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Revista Mexicana de Física

The conserved operators generated by a solution of the Schrödinger equation

G. F. Torres del Castillo, E. Navarro Morales

Published May 9, 2012• 0 citations
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Abstract

It is shown that, in a similar manner as a complete solution of the Hamilton-Jacobi equation for a system with n degrees of freedom yields 2n constants of motion, each solution of the Schrödinger equation containing n parameters leads to 2n operators that are constants of motion; these 2n operators form two sets of n mutually commuting operators.

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•Dec 1, 2019•0 citations
It is shown that, in a similar manner as a complete solution of the Hamilton-Jacobi equation for a system with n degrees of freedom yields 2n constants of motion, each solution of the Schrödinger equation containing n parameters leads to 2n operators that are constants of motion; these 2n operators form two sets of n mutually commuting operators.
We find the form of the potential depending on the coordinates and the time such that a solution, SSS, of the Hamilton--Jacobi equation yields an exact solution, exp⁡(iS/ℏ)\exp ({\rm i} S/\hbar)exp(iS/ℏ), of the corresponding Schrödinger equation.
An ordinary unambiguous integral representation for the finite propagator of a quantum system is found by starting of a privileged skeletonization of the functional action in phase space, provided by the complete solution of the Hamilton-Jacobi equation. This representation allows to regard the propagator as the sum of the contributions coming from paths where the momenta generated by the complete solution of the Hamilton-Jacobi equation are conserved -as it does happen on the classical trajectory-, but are not restricted to having the classical values associated with the boundary conditions for the original coordinates.
In quantum mechanics, as well as in classical mechanics, the number of variables involved in the equations of motion may be reduced by utilizing the constants of motion of the system. For a set of N particles moving in a field-free space there are six integrals if N is greater than two. These correspond to the laws of conservation of linear and of angular momentum. It is the purpose of this paper to eliminate the coordinates of rotation and of translation of the total system from the Schrodinger equation and to obtain a new set of differential equations involving only 3N-6 variables.1
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.