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A Semiclassical Coherent-State Propagator via Path Integrals with Intermediate States of Variable Width

We derive a semiclassical approximation for the coherent state propagator $\langle z”|e^{-iHt/\hbar}|z’\rangle$ 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 ${\cal H}(q, p, b)= \langle z|\hat{H}|z \rangle$ where $b$, the width of the coherent state $|z\rangle$, is a free function that can be chosen conveniently. The generality of this formalism enable us to derive a Gaussian Initial Value Representation (GIVR) which contains as particular cases some of the GIVR’s known in 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 $\sigma’$ or $\sigma”$ of the initial and final states $| z’ \rangle$ and $| z” \rangle$.

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