Quantum confinement in 1D systems through an imaginary-time evolution method

TL;DR

This paper introduces a reliable imaginary-time evolution method to numerically solve the Schrödinger equation for quantum confinement in 1D systems, accurately computing eigenstates and energies for various potentials and confinement geometries.

Contribution

It presents a novel application of imaginary-time evolution combined with expectation value minimization for quantum confinement problems, achieving high accuracy and broad applicability.

Findings

Accurate eigenvalues and eigenfunctions for harmonic, repulsive, and quartic oscillators.

Effective handling of symmetric and asymmetric confinement scenarios.

Excellent agreement with existing literature results.

Abstract

Quantum confinement is studied by numerically solving time-dependent Schr\"odinger equation. An imaginary-time evolution technique is employed in conjunction with the minimization of an expectation value, to reach the global minimum. Excited states are obtained by imposing the orthogonality constraint with all lower states. Applications are made on three important model quantum systems, namely, harmonic, repulsive and quartic oscillators; enclosed inside an impenetrable box. The resulting diffusion equation is solved using finite-difference method. Both symmetric and asymmetric confinement are considered for attractive potential; for others only symmetrical confinement. Accurate eigenvalue, eigenfunction and position expectation values are obtained, which show excellent agreement with existing literature results. Variation of energies with respect to box length is followed for small, intermediate and large sizes. In essence a simple accurate and reliable method is proposed for confinement in quantum systems.