Integral Coalescence Conditions In $D \geq 2$ Dimension Space

TL;DR

This paper derives integral and differential coalescence conditions for wavefunctions of charged particles in D-dimensional space, generalizing known 3D results and applying to fractional Quantum Hall Effect and electron pair correlations.

Contribution

It provides the first derivation of integral coalescence conditions in arbitrary D-dimensional space, extending known 3D results and linking them to pair-correlation functions.

Findings

Integral form of coalescence conditions derived for D ≥ 2.

Differential form reduces to known 3D conditions.

Laughlin wavefunction satisfies the node coalescence condition.

Abstract

We have derived the integral form of the cusp and node coalescence conditions satisfied by the wavefunction at the coalescence of two charged particles in $D \geq 2$ dimension space. From it we have obtained the differential form of the coalescence conditions. These expressions reduce to the well-known integral and differential coalescence conditions in $D = 3$ space. It follows from the results derived that the approximate Laughlin wavefunction for the fractional Quantum Hall Effect satisfies the node coalescence condition. It is further noted that the integral form makes evident that unlike the electron-nucleus coalescence condition, the differential form of the electron-electron coalescence condition cannot be expressed in terms of the electron density at the point of coalescence. From the integral form, the integral and differential coalescence conditions for the pair-correlation function in $D\geq 2$ dimension space are also derived. The known differential form of the pair function cusp condition for the uniform electron gas in dimensions $D = 2,3$ constitute a special case of the result derived.