On the exact discretization of Schr\"odinger equation

TL;DR

This paper derives the exact discrete form of the Schr"odinger equation from the Hamiltonian of a Schr"odinger field theory using discrete Fourier transforms, comparing it with the standard discretization through numerical simulations.

Contribution

It introduces an exact discretization method for the Schr"odinger equation based on Hamiltonian operators and Fourier transforms, highlighting differences from the standard finite difference approach.

Findings

Exact discretization derived from Hamiltonian and Fourier transform.

Standard finite difference discretization describes a different theory.

Both discretizations produce accurate results, with the exact method being more computationally intensive.

Abstract

We show that the exact discrete analogue of Schr\"odinger equation can be derived naturally from the Hamiltonian operator of a Schr\"odinger field theory by using the discrete Fourier transform that transforms the operator from momentum representation into position representation. The standard central difference equation that is often used as the discretized Schr\"odinger equation actually describes a different theory since it is derived from a different Hamiltonian operator. The commutator relation between the position and momentum operators in discrete space is also derived and found to be different from the conventional commutator relation in continuous space. A comparison between the two discretization formulas is made by numerically studying the transmission probability for a wave packet passing through a square potential barrier in one dimensional space. Both discretization formulas are shown to give sensible and accurate numerical results as compared to theoretical calculation, though it takes more computation time when using the exact discretization formula. The average wave number $k_0$ of the incident wave packet must satisfy $|k_0\ell| < 0.35$, where $\ell$ is the lattice spacing in position space, in order to obtain an accurate numerical result by using the standard central difference formula.