Exact solutions of the Schrödinger equation for some quantum-mechanical many-body systems

Abstract

We have solved exactly the Schr\"odinger equation in hyperspherical coordinates for some quantum-mechanical many-body systems. These systems, which interact with Coulomb, inverse-square, harmonic-oscillator, dipole, or Yukawa potentials, etc., can be solved in a similar way. Wave functions are expanded into orthonormal complete basis sets of the hyperspherical harmonics of hyperangles and generalized Laguerre polynomials of the hyperradius. The eigenvalues can be obtained explicitly by solving a simple secular equation. The results of practical calculations of Coulomb potential systems, such as H, He, and ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$, etc., agree well with the reported exact ones.