A basis for variational calculations in d dimensions

TL;DR

This paper derives general formulas for matrix elements in variational calculations for quantum systems in multiple dimensions, enabling efficient analysis of complex potentials including singular and perturbed Coulomb types.

Contribution

It provides explicit expressions for matrix elements using Gamma functions for arbitrary dimensions, expanding the toolkit for variational methods in quantum mechanics.

Findings

Derived matrix element formulas for d > 1 dimensions

Applied formulas to various potentials including singular and perturbed Coulomb

Discussed significance of parameters t, p, and scale s in calculations

Abstract

In this paper we derive expressions for matrix elements (\phi_i,H\phi_j) for the Hamiltonian H=-\Delta+\sum_q a(q)r^q in d > 1 dimensions. The basis functions in each angular momentum subspace are of the form phi_i(r)=r^{i+1+(t-d)/2}e^{-r^p/2}, i >= 0, p > 0, t > 0. The matrix elements are given in terms of the Gamma function for all d. The significance of the parameters t and p and scale s are discussed. Applications to a variety of potentials are presented, including potentials with singular repulsive terms of the form b/r^a, a,b > 0, perturbed Coulomb potentials -D/r + B r + Ar^2, and potentials with weak repulsive terms, such as -g r^2 + r^4, g > 0.