Integral coalescence conditions in D⩾2, dimension space

Abstract

We have derived the integral form of the cusp and node coalescence conditions satisfied by the wave function at the coalescence of two charged particles in D⩾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 wave function 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⩾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 constitutes a special case of the result derived.