An exact analytical scheme using a new potential to solve one-dimensional quantum systems

TL;DR

This paper introduces an exact analytical method for solving one-dimensional Schrödinger equations by representing arbitrary potentials as collections of short-width potentials, enabling efficient computation of eigenenergies and transmission properties.

Contribution

The paper presents a novel solvable potential based on wavefunction expansion, improving computational efficiency and analytical solvability for 1D quantum systems.

Findings

Successfully reproduces rectangular potential results

Offers computational advantages over existing schemes

Provides a Mathematica code for arbitrary potentials

Abstract

We propose an exact method for solving a one-dimensional Schr\"odinger equation. An arbitrary potential is represented by the collection of short-width potentials. For building the collection scheme, a new solvable potential is introduced. It is based on the simple expansion of the wavefunction of the introduced potential. The illustration of the scheme is done by reproducing the results of the rectangular potential. The scheme has computational advantages and the transmission properties, eigenenergies can be calculated efficiently. The presented scheme is compared with the other similar schemes in terms of computational complexity, analytical solubility, etc.. A \textit{Mathematica} code is provided in the supplementary file that solves the Schr\"odinger equation with arbitrary potential function $V(x)$ and effective mass $m(x)$.