Many-Body Problem: Pairwise Harmonic Forces

Frederick William Schmidlin

James S. Bell

Francesco Calogero

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.

John W. Norbury

A solution to quantum mechanical problems of two, three, and four bodies is discussed from the point of view of two-body harmonic oscillator forces. An exact solution to the three- and four-body problems is found for harmonic oscillator forces, assuming equal masses. The aim of the discussion is to make students aware of the real difficulties of the few-body problem.

G. E. Brown

Kawon Lee, Sangeun Oh, Young Woo Choi +1 more

Exchange statistics endows identical particles with an effective "statistical potential", whose familiar exact form is two-body and purely repulsive for fermions. Here we construct an exact collective many-body form: the thermodynamics of $N$ trapped ideal fermions maps onto classical distinguishable particles governed by a single potential -- exactly for harmonic confinement at all temperatures, and to leading semiclassical order for arbitrary potentials. The associated force separates canonically into pairwise contributions, which for $N\geq 3$ can turn attractive, governed by a simple geometric criterion. Classical minimization reproduces observed few-body Pauli-crystal symmetries and agrees with the $N=55$ ground-state probability maximum at sub-percent shell accuracy. Heating drives discrete structural transitions accompanied by a crossover of the strongest force from attractive to repulsive. Both the potential and its forces are directly computable from existing single-shot imaging data, turning quantum exchange into measurable classical mechanics.