Scale Dependence of the Retarded van der Waals Potential

TL;DR

This paper investigates how the van der Waals potential between two hydrogen atoms transitions from the non-retarded to the retarded regime as their separation increases, explicitly characterizing the crossover function in the quantum electrodynamics framework.

Contribution

It explicitly determines the crossover function from the $R^{-6}$ to the $R^{-7}$ potential in the quantum electrodynamics setting, considering the limit of small coupling constant and increasing interatomic distance.

Findings

Explicit form of the crossover function between van der Waals and retarded potentials.

Identification of the scale at which the transition occurs, proportional to $ ext{alpha}^{-2} R$.

Analysis of the ground state energy in the combined limit of small coupling and large separation.

Abstract

We study the ground state energy for a system of two hydrogen atoms coupled to the quantized Maxwell field in the limit $\alpha \to 0$ together with the relative distance between the atoms increasing as $\alpha^{-\gamma} R$, $\gamma > 0$. In particular we determine explicitly the crossover function from the $R^{-6}$ van der Waals potential to the $R^{-7}$ retarded van der Waals potential, which takes place at scale $\alpha^{-2} R$.