Semiclassical coherent-state propagator via path integrals with intermediate states of variable width

Abstract

We derive a semiclassical approximation for the coherent state propagator $〈{z}^{\ensuremath{''}}|{e}^{\ensuremath{-}iHt/\ensuremath{\Elzxh}}|{z}^{\ensuremath{'}}〉$ using a path integral formulation in which the intermediate coherent states can have arbitrary widths. Our semiclassical formula involves complex trajectories of the smoothed Hamiltonian $\mathcal{H}(q,p,b)=〈z|\ifmmode \hat{H}\else \^{H}\fi{}|z〉$ where b, the width of the coherent state $|z〉,$ is a free function that can be chosen conveniently. The generality of this formalism enables us to derive a semiclassical approximation which contains, as particular cases, other similar approximations known in the literature, providing a natural link between them. We present numerical results showing that the semiclassical propagation can be very sensitive to the choice of b and we suggest an energy dependent value ${b=b}_{E}$ that results in considerable improvement over other choices. This value for the width will be generally different from the widths ${\ensuremath{\sigma}}^{\ensuremath{'}}$ or ${\ensuremath{\sigma}}^{\ensuremath{''}}$ of the initial and final states $|{z}^{\ensuremath{'}}〉$ and $|{z}^{\ensuremath{''}}〉.$