Numerical approach to the Schrodinger equation in momentum space

TL;DR

This paper introduces a numerical method for solving the time-independent Schrödinger equation in momentum space by discretizing the Hamiltonian, enabling exploration of bound states in localized potentials.

Contribution

It presents a novel numerical approach to the momentum-space Schrödinger equation, which is less commonly taught and analytically challenging.

Findings

Effective for calculating bound states in localized potentials

Complements real-space Schrödinger equation methods

Demonstrated through multiple example cases

Abstract

The treatment of the time-independent Schrodinger equation in real-space is an indispensable part of introductory quantum mechanics. In contrast, the Schrodinger equation in momentum space is an integral equation that is not readily amenable to an analytical solution and is rarely taught. We present a numerical approach to the Schrodinger equation in momentum space. After a suitable discretization process, we obtain the Hamiltonian matrix and diagonalize it numerically. By considering a few examples, we show that this approach is ideal for exploring bound-states in a localized potential and complements the traditional (analytical or numerical) treatment of the Schrodinger equation in real-space.