Historical research guide · 1122 · 1948

The Influence of Retardation on the London-van der Waals Forces

H. B. G. Casimir · D. Polder

In one minute

This is the paper that turned the van der Waals force into a vacuum effect, and it is the one Casimir wrote immediately before the parallel-plate result the whole Casimir programme is named for. London had already explained the weak attraction between two neutral atoms quantum-mechanically, with an energy falling as the inverse sixth power of the separation. Casimir and Polder ask what happens when the separation is large enough that the light carrying the interaction takes time to cross it. They work the problem in quantum electrodynamics, in two parts: first a single neutral atom facing a perfectly conducting plane, then the harder case of two atoms. Both give the same shape of answer. Retardation multiplies the unretarded energy by a correction factor that falls monotonically with distance — equal to one while the separation is short compared with the atomic wavelengths, and falling off as one over the separation once it is long. The force therefore weakens faster than London's law at long range, and the two limiting energies come out as clean closed formulas.

Connection

Chapter 2 needs the step where a chemistry-scale attraction between molecules becomes a property of the electromagnetic field in the space between bodies. This is that step: once the interaction is carried by a field with a finite signal speed, the answer depends on the modes in the gap, and the parallel-plate force follows from the same reasoning within the year.

From the source

The influence of retardation on the energy of interaction between two neutral atoms is investigated by means of quantum electrodynamics — that is, the problem is treated as a question about the electromagnetic field between the bodies, not as a question about the molecules alone.Abstract, first sentence
As a preliminary step, Part I treats the interaction between a neutral atom and a perfectly conducting plane, and finds that retardation reduces the interaction energy by a correction factor which decreases monotonically with increasing separation.Abstract, second sentence
That correction factor is equal to unity for separations small compared with the wavelengths corresponding to the atomic frequencies, and is proportional to the inverse of the separation for distances large compared with those wavelengths — so the short-range answer is the familiar unretarded one and the long-range answer is weaker.Abstract, third sentence
In the long-distance limit the atom-plane interaction energy is minus three h-bar c alpha divided by eight pi times the fourth power of the separation, where alpha is the static polarizability of the atom.Abstract, fourth sentence
The two-atom problem of Part II is much more difficult to handle mathematically, but the results are very similar: again a monotonically decreasing correction factor, unity at small distances and proportional to the inverse separation at large ones.Abstract, fifth and sixth sentences
In that limit the energy of interaction between two atoms is minus twenty-three h-bar c times the product of the two polarizabilities, divided by four pi times the seventh power of the separation — the inverse seventh power that replaces London's inverse sixth power once retardation is taken into account.Abstract, closing sentence

The way in

https://doi.org/10.1103/PhysRev.73.360

Historical source-access record. This guide links to the original; it does not reproduce the paper text.

SOURCE NOT REACHED IN FULL. The article is held by the American Physical Society and is closed: on 2026-09-11 OpenAlex reported open-access status closed with no repository location, and no preprint or repository copy was found — the paper predates every preprint server. What is reproduced below is the authors’ own complete abstract as OpenAlex carries it, and the summary and every claim are written from that abstract, which is why each locator says Abstract. FORMULAE. The bibliographic record stores the two closing results as TeX markup; they are set out below in plain reading order, with the same symbols the authors use — R for the separation, alpha for the static polarizability of an atom, h-bar for the reduced Planck constant and c for the speed of light. The numerical coefficients, 3 over 8 pi and 23 over 4 pi, are the record’s and are the ones the paper is known for. AUTHORS. Hendrik Brugt Gerhard Casimir and Dirk Polder, both then at the Philips Research Laboratories in Eindhoven.

How to cite it

H. B. G. Casimir, D. Polder (1948) The Influence of Retardation on the London-van der Waals Forces doi:10.1103/PhysRev.73.360

Where it sits in the curriculum

What the vacuum is