodinger equation, and to the quantum Hamilton-Jacobi equation. The boundary values are obtained by integrating the one-dim equations on the caustics arcs enveloping the classical trajectories. This approach gives the same results as the usual methods, and furthermore clarifies the links between quantum and classical mechanics.
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.
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.
We present a classical and quantum analysis of a particle confined in a three-dimensional paraboloidal cavity formed by two confocal paraboloids. Classically, the system is integrable and presents three independent constants of motion, namely, the energy, the z-component of the angular momentum, and a third dynamical constant associated with the paraboloidal geometry, which can be derived from the separability of the Hamilton--Jacobi equation. We derive closed-form analytical expressions for the actions, which allow us to determine the two conditions to get periodic closed trajectories. We classify these trajectories through the indices (s,t,ℓ). The caustic paraboloids that bound the motion provide a complete geometric characterization of admissible trajectories. Quantum mechanically, separability of the Schrödinger equation in parabolic coordinates yields eigenmodes described by Whittaker functions. We determine the energy spectrum and identify degeneracies arising not only from azimuthal symmetry but also from specific cavity deformations. A direct correspondence between classical trajectories and quantum eigenstates reveals that probability densities concentrate in the classically allowed region with controlled penetration into forbidden zones.
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.
By means of numerical solutions of the quantum Hamilton Jacobi equation, a general WKB-like representation for one-dimensional wave functions is obtained. This representation is unique in the classically forbidden regions, while in the allowed one, each wave function corresponds to a one parameter family of solutions of the QHJE. The method has been applied to various systems, with different energies and initial conditions. In all investigated cases, the wave functions so obtained accurately reproduce the solutions of the Schrödinger equation, analytically or numerically computed by other ways. Some results for harmonic oscillator and radial Coulomb motion are presented.