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Journal of Mathematical Physics

Ground State of a One-Dimensional N -Body System

Francesco Calogero

Published Dec 1, 1969• 550 citations
Publisher

Abstract

The problem of N quantum-mechanical equal particles interacting pairwise by inverse-cube forces (``centrifugal potential'') in addition to linear forces (``harmonical potential'') is considered in a onedimensional space. An explicit expression for the ground-state energy and for the corresponding wavefunction is exhibited. A class of excited states is similarly displayed.

DOI
10.1063/1.1664821

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American Journal of Physics•Mar 1, 1972•14 citations
The problem of N quantum-mechanical equal particles interacting pairwise by inverse-cube forces (centrifugal potential'') in addition to linear forces (harmonical potential'') is considered in a onedimensional space. An explicit expression for the ground-state energy and for the corresponding wavefunction is exhibited. A class of excited states is similarly displayed.
We exhibit the exact ground state and the corresponding binding energy of the system composed of NNN (nonidentical) quantum-mechanical particles interacting in one dimension via the potential ${V}_{\mathrm{ij}}({x}_{i}\ensuremath{-}{x}_{j})=g{({x}_{i}\ensuremath{-}{x}_{j})}^{\ensuremath{-}2}+({f}_{i}\ensuremath{-}{f}_{j}){({x}_{i}\ensuremath{-}{x}_{j})}^{\ensuremath{-}1}$. We discuss some special choices of the constants fi{f}_{i}fi​ and ggg that yield examples of possible physical interest.
The quantum-mechanical problems of N 1-dimensional equal particles of mass m interacting pairwise via quadratic (``harmonical'') and/or inversely quadratic (``centrifugal'') potentials is solved. In the first case, characterized by the pair potential ¼mω2(xi − xj)2 + g(xi − xj)−2, g > −ℏ2/(4m), the complete energy spectrum (in the center-of-mass frame) is given by the formula E=ℏω(12N)12[12(N−1)+12N(N−1)(a+12)+ ∑ l=2Nlnl],with a = ½(1 + 4mgℏ−2)½. The N − 1 quantum numbers nl are nonnegative integers; each set {nl; l = 2, 3, ⋯, N} characterizes uniquely one eigenstate. This energy spectrum can also be written in the form Es = ℏω(½N)½ [½(N − 1) + ½N(N − 1)(a + ½) + s], s = 0, 2, 3, 4, ⋯, the multiplicity of the sth level being then given by the number of different sets of N − 1 nonnegative integers nl that are consistent with the condition s=∑l=2Nlnl. These equations are valid independently of the statistics that the particles satisfy, if g ≠ 0; for g = 0, the equations remain valid with a = ½ for Fermi statistics, a = −½ for Bose statistics. The eigenfunctions corresponding to these energy levels are not obtained explicitly, but they are rather fully characterized. A more general model is similarly solved, in which the N particles are divided in families, with the same quadratic interaction acting between all pairs, but with the inversely quadratic interaction acting only between particles belonging to the same family, with a strength that may be different for different families. The second model, characterized by the pair potential g(xi − xj)−2, g > −ℏ2/(4m), contains only scattering states. It is proved that an initial scattering configuration, characterized (in the phase space sector defined by the inequalities xi ≥ xi.1, i = 1, 2, ⋯, N = 1, to which attention may be restricted without loss of generality) by (initial) momenta pi, i = 1, 2, ⋯, N, goes over into a final configuration characterized uniquely by the (final) momenta pi′, with pi′=pN+1−i. This remarkably simple outcome is a peculiarity of the case with equal particles (i.e., equal masses and equal strengths of all pair potentials).
We construct a new many-body Hamiltonian with two- and three-body interactions in two space dimensions and obtain its exact many-body ground state for an arbitrary number of particles. This ground state has a novel pairwise correlation. A class of exact solutions for the excited states is also found. These excited states display an energy spectrum similar to the Calogero-Sutherland model in one dimension. The model reduces to an analog of the well-known trigonometric Sutherland model when projected on to a circular ring.