Wide effectiveness of a sine basis for quantum-mechanical problems in d dimensions

TL;DR

This paper demonstrates that sine basis functions, originally from quantum particle-in-a-box problems, are highly effective for a wide range of quantum systems, including those with complex potentials and singularities, across multiple dimensions.

Contribution

It introduces a scaled sine basis with variational parameters as a versatile tool for solving diverse quantum-mechanical problems beyond traditional confinement scenarios.

Findings

Sine basis functions are effective for general quantum problems.

Scaling with variational parameters improves accuracy.

Applicable to systems with singular potentials and multiple dimensions.

Abstract

It is shown that the spanning set for L^2([0, 1]) provided by the eigenfunctions {sqrt{2} sin(n\pi x)}_{n=1}^{\infty} of the particle-in-a-box in quantum mechanics provide a very effective variational basis for more general problems. The basis is scaled to [a,b], where a and b are then used as variational parameters. What is perhaps a natural basis for quantum systems confined to a spherical box in R^d, turns out to be appropriate also for problems that are softly confined by U-shaped potentials, including those with strong singularities at r=0. Specific examples are discussed in detail, along with some bound N-boson systems