On the four-body problem in the Born-Oppenheimer approximation

C. A. Escobar; A. Martín-Ruiz

doi:10.1016/j.aop.2020.168311

paper

On the four-body problem in the Born-Oppenheimer approximation

arXiv:2007.14948 · doi:10.1016/j.aop.2020.168311

Abstract

The quantum problem of four particles in $R^{d}$ ( $d \geq 3$ ), with arbitrary masses $m_{1}, m_{2}, m_{3}$ and $m_{4}$ , interacting through an harmonic oscillator potential is considered. This model allows exact solvability and a critical analysis of the Born-Oppenheimer approximation. The study is restricted to the ground state level. We pay special attention to the case of two equally heavy masses $m_{1} = m_{2} = M$ and two light particles $m_{3} = m_{4} = m$ . It is shown that the sum of the first two terms of the Puiseux series, in powers of the dimensionless parameter $σ = \frac{m}{M}$ , of the exact phase $Φ$ of the wave function $ψ_{0} = e^{- Φ}$ and the corresponding ground state energy $E_{0}$ , coincide exactly with the values obtained in the Born-Oppenheimer approximation. A physically relevant rough model of the $H_{2}$ molecule and of the chemical compound $H_{2} O_{2}$ (Hydrogen peroxide) is described in detail. The generalization to an arbitrary number of particles $n$ , with $d$ degrees of freedom ( $d \geq n - 1$ ), interacting through an harmonic oscillator potential is briefly discussed as well.