The oscillator representation and the stability of three-body Coulomb systems

Abstract

Within nonrelativistic quantum mechanics the Wick-ordering method, called the oscillator representation, is suggested for calculating the energy spectrum for a wide class of potentials allowing the existence of a bound state. As test cases, anharmonic (V(r)=r 2σ) and screened Coulomb potentials are considered. In particular, the method is applied to three-body Coulomb systems to obtain the dependence of the bound-state energy on the masses and charges of the particles. The calculations of the bound-state energies for the moleculesH −=(pee),H 2 + =(ppe), (e −e−e+) and (ppμ), (ddμ), (dtμ) prove the accuracy of the zeroth approximation to be better than one per cent. For the three-body Coulomb system with charges +, −, − and arbitrary masses the region of stability is determined. For the systems (pe −C+), (A +e−e+), and (pB −e−) the critical masses are calculated to beM c=1.945me,M A=4.350me andM B=1.575me. It turns out that the system (pe −e+) is unstable.