Split Kinetic Energy Method for Quantum Systems with Competing Potentials

TL;DR

This paper introduces a novel kinetic energy dissection method for solving quantum systems with competing potentials, offering faster convergence and improved accuracy over traditional perturbation techniques.

Contribution

A new energy dissection approach that partitions kinetic energy to better handle quantum systems with competing potentials, enhancing convergence and accuracy.

Findings

Faster convergence to exact solutions compared to perturbation theory.

Effective for systems with strong coupling, such as charged harmonic oscillators and hydrogen molecules.

Applicable to a broad class of quantum problems with competing potentials.

Abstract

For quantum systems with competing potentials, the conventional perturbation theory often yields an asymptotic series and the subsequent numerical outcome becomes uncertain. To tackle such kind of problems, we develop a general solution scheme based on a new energy dissection idea. Instead of dividing the potential energy into "unperturbed" and "perturbed" terms, a partition of the kinetic energy is performed. By distributing the kinetic energy term in part into each individual potential, the Hamiltonian can be expressed as the sum of the subsystem Hamiltonians with respective competing potentials. The total wavefunction is expanded by using a linear combination of the basis sets of respective subsystem Hamiltonians. We first illustrate the solution procedure using a simple system consisting of a particle under the action of double delta-function potentials. Next, this method is applied to the prototype systems of a charged harmonic oscillator in strong magnetic field and the hydrogen molecule ion. Compared with the usual perturbation approach, this new scheme converges much faster to the exact solutions for both eigenvalues and eigenfunctions. When properly extended, this new solution scheme can be very useful for dealing with strongly coupling quantum systems.