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

## TL;DR

This paper presents an exact solution for a two-particle quantum system in three dimensions with harmonic confinement and finite-range soft-core interaction, using separation of variables and special functions.

## Contribution

It introduces an analytical approach to solve the Schrödinger equation for two interacting particles in 3D harmonic traps with finite-range interactions, extending previous models.

## Key findings

- Solutions expressed via confluent hypergeometric functions.
- Mapping to 1D harmonic oscillator in absence of central potential.
- Eigenvalue equations derived from Weber's differential equation.

## 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.