This paper presents a semiclassical theory for the computation of matrix elements of the type 〈u‖v〉 when either one or both ‖u〉 and ‖v〉 are coherent states (in different representations). Our results can be considered as an extension of Miller’s semiclassical theory [Adv. Chem. Phys. 25, 69 (1974)]. Such an extension has been presented also by Heller [J. Chem. Phys. 66, 5777 (1977)]. We were able to simplify considerably some of Heller’s results by exploiting the canonical properties of the classical coherent variables. This enabled us to relate the elements 〈u‖v〉 to certain generalized, complex generator functions in a manner that is very similar to the relations that appear in the original Miller’s theory. The advantages that are inherent in the coherent states representation are illustrated in a few elementary examples. We were able to derive an excellent approximation to the eigenstates of the harmonic oscillator which is valid even for the ground state. Furthermore, we have demonstrated that it is possible to describe semiclassically dynamical processes that are classically forbidden by real time trajectories in a certain generalized phase space.
Articles you may be interested inTunneling dynamics with a mixed quantum-classical method: Quantum corrected propagator combined with frozen Gaussian wave packetsIn this paper we consider some of the consequences of classical stochasticity phenomena in the quantum mechanical Henon-Heiles and Barbanis systems. To explore classical--quantum analogies we have introduced a quantum mechanical phase space based on the coherent state representation. Quantum Poincare maps (QPMs) were constructed from contour plots of the stationary phase-space densities. We have observed a correlation between the topological features of a QPM at an eigenenergy E and the sensitivity IdE /d€ I of that energy to the strength E of the nonlinear coupling. States with extreme values of IdE/d€l, corresponding to both high and to low values of IdE/d€l, show regular regions in the quantum phase space, while states with intennediate values of IdE/dEl span large parts of the quantum phase-space regions. These general features of the QPMs apply for almost each energy multiplet, and no qualitative change in the character of the QPMs was observed above the classical critical energy E, for the stochastic transition. To investigate the quantum analog of classical trajectories, we have studied the dynamics of initially coherent Gaussian wave packets. We have discovered two limiting types of time evolution of intially coherent wave packets, which exhibit quasiperiodic time evolution or rapid dephasing. Quasiperiodic quantum dynamics is analogous to quasiperiodic classical trajectory dynamics, while rapid dephasing of the wave packet reflects the effects of spreading of the wave packet or/and stochastic classical dynamics. We were able to establish a correspondence between the topology of the quantum phase-space density and the wave packet dynamics.Quasiperiodic time evolution and rapid dephasing are exhibited by wave packets initially located in the regular and in the irregular regions of the quantum phase space, respectively.
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