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The Journal of Chemical Physics

Wave model for conservative bound systems

Alexandru Popa

Published Jun 22, 2005• 10 citations
Publisher

Abstract

In the hidden variable theory, Bohm proved a connection between the Schrodinger and Hamilton-Jacobi equations and showed the existence of classical paths, for which the generalized Bohr quantization condition is valid. In this paper we prove similar properties, starting from the equivalence between the Schrodinger and wave equations in the case of the conservative bound systems. Our approach is based on the equations and postulates of quantum mechanics without using any additional postulate. Like in the hidden variable theory, the above properties are proven without using the approximation of geometrical optics or the semiclassical approximation. Since the classical paths have only a mathematical significance in our analysis, our approach is consistent with the postulates of quantum mechanics.

DOI
10.1063/1.1943387

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In the hidden variable theory, Bohm proved a connection between the Schrodinger and Hamilton-Jacobi equations and showed the existence of classical paths, for which the generalized Bohr quantization condition is valid. In this paper we prove similar properties, starting from the equivalence between the Schrodinger and wave equations in the case of the conservative bound systems. Our approach is based on the equations and postulates of quantum mechanics without using any additional postulate. Like in the hidden variable theory, the above properties are proven without using the approximation of geometrical optics or the semiclassical approximation. Since the classical paths have only a mathematical significance in our analysis, our approach is consistent with the postulates of quantum mechanics.
It is well known that the Schrödinger equation is equivalent to the wave equation for conservative bound quantum systems. Consequently, the motion of such a system is represented—from a mathematical point of view—by the motion of the characteristic surface of the wave equation. In this paper we present a demonstration of the periodic motion of the characteristic surface. It results that the normal curves of the characteristic surface are periodic solutions of the Hamilton–Jacobi equation written for the same system. This leads to a direct connection between the periodic solutions of the Hamilton–Jacobi equation and the wave properties of the system. The constants of motion corresponding to the above periodic solutions of the Hamilton–Jacobi equation are identical to the eigenvalues of the Schrödinger equation. These properties are proved without any approximation and they are valid for all the values of the principal quantum number.
We have demonstrated in previous papers the following property of closed, conservative and bounded systems: The energy which results from the Schrödinger equation can be rigorously calculated by line integrals of analytical functions, if the Hamilton-Jacobi equation, written for the same system, is satisfied in the space of coordinates by a periodical trajectory. In the present article, we show that this property is connected to the intrinsic wave properties of the system. This results from the equivalence between the Schrödinger equation and the wave equation, valid for conservative systems. As a consequence of the wave properties of the system, we show that the Hamilton-Jacobi equation has always periodical solutions, whose constants of motion are identical to the eigenvalues of the Schrödinger equation, written for the same system. It results that the calculation model presented in previous papers is generally valid in the case of closed, conservative and bounded systems. We present the applications of the model to the nitrogen and oxygen atoms, to the ions with the same structure, and to the He 2 , Be 2 and B 2 molecules.
Striking parallelism between classical mechanics and quantum mechanics is evidenced for several bound system in a rather general form by introducing reasonably “quantum-mechanical momentum” from the Schrödinger equation. The following are proved: The “quantum-mechanical momentum” which is nearly equal to the corresponding classical momentum in “classical region” can be defined in the whole range of validity of the coordinate for the bound systems. In the same way as in classical mechanics, in quantum mechanics one can construct an associated wave accompanying a motion of a quantum particle and obeying the Schrödinger equation. The uniqueness condition on the associated wave throughout an orbit leads to the boundary condition on the wave function which is composed of the associated wave and to “quantum-mechanical quantization condition”, equivalently. In addition, all these quantum-mechanical quantities are approximately equal to the corresponding classical ones in the “classical region” of the coordinate. In particular it is demonstrated that even for a bound system of a singular (central) potential exists the ground state with the lowest energy eigenvalue. It is proved in one dimension that to any physical system obeying the Schrödinger equation always exists the corresponding classical system.
It is shown how the time-dependent Schrödinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton–Jacobi equation of classical mechanics. Schrödinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton–Jacobi equation, are also reviewed. The derivation of the time-dependent equation is based on an a priori assumption equivalent to Feynman's dynamical postulate. de Broglie's concepts of ‘matter waves’ and their phase and group velocities are also critically discussed.
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.
Abstract 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.