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Elsevier eBooks

WITHDRAWN: Study of Exactly Soluble One-Dimensional N-Body Problems

J. B. McGuire

Published Jan 1, 1966• 0 citations
Publisher
DOI
10.1016/b978-0-12-448750-5.50045-1

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Study of Exactly Soluble One-Dimensional N-Body Problems

James McGuire

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.

Journal of Mathematical Physics•May 1, 1964•745 citations
We introduce and solve a lattice version of an easily soluble one-dimensional continuum quantum system with a periodic potential. The relationship of the one problem to the other is identical to the relationship of the one-dimensional quantum many-body lattice gas recently introduced by Shastry [Phys. Rev. Lett. 60, 639 (1988)] and by Haldane [Phys. Rev. Lett. 60, 635 (1988)] to the one-dimensional quantum many-body continuum gas earlier solved by Sutherland. Thus it is hoped that by understanding the "missing" states in this simple case, we might better understand the missing states in the more complicated many-body case, which in many instances comprise all the states. The results are not encouraging.
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.