Ground-State Energy and Effective Mass of the Polaron

Abstract

The polaron Hamiltonian would be easily soluble were it not for the quartic term appearing therein. It is proposed to substitute for the quartic term a quadratic term having roughly the same properties, and in such a way that the ground-state energy of the new Hamiltonian is rigorously a lower bound for the true energy. With a very small amount of work one can obtain a lower bound as a continuous function of $\ensuremath{\alpha}$ for all values of $\ensuremath{\alpha}$. The result agrees fairly well with the results obtained by other methods. Using the equivalent Hamiltonian one can also obtain an analytic expression for the effective mass, although one cannot say it is a bound for the true effective mass. Futhermore, once one has obtained a lower bound for the energy as a continuous function of the parameters of the Hamiltonian, one can rigorously derive upper and lower bounds for the ground-state expectation values of various operators. For example, it can be shown that for large $\ensuremath{\alpha}$ and large k, $〈{{a}_{\mathrm{k}}}^{*}{a}_{\mathrm{k}}〉\ensuremath{\sim}{\mathrm{k}}^{\ensuremath{-}6}$ and not ${\mathbf{k}}^{\ensuremath{-}2}$ exp(-${\mathbf{k}}^{\ensuremath{-}2}$) as in Pekar's solution. Because of its simplicity, it is possible that this method may have application to other ground-state problems.