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Acta Phys. Polon.

THE PROBLEM OF THE SUBSIDIARY CONDITION IN THE ADDITIONAL VARIABLES METHOD FOR ARBITRARY CENTRAL INTERACTIONS

Zygmunt M. Galasiewicz

Published Jan 1, 1958• 0 citations
OpenAlex

Abstract

The method of Bohm and Pines is generalized for the case of arbitrary central forces. It is shown that subsidiary conditions imposed on the system wave funetion result from the equations of motion. In addition, the equivalence of the generalized method and the method of Zubarev is shown. From this follows the physical significance of the subsidiary conditions imposed on the system wave function in this method. (auth)

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The method of Bohm and Pines is generalized for the case of arbitrary central forces. It is shown that subsidiary conditions imposed on the system wave funetion result from the equations of motion. In addition, the equivalence of the generalized method and the method of Zubarev is shown. From this follows the physical significance of the subsidiary conditions imposed on the system wave function in this method. (auth)
We develope a collective description of the many-particle system interacting through an arbitrary two-body potential. The description is given both in terms of the second quantized form and in terms of an auxiliary field analogous to the longitudinal electromagnetic field. As Bohm-Pines theory of the collective description has some unsatisfactory features concerning the subsidiary condition imposed on the auxiliary field, we propose here a strict method of treatment on the subsidiary condition. To obtain the useful result of the collective representation, we have to take the three steps of the canonical transformations. Though the first two transformations are the same as those of B-P, the third transformation, which is not applied by B-P and is characteristic of our theory, is necessary for obtaining a meaningful result. By using both the strict subsidiary condition and the three canonical transformations, the collective field coordinate and the particle coordinate are separated not only in the total Hamiltonian but also in the subsidiary condition. Moreover, we are able to clarify the correct meaning of the subsidiary condition and clarify also the connections between our theory and the theory of Sawada-Brueckner-Fukuda-Brout.
A method for describing the motion of many-body systems that interact solely through two-body forces is introduced. The general form of the Hamiltonian is found, and applied to the case of particles interacting through Hooke's law forces. The general form of the wave function is exhibited and the means of antisymmetrization discussed. It is shown that the kinetic energy of the center of mass can be separated from the Hamiltonian by transforming to a new set of external (pair c.m.) coordinates. A particular approximation to the system boundary conditions is applied to light nuclei, and some of the implications of the use of current nucleon-nucleon forces discussed.
The normalization problem of a wave function for an arbitrary one-dimensional field is considered. It is shown, that for an infinite motion type the magnitude of the normalization constant does not depend on the form of a scattering potential and it is determined by the boundary conditions imposed on the character of the investigated motion. The general formula for the normalization constant as a function of amplitudes of the waves converging to the potential is derived. Keyword: Normalization problem, wave function, one-dimensional motion, arbitrary potential PACS: 03.65.Nk, 02.30.Em
A study is made of a system of two particles with spin in relativistic functional theory. Equation of the barycentric wave, equations of the relative motion, and the reduced equation are obtained. In the states of the system, in some cases there exist states where the relative waves are constants. This case corresponds to vanishing relative motion. In the case of two particles of spin 1/ 2 of the same mass, a fused system is obtained which is equivalent to a particle with spin 1. From this result, by the approximation giving the usual wave mechanics, the equations given by the fusion method are obtained. The fusion method is thus justified and its physical significance is apparent. (auth.)