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Journal of Mathematical Physics

An exactly solvable one-dimensional three-body problem with hard cores

Robert Nyden Hill

Published May 1, 1980• 4 citations
Publisher

Abstract

Three identical particles in one dimension interact via a potential which is infinite whenever one or more of the interparticle separations is less than a or greater than b, and zero when all interparticle separations lie between a and b. Their Schrödinger equation is solved by reducing it to the exactly solvable problem of the two dimensional Helmholtz equation inside an equilateral triangle.

DOI
10.1063/1.524556

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Three identical particles in one dimension interact via a potential which is infinite whenever one or more of the interparticle separations is less than a or greater than b, and zero when all interparticle separations lie between a and b. Their Schrödinger equation is solved by reducing it to the exactly solvable problem of the two dimensional Helmholtz equation inside an equilateral triangle.
An exactly soluble one-dimensional three-body problem is presented, in which the interaction between the particles consists of local two-body potentials between each two particles. Infinitely high step functions are chosen for the form of the three potential functions. This interaction allows only three-body bound states and no continuum states. We have considered three different choices of the mass ratios of the three particles and we give formulas in closed form for the energies and for the wavefunctions of all states.
We extend to the N-body case previous techniques for solving three-body problems with repulsive interparticle potentials and periodic boundary conditions on each particle. For clarity, we begin with one-dimensional problems, although the techniques are not peculiar to them and can be generalized to three dimensions. We decompose the wavefunction ψ into N(N − 1)/2 parts, according to Ψ= ∑ i<j=1Nψij;from the Schrödinger equation we find a basic equation for a typical ψij. As a test of this equation, we apply it for N = 4 and the case of δ-function interactions, and solve it numerically to find close agreement between the energy per particle of a four-body system and a system of the same density in which N → ∞. The numerical results are consistent with the analytic one that Eg(∞), which is the ground-state energy per particle of the system with an infinite number of particles, is related to Eg(N), which is that energy for the N-body system of the same density, by Eg(N) = Eg(∞)[1 − (1/N)] for weak repulsion and by Eg(N) = Eg(∞)[1 − (1/N2)] for very strong repulsion. We begin a similar comparison for the three-dimensional hard-sphere case by working out the problem of two hard spheres with periodic boundary conditions over a length L2 such that the density is the same as for an N-body problem in volume V=LN3:2/L23=N/LN3. We find that we get the analog of the one-dimensional result, for which case Eg(2) = Eg(∞)/2 for N = 2. That is, we find Eg(2) = πa2h/mv, which is just one-half of the many-body theory result. We also calculate the two-body correlation function and find good agreement with the many-body one, as calculated by Lee, Huang, and Yang.
We consider the quantum-mechanical problem of a particle bound to a configuration of spherical potentials, each of finite range. If the Schr\"odinger equation can be solved for each potential by itself, then it is shown how to solve it for the configuration, provided the potentials do not overlap. The energy levels are the zeros of a determinant of formally infinite order, but in practice this is always well approximated by a finite determinant and often by one of small order. As illustrative examples we consider some states of a particle bound to three square wells with the configuration of an equilateral triangle, and to two truncated Coulomb potentials. The possibility of extending an approximate version of the method to overlapping potentials is pointed out.