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American Journal of Physics

An Exactly Soluble Quantum-Mechanical Three-Body Problem

T. K. Lim

Published Aug 1, 1971• 6 citations
Publisher

Abstract

An exact solution is obtained for a model three-body system in which the particles interact through pairwise, harmonic oscillator, and spin-dependent potentials, the solution being achieved by a simple transformation to an appropriate coordinate system. In particular, it is shown that the S-state wave functions of the ground state of the triton span the two-dimensional representation of the symmetric group S3, are orthogonal, and resemble closely the Irving wave forms found so successful in variational calculations.

DOI
10.1119/1.1986328

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An exact solution is obtained for a model three-body system in which the particles interact through pairwise, harmonic oscillator, and spin-dependent potentials, the solution being achieved by a simple transformation to an appropriate coordinate system. In particular, it is shown that the S-state wave functions of the ground state of the triton span the two-dimensional representation of the symmetric group S3, are orthogonal, and resemble closely the Irving wave forms found so successful in variational calculations.
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.
The wave functions of boson and fermion gases are known even when the particles have harmonic interactions. Here we generalize these results by solving exactly the N–body Schrödinger equation for potentials V that can be any function of the sum of the squares of the distances of the particles from one another in three dimensions. For the harmonic case that function is linear in r2. Explicit N–body solutions are given when U(r) = –2Mh–2V(r) = 𝛇r–1 – 𝛇2r–2. Here M is the sum of the masses and r2 = 1/2 M–2ΣmImJ(xI – xJ)2. For general U(r) the solution is given in terms of the one– or two–body problem with potential U(r) in three dimensions. The degeneracies of the levels are derived for distinguishable particles, for bosons of spin zero and for spin 1/2 fermions. The latter involve significant combinatorial analysis which may have application to the shell model of atomic nuclei. For large N the fermionic ground state gives the binding energy of a degenerate white dwarf star treated as a giant atom with an N–body wave function. The N–body forces involved in these extraordinary N–body problems are not the usual sums of two–body interactions, but nor are forces between quarks or molecules. Bose–Einstein condensation of particles in three dimensions interacting via these strange potentials can be treated by this method.