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Van der Waals interaction at short and long distances: a pedagogical path from stationary to time-dependent perturbation theory

L. Saba, C. D. Fosco

TL;DR

The paper addresses how to unify the London ($R^{-6}$) and Casimir–Polder ($R^{-7}$) regimes of van der Waals interactions within a single framework by recasting stationary perturbation theory in terms of time-ordered correlation functions. It introduces an imaginary-time, time-dependent formulation that sums all orders via a generating functional, while staying consistent with a Coulomb-gauge, instantaneous description and a dipole/multipole expansion. The approach recovers London’s result at short range and, with retardation included through photon propagators, yields the Casimir–Polder asymptotics, providing a coherent bridge between regimes and a pedagogical tool for advanced quantum mechanics courses. The work offers a unified, systematic method to treat dispersion forces across regimes and highlights the conceptual clarity gained by treating higher-order corrections through time-ordered correlations.

Abstract

The van der Waals interaction between neutral atoms is typically studied using stationary perturbation theory for the short-distance (London) limit, while long-distance (Casimir-Polder) results are usually derived via semiclassical, time-dependent approaches. In this pedagogical article, we demonstrate that reformulating stationary perturbation theory calculations in terms of time-ordered correlation functions significantly simplifies the mathematical treatment. This reformulation is particularly advantageous for higher-order calculations required in the long-distance regime, where retardation effects become important. Our approach provides a unified framework that connects both limiting cases while offering a clear conceptual picture suitable for advanced quantum mechanics courses.

Van der Waals interaction at short and long distances: a pedagogical path from stationary to time-dependent perturbation theory

TL;DR

The paper addresses how to unify the London () and Casimir–Polder () regimes of van der Waals interactions within a single framework by recasting stationary perturbation theory in terms of time-ordered correlation functions. It introduces an imaginary-time, time-dependent formulation that sums all orders via a generating functional, while staying consistent with a Coulomb-gauge, instantaneous description and a dipole/multipole expansion. The approach recovers London’s result at short range and, with retardation included through photon propagators, yields the Casimir–Polder asymptotics, providing a coherent bridge between regimes and a pedagogical tool for advanced quantum mechanics courses. The work offers a unified, systematic method to treat dispersion forces across regimes and highlights the conceptual clarity gained by treating higher-order corrections through time-ordered correlations.

Abstract

The van der Waals interaction between neutral atoms is typically studied using stationary perturbation theory for the short-distance (London) limit, while long-distance (Casimir-Polder) results are usually derived via semiclassical, time-dependent approaches. In this pedagogical article, we demonstrate that reformulating stationary perturbation theory calculations in terms of time-ordered correlation functions significantly simplifies the mathematical treatment. This reformulation is particularly advantageous for higher-order calculations required in the long-distance regime, where retardation effects become important. Our approach provides a unified framework that connects both limiting cases while offering a clear conceptual picture suitable for advanced quantum mechanics courses.
Paper Structure (10 sections, 82 equations, 2 figures)

This paper contains 10 sections, 82 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The system
  3. Coulomb gauge and instantaneous approximation
  4. Coulomb gauge
  5. The instantaneous approximation
  6. Perturbative corrections to the vacuum energy
  7. Introducing a time-dependent formulation for higher orders
  8. Including retardation
  9. Conclusions
  10. Vacuum energy in Rayleigh--Schrödinger perturbation theory

Figures (2)

  • Figure 1: Numerical evaluation of Eq. (\ref{['eq:E_I_result']}) shown on a log--log scale. The thick curve plots the dimensionless combination $-E_I(r)\, r^6/A^2$ versus $r\equiv\Omega R/c$, where $A\equiv q^2\Omega/(4\pi m)$. Thin dashed lines show the expected behavior if $E_I$ followed pure power laws: constant for $E_I \propto r^{-6}$ (London regime, short distances) and $\propto r^{-1}$ for $E_I \propto r^{-7}$ (Casimir--Polder regime, large distances).
  • Figure 2: Logarithmic slope extracted from Fig. \ref{['fig:E4']}: $\mathrm{d}\log[-E_I(r)\,r^6]/\mathrm{d}\log r$. The curve interpolates between the London exponent $-6$ for $r\ll 1$ and the Casimir--Polder exponent $-7$ for $r\gg 1$.