Interaction energy for a pair of quantum wells

We present a calculation of the interaction energy and force between two quantum wells in which the wells are treated as strictly two-dimensional metallic sheets. Numerical results are presented for separation values ranging from 1 to ${10}^{12}\AA .$ Both nonretarded and retarded calculations are presented. Three distinct ranges with different separation dependences are clearly revealed in the results; the Casimir limit for large separations; the van der Waals range for intermediate separations; the small separation limit where also the single-particle continuum contributes. The result in the Casimir limit coincides with the corresponding result for two metal half-spaces but the van der Waals result is different from both the half-space result and that of two thin films. In the present problem the energy and force varies as $d^{-5/2}$ and $d^{-7/2},$ respectively.