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Journal of the Physical Society of Japan

Applications of a Property of the Schrödinger Equation to the Modeling of Conservative Discrete Systems. III

Alexandru Popa

Published Sep 15, 1999• 12 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.68.2923

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Stationary states General properties We shall consider first some general characteristics of the ‘stationary states’. These are eigenfunctions of the Hamiltonian operator: The requirement that ψ satisfies the Schrödinger equation fixes its time dependence: Notice that (4.1.2) can only be a solution if the external potential V is independent of the time: V = V ( x ). The initial function ψ 0 is not arbitrary but must satisfy the time-independent Schrödinger equation (4.1.1). Writing ψ = R e iS /ħ one thus has The following deductions may be made from (4.1.3): ( a ) The probability density is independent of the time: |ψ| 2 = R 2 0 ( X ). ( b ) The quantum potential Q = (- ħ2/2 mR 0 )∇ 2 R 0 is time-independent. Therefore so is the total effective potential: ( c ) The velocity field v = (l/ m )∇ S 0 is independent of time and hence the trajectories, found by solving m = v ( x )| x = x ( t ) , are solutions of Moreover, ψ 0 ( x ) is often a real function for many bound state problems of interest (although not for scattering states) which implies that the velocity is zero. The particle is at rest where one would classically expect it to move since the quantum force (- ∇ Q ) cancels the classical force (- ∇ K ). ( d ) The energy of all particles in the ensemble, -∂ S /∂ t , is a constant of the motion and equal to the energy eigenvalue: with (4.1.4), we have an example of the conservation of energy (cf. §3.9.3). ( e ) The total field energy is conserved: 〈Ĥ〉 = E . […]

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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.
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.
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.
In previous papers we proved the existence of exact connections between quantum and classical equations in the cases of two different quantum systems, namely the stationary atomic and molecular systems and the systems composed by electrons and very intense electromagnetic field. We proved that the geometrical elements of the wave described by the Schrodinger equation, namely the characteristic surfaces and curves, denoted, respectively, by Σ and C, are solutions of the HamiltonJacobi equations, written for the same system, in the case of multidimensional stationary systems. The Σ surfaces, which have the significance of wave surfaces, and the C curves, which are their normals, correspond to the same constants of motion as the eigenvalues of the Schrodinger equation. In three recent papers we presented a central field method for the calculation of the C curves, and of the corresponding energetic values. We have verified this method for a large number of atoms and diatomic molecules, and we found that its accuracy is comparable to the accuracy of the Hartree-Fock method applied to the same atoms and diatomic molecules. In another papers we proved that the Klein-Gordon equation for the system electron-very intense electromagnetic field is verified exactly by the function associated to the classical motion of the electron. An exact connection follows between the solutions of the Klein-Gordon and the relativistic Hamilton-Jacobi equations, for that system. This result explains the accuracy of numerous classical models from the literature, belonging to the new field studying the interactions between very intense laser beams and electron plasmas. We present now a short review of these results.
It is well known in classical mechanics that the frequencies of a periodic system can be obtained rather easily through the action variable, without completely solving the equation of motion. Analogously, the equivalent quantum action variable appearing in the quantum Hamilton–Jacobi formalism can provide the energy eigenvalues of a bound state problem, without the necessity of solving the corresponding Schrödinger equation explicitly. This elegant and useful method is elucidated here in the context of some known and not so well-known solvable potentials. It is also shown how this method provides an understanding as to why approximate quantization schemes such as ordinary and supersymmetric WKB can give exact answers for certain potentials.
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.
Stationary states General properties We shall consider first some general characteristics of the ‘stationary states’. These are eigenfunctions of the Hamiltonian operator: The requirement that ψ satisfies the Schrödinger equation fixes its time dependence: Notice that (4.1.2) can only be a solution if the external potential V is independent of the time: V = V ( x ). The initial function ψ 0 is not arbitrary but must satisfy the time-independent Schrödinger equation (4.1.1). Writing ψ = R e iS /ħ one thus has The following deductions may be made from (4.1.3): ( a ) The probability density is independent of the time: |ψ| 2 = R 2 0 ( X ). ( b ) The quantum potential Q = (- ħ2/2 mR 0 )∇ 2 R 0 is time-independent. Therefore so is the total effective potential: ( c ) The velocity field v = (l/ m )∇ S 0 is independent of time and hence the trajectories, found by solving m = v ( x )| x = x ( t ) , are solutions of Moreover, ψ 0 ( x ) is often a real function for many bound state problems of interest (although not for scattering states) which implies that the velocity is zero. The particle is at rest where one would classically expect it to move since the quantum force (- ∇ Q ) cancels the classical force (- ∇ K ). ( d ) The energy of all particles in the ensemble, -∂ S /∂ t , is a constant of the motion and equal to the energy eigenvalue: with (4.1.4), we have an example of the conservation of energy (cf. §3.9.3). ( e ) The total field energy is conserved: 〈Ĥ〉 = E . […]