Exact Analytical Solution of the Schroedinger Equation for an N-Identical Body-Force System

摘要

A mathematical method is presented for solving the Schroedinger equation for a system of identical body forces. The N-body forces are more easily introduced and treated within the hyperspherical harmonics. The problem of the N-body potential has been used at the level of both classical and quantum mechanics. The hypercentral interacting potential is assumed to depend on the hyper-radius x — (ξ_1~2 + ξ_2~2 + … + ξ_(N-1)~2)~(1/2) only, where ξ_1, ξ_2, …, ξ_(N-1) are Jacobi relative coordinates which are functions of N-particle relative positions r_(12), r_(23), …, R_(N1). The problem of the harmonic oscillator and the Coulomb-type potential has been widely studied in different contexts. Using the N-body potential V(x) = ax~2 + bx — (c/x) as an example, and assuming an ansatz for the eigenfunction, an exact analytical solution of the Schroedinger equation for an N-body system in three dimensions is obtained. This method is also applicable to some other types of potentials for N-identical interacting particles.