Deciphering the coalescence behavior of Coulomb-Schr\"odinger atomic wave functions from an operator product expansion

TL;DR

This paper uses operator product expansion within nonrelativistic effective field theory to analyze atomic wave function coalescence, reproducing known cusp conditions and enabling extensions to multi-particle and relativistic cases.

Contribution

It introduces a formalism based on OPE to rigorously analyze coalescence behavior of atomic wave functions, including the Kato cusp condition, with potential for broader applications.

Findings

Reproduces Kato's cusp condition using OPE.

Proves an exact OPE relation to all orders in perturbation theory.

Framework can be extended to multi-particle and relativistic effects.

Abstract

We revisit the coalescence behavior of the atomic Schr\"odinger wave functions from the angle of an operator product expansion (OPE) within the nonrelativistic Coulomb-Schr\"odinger effective field theory. We take the electron-nucleus coalescence as an explicit example to demonstrate our formalism, where the celebrated Kato's cusp condition can be easily reproduced. An exact OPE relation is rigorously proved to all orders in perturbation theory. Our approach can be readily extended to ascertain the multi-particle coalescence behaviors of atomic wave functions, as well as to take relativistic effects into account.