= Electric-dipole approximation
{title2=$H_{\rm int}\simeq-\mathbf d\cdot\mathbf E(\mathbf R,t)$}
For a localized charge distribution in an external field varying little across its extent, expand the external scalar potential about a reference point $\mathbf R$. The constant term couples to total charge; the first spatially varying term gives the <electric-dipole interaction> $-\mathbf d\cdot\mathbf E(\mathbf R,t)$, where $\mathbf d$ is the <electric dipole moment>. Higher spatial multipoles are neglected.
For a passing charge at minimum distance $b$, the atomic extent must be small compared with $b$. For a plane electromagnetic wave, the analogous long-wavelength condition is $ka\ll1$, as in the <electric-dipole approximation for radiation>. A short interaction time and a spatially uniform field across the atom are different assumptions; neither implies the other.
Back to article page
