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Investigations of a one dimensional system of "charged" bosons undergoing delta-function interactions.

T. W. Craig

Published Jan 1, 1994• 0 citations
OpenAlex

Abstract

A one-dimensional system consisting of equal numbers of two types of bosons undergoing interactions through delta function potentials is examined. All particles have the same mass but each boson carries a "charge" that distinguishes one boson type from the other. The interactions are such that like particles repel and unlike attract. The amount of attraction and/or repulsion is determined by two adjustable parameters so that the amount of repulsion is not necessary equal to that of attraction.

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A one-dimensional system consisting of equal numbers of two types of bosons undergoing interactions through delta function potentials is examined. All particles have the same mass but each boson carries a "charge" that distinguishes one boson type from the other. The interactions are such that like particles repel and unlike attract. The amount of attraction and/or repulsion is determined by two adjustable parameters so that the amount of repulsion is not necessary equal to that of attraction.
For NNN particles with equal mass, interacting with repulsive or attractive $\ensuremath{\delta}$-function interaction of the same strength, the SSS matrix is explicitly given and shown to be symmetrical and unitary. The incoming and outgoing states may consist of bound compounds as well as single particles. The momenta of the particles and compounds are not changed in the scattering, but particles are exchanged, such as $ABC+DE\ensuremath{\rightarrow}ADC+BE$. Only distinguishable particles are considered.
For NNN particles with equal mass, interacting with repulsive or attractive $\ensuremath{\delta}$-function interaction of the same strength, the SSS matrix is explicitly given and shown to be symmetrical and unitary. The incoming and outgoing states may consist of bound compounds as well as single particles. The momenta of the particles and compounds are not changed in the scattering, but particles are exchanged, such as $ABC+DE\ensuremath{\rightarrow}ADC+BE$. Only distinguishable particles are considered.
In this paper it is shown that several cases of one-dimensional N-body problems are exactly soluble. The first case describes the motion of three one-dimensional particles of arbitrary mass which interact with one another via infinite-strength, repulsive delta-function potentials. It is found in this case that the stationary-state solution of the scattering of the three particles is analogous to an electro-magnetic diffraction problem which has already been solved. The solution to this analogous electro-magnetic problem is interpreted in terms of particles. Next it is shown that the problem of three particles of equal mass interacting with each other via finite- but equal-strength delta-function potentials is exactly soluble. This example exhibits rearrangement and bound-state effects, but no inelastic processes occur. Finally it is shown that the problem of N particles of equal mass all interacting with one another via finite- but equal-strength delta functions is exactly soluble. Again no inelastic processes occur, but various types of rearrangements and an N-particle bound state do occur. These rearrangements and the N-particle bound state are illustrated by means of a series of sample calculations.