Two quantum particles trapped in three dimensions harmonic oscillator and interacting via finite range soft-core interaction

Abstract: We study the exactly solvable quantum system of two particles confined in a three-dimensional harmonic trap and interacting via finite-range soft-core interaction by means of the separation of variables and ansatz method. Supposing the solution in the form of ansatz $\Psi(r)=r^{a}e^{-\lambda r^2} \psi(r)$ we transform the time independent Schr\"{o}dinger equation into Kummer's differential equation whose solution are given in the form of confluent hypergeometric function. We also discuss that in the absence of central force potential, the quantum system map into the problem of two quantum particle trapped in one-dimension harmonic oscillator and interacting through finite distance soft-core potential. In such special case the eigen value equation become the Weber's differential equation and its solution are also given in the form of confluent hypergeometric function.