Ground-state energy of a polaron in<i>n</i>dimensions

Abstract

The Fr\"ohlich Hamiltonian is generalized to the case of an electron moving in n space dimensions. For n=2 and n=3 the familiar Fr\"ohlich Hamiltonian is reobtained. The polaron ground-state energy is calculated up to fourth order in perturbation theory. We found that within the Feynman two-particle polaron model approximation the polaron ground-state energy satisfies the scaling relation ${E}_{n\mathrm{D}(\mathrm{\ensuremath{\alpha}})}$=(n/3)${\mathrm{E}}_{3\mathrm{D}}$({\ensuremath{\Gamma}[ (n-1)/2]3 \ensuremath{\surd}\ensuremath{\pi} /2n\ensuremath{\Gamma}(n/2)}\ensuremath{\alpha}), where ${E}_{n\mathrm{D}}$ is the Feynman polaron ground-state energy for the polaron in n dimensions and ${E}_{3\mathrm{D}}$ the energy in three dimensions.