Path integrals for the Liouville equation (in Ukrainian)

L. F. Blazhievskii

L~v vs~ki deravni un versitet men vana Franka, kafedra teoretiqnoÝ f ziki UkraÝna, 290005, L~v v, vul.Dragomanova, 12 (Otrimano 17 ltogo 1997) Zaproponovano funkc onal~ne peretvorenn dl T -vpordkovanih eksponentnih funkc v d dobutku dvoh nekomutuqih operator v. Na c osnov otriman kontinual~n rozv'zki r vnnn L uv ll u vigld nteral v za tra ktor mi v gam l~tonovih ta lagranevih zm nnih.

Ioannis E. Antoniou, Oleg Georgievich Smolyanov

We show that the Liouville equation for any Hamiltonian system can be identified with the Schrödinger equation obtained by the Schrödinger-Dirac quantization of some “extended” Hamiltonian system whose phase space is the product of two copies of the initial phase space and hence whose configuration space coincides with the initial phase space. This identification allows us to represent some solutions of the Cauchy problem for the Liouville equation by Feynman path integrals [ 1 ] over trajectories in the phase space of the extended Hamiltonian systems.

Frank J. McLafferty

We continue our study of Wigner path integrals. We first define a Wigner path integral for the kernel of the Wigner–Bloch equation, solve it for the free particle and harmonic oscillator cases, discuss the physical interpretation which is new and then we apply it (along with the path integral representation of the Liouville equation) to obtain a path integral representation in phase space of the quantum transition state theory of Miller. We also obtain an expression involving products of Wigner kernels for the correlation function form of the exact bimolecular rate constant.

Oliver Mülken, Lothar Mühlbacher, Tobias Schmid +1 more

The trapping of excitations in systems coupled to an environment allows one to study the quantum to classical crossover by different means. We show how to combine the phenomenological description by a non-Hermitian Liouville-von Neumann equation (LvNE) approach with the numerically exact path integral Monte Carlo (PIMC) method, and exemplify our results for a system of two coupled two-level systems. By varying the strength of the coupling to the environment we are able to estimate the parameter range in which the LvNE approach yields satisfactory results. Moreover, by matching the PIMC results with the LvNE calculations, we have a powerful tool to extrapolate the numerically exact PIMC method to long times.