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Journal of Mathematical Physics

The Scattering of Three Impenetrable Particles in One Dimension

James McGuire, C. A. Hurst

Published Oct 1, 1972• 29 citations
Publisher

Abstract

A special one-dimensional quantum mechanical three-body problem with an interaction among the particles is solved exactly. The interaction is introduced in such a way that the particles retain their order along a line, and hence the particles are impenetrable. The assumption of impenetrability is consistant with the existence of bound states in all accessible channels, and the probabilities for direct and exchange scattering of one bound pair and one free particle are calculated both below and above the breakup threshold. Above the breakup threshold, the probabilities for breakup and free particle scattering are also found. The absolute squares of all elements of the scattering matrix are given explicitly, and the scattering matrix is shown to be unitary.

DOI
10.1063/1.1665884

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A special one-dimensional quantum mechanical three-body problem with an interaction among the particles is solved exactly. The interaction is introduced in such a way that the particles retain their order along a line, and hence the particles are impenetrable. The assumption of impenetrability is consistant with the existence of bound states in all accessible channels, and the probabilities for direct and exchange scattering of one bound pair and one free particle are calculated both below and above the breakup threshold. Above the breakup threshold, the probabilities for breakup and free particle scattering are also found. The absolute squares of all elements of the scattering matrix are given explicitly, and the scattering matrix is shown to be unitary.
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.
The exact solution is presented of the scattering problem of three equal particles interacting in one-dimension via two- and/or three-body inverse-square potentials. Both the classical and the quantal problems are treated. It is shown that the outcome of this scattering problem is an extremely simple relation between initial and final momenta, the latter being univocally determined by the former even in the quantal case. The solvability of the problem, and the simple results just mentioned, are peculiar to the equal particle case.
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.
So far, we have dealt with one- and two-body systems in a variety of settings. These are quite simple, at least a priori, since the equations needed to solve one- and twoparticle scattering problems, as well as bound states, are usually easy enough to handle. The situation becomes much more intricate in the three-body problem, as we shall see in this chapter. Firstly, while two particles in free space can either scatter or bind together, three bodies have extra channels (for attractive interactions) corresponding to two bound particles scattering off a third. Moreover, the energies of these extra channels can be degenerate with those of three unbound particles, and the total momentum of a pair within the triplet is not conserved. Fortunately, we will see that getting our hands dirty with the three-body problem pays off, since there are a number of exotic phenomena associated with the quantum three-body problem, for both bound and scattering states. In this chapter, we introduce the formalism necessary to investigate the three-body problem and study in detail some of the most striking phenomenology known to date associated with, at least, the simplest interactions.