The van der Waals interaction

The interaction between charged particles via the Coulomb interaction is one of the most important features in physics and is familiar to any student of the subject. The way in which electrons and protons bind to form the hydrogen atom is also well known and is a staple of any quantum mechanics course. However, less familiar is the interaction between such bound systems at separation R, which is the so-called van der Waals force and is of a completely different character from its Coulombic analog. That this must be the case is clear from the fact that the hydrogen atom is neutral, so that to lowest order there is no interaction. On the other hand the system is polarizable, and thus can interact with the other polarizable system, leading to a short-ranged attraction which varies as 1/R, and this feature is discussed by a number of quantum mechanical references. Somewhat less well known is the fact that at larger distances the character of the interaction changes and varies as 1/R—discussion of this feature can be found, e.g., in the quantum field theory book by Itzykson and Zuber. It is clear that the origin of this change is retardation, i.e., the finite propagation time of signals connecting the two systems, but the precise way in which this modification comes about is not so easy to calculate and is not generally presented. The nature of the van der Waals force is quite topical at present due to the possible importance of such effects for the interactions of small color dipoles such as charmonium or bottomonium, so it is useful to examine the physics of this effect. In the next section, then, we review the usual textbook discussion leading to the London ;1/R interaction. Then in Sec. III, we show how retardation effects modify the character of the force and change its asymptotic dependence to the Casimir–Polder form ;1/R7. In a brief concluding section we summarize our findings and discuss the relevance to modern particle and nuclear physics.