# Note on the Retarded van der Waals Potential within the Dipole Approximation

## TL;DR

This paper rigorously analyzes the retarded van der Waals potential within the dipole approximation in nonrelativistic QED, confirming the $R^{-7}$ behavior of the binding energy at large distances, supporting Casimir-Polder's conjecture.

## Contribution

It provides a rigorous diagonalization of the dipole approximated Hamiltonian and proves the $R^{-7}$ decay of the binding energy for large atomic separations.

## Key findings

- Binding energy behaves as $R^{-7}$ at large distances
- Employs Feynman's representation for rigorous diagonalization
- Supports the Casimir-Polder conjecture

## Abstract

We examine the dipole approximated Pauli-Fierz Hamiltonians of the nonrelativistic QED. We assume that the Coulomb potential of the nuclei together with the Coulomb interaction between the electrons can be approximated by harmonic potentials. By an exact diagonalization method, we prove that the binding energy of the two hydrogen atoms behaves as $R^{-7}$, provided that the distance between atoms $R$ is sufficiently large. We employ the Feynman's representation of the quantized radiation fields which enables us to diagonalize Hamiltonians, rigorously. Our result supports the famous conjecture by Casimir and Polder.