The van der Waals interaction in one, two and three dimensions

The van der Waals interaction between two polarizable atoms is considered. In three dimensions the standard form with an attractive $1\mathrm{/}\mathrm{\mid }R{\mathrm{\mid }}^6$ potential is obtained from second-order quantum perturbation theory. When the electron motion is restricted to lower dimensions (but the $1\mathrm{/}\mathrm{\mid }R\mathrm{\mid }$ Coulomb potential is retained), new terms in the expansion appear and alter both the sign and the $\mathrm{\mid }R\mathrm{\mid }$ dependence of the interaction.