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

TL;DR

This paper presents a unified pedagogical framework that reformulates stationary perturbation theory using time-ordered correlation functions, simplifying calculations of van der Waals interactions at both short and long distances.

Contribution

It introduces a reformulation of stationary perturbation theory in terms of time-ordered correlation functions, facilitating higher-order calculations and unifying short- and long-distance van der Waals interactions.

Findings

Simplifies mathematical treatment of van der Waals interactions.

Provides a unified framework connecting London and Casimir-Polder regimes.

Enhances conceptual understanding for advanced quantum mechanics education.

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.