Analysis of the Pechukas description of mixed quantum-classical dynamics
| Autor: | J. Liam McWhirter |
|---|---|
| Rok vydání: | 1999 |
| Předmět: |
Differential equation
General Physics and Astronomy Semiclassical physics Propagator symbols.namesake Uniqueness theorem for Poisson's equation Quantum mechanics Path integral formulation symbols Piecewise Statistical physics Physical and Theoretical Chemistry Hamiltonian (quantum mechanics) Eigenvalues and eigenvectors Mathematics |
| Zdroj: | The Journal of Chemical Physics. 110:4184-4198 |
| ISSN: | 1089-7690 0021-9606 |
| DOI: | 10.1063/1.478300 |
| Popis: | We review the Pechukas stationary phase analysis which yields the semiclassical reduced propagator. This propagator describes the dynamics of a mixed quantum-semiclassical system. In addition, we review a variant of this analysis, presented by us in an earlier article [J. Chem. Phys. 108, 5683 (1998)], which yields a semiclassical reduced propagator written in terms of connected and partially connected “classical” paths. These paths are obtained by solving the concatenation of several short time interval Pechukas equations. We argue and then demonstrate numerically that the “energy” along these paths is generally piecewise conserved: conserved across one short time interval, but not across several such intervals. In our review of these analyses, we relax the assumption made by Pechukas that the magnitude of the transition amplitude associated with the quantum subsystem varies much more slowly with changes in the classical subsystem’s trajectory than its phase. As our analyses demonstrate, this assumption serves to simplify the evaluation of the normalization path integral; the stationary phase paths are not affected by the making of this assumption. Solving the Pechukas equation subject to the initial configuration and velocity of the classical subsystem yields a collection of “classical” paths; the solution is nonunique. We provide a short time uniqueness theorem pertaining to the class of functional differential equations to which the Pechukas equation belongs; then, we review the sample problem that Pechukas used to first demonstrate this nonuniqueness. The theorem and the sample problem allow us to identify which parts within the Pechukas equation’s structure are responsible for the nonuniqueness in its solution. This nonuniqueness is verified numerically. Here, we show that the “energy” is conserved along each of the “classical” paths in the collection. However, the “energies” of any two paths in this collection will be the same only if the initial state specified in the Pechukas equation is an instantaneous adiabatic eigenstate of the quantum subsystem Hamiltonian which contains the interaction potential between the quantum and classical subsystems. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|