Improvement of the matching of the exact solution and variational approaches in an interacting two-fermion system

TL;DR

This paper develops an improved trial wave function for a two-fermion system in a harmonic potential, achieving better agreement with analytical solutions by refining boundary conditions and variational parameters.

Contribution

It introduces a more realistic trial wave function with continuous boundary conditions and practical constraints, enhancing the accuracy of variational approximations for the system.

Findings

Energy estimates are closer to analytical solutions.

Wave function matches boundary conditions more accurately.

Method improves variational approach for interacting fermions.

Abstract

A more reasonable trial ground state wave function is constructed for the relative motion of an interacting two-fermion system in a 1D harmonic potential. At the boundaries both the wave function and its first derivative are continuous and the quasi-momentum is determined by a more practical constraint condition which associates two variational parameters. The upper bound of the ground state energy is obtained by applying the variational principle to the expectation value of the Hamiltonian of relative motion on the trial wave function. The resulted energy and wave function show better agreement with the analytical solution than the original proposal.