The kinetic energy partition method applied to a confined quantum harmonic oscillator in a one-dimensional box

Abstract

A new, physically motivated, basis set expansion method for solving quantum eigenvalue problems with competing interaction potentials is presented. In contrast to the usual dissection of the potential energy into unperturbed and perturbing terms, we divide the kinetic energy into partial terms by modifying the mass factor. The partition scheme results in partial kinetic energies with their effective mass factors. By distributing each partial kinetic energy to a respective potential energy to form a subsystem, the total Hamiltonian is written as the sum of subsystem Hamiltonians. Using a linear combination of the subsystem wave-functions to represent the system wave-function we obtain a set of coupled equations for the expansion coefficients, by solving these energies and wave-functions can be obtained. We demonstrate the solution scheme with a standard model system: a confined harmonic oscillator in a one-dimensional box. With only a few (less than ten) basis functions from each subsystem, we can reproduce the exact solutions very accurately, thus showing the applicability of this method.