Time-Dependent Conformal Transformations and the Propagator for Quadratic Systems

The method proposed by Inomata and his collaborators allows us to transform a damped Caldirola–Kanai oscillator with a time-dependent frequency to one with a constant frequency and no friction by redefining the time variable, obtained by solving an Ermakov–Milne–Pinney equation. Their mapping “Eisenhart–Duval” lifts as a conformal transformation between two appropriate Bargmann spaces. The quantum propagator is calculated also by bringing the quadratic system to free form by another time-dependent Bargmann-conformal transformation, which generalizes the one introduced before by Niederer and is related to the mapping proposed by Arnold. Our approach allows us to extend the Maslov phase correction to an arbitrary time-dependent frequency. The method is illustrated by the Mathieu profile.

Feynman’s propagator in Schwinger’s picture of Quantum Mechanics

A novel derivation of Feynman’s sum-over-histories construction of the quantum propagator using the groupoidal description of Schwinger picture of Quantum Mechanics is presented. It is shown that such construction corresponds to the GNS representation of a natural family of states called Dirac–Feynman–Schwinger (DFS) states. Such states are obtained from a q-Lagrangian function [Formula: see text] on the groupoid of configurations of the system. The groupoid of histories of the system is constructed and the q-Lagrangian [Formula: see text] allows us to define a DFS state on the algebra of the groupoid. The particular instance of the groupoid of pairs of a Riemannian manifold serves to illustrate Feynman’s original derivation of the propagator for a point particle described by a classical Lagrangian L.

Numerical Simulation of Multistage Bimolecular Photoreactions in Liquids: Algorithms and Software

In this paper we consider the tools for numerical simulation of multistage bimolecular photoreactions assisted by diffusive mobility of reactant molecules in viscous solutions. To describe these processes, the differential encounter theory (DET) is extended to account for the coherent dynamics of certain degrees of freedom (for example, electronic spins of reactants and intermediates). The model involves diffusion of reactants in solution and multistage/multichannel physicochemical processes proceeding both at the level of individual molecules and encounter complexes. Algorithms for numerical solution of model equations are proposed, which are related to evaluation of evolution operators. The algorithm for computing the quantum propagator for the density matrix based on the Trotter splitting is presented. A software package for simulation of multistage photoreactions has been developed using the suggested numerical approaches. The structure of the key software components is presented, examples of the program model construction are presented. A software testing has been carried out, showing good correspondence between the numerical results and exact solutions of the model equations in certain particular cases. As an example, a photoreaction with participation of 9,10-dimethylanthracene and 1,3-dicyanobenzene in acetonitrile solution has been considered, and basic procedures for configuring and simulating multistage bimolecular photoprocesses are shown. An importance of coherent description of the electronic spin evolution at the radical stage is shown.

Quantum Propagator Derivation for the Ring of Four Harmonically Coupled Oscillators

The ring model of the coupled oscillator has enormously studied from the perspective of quantum mechanics. The research efforts on this system contribute to fully grasp the concepts of energy transport, dissipation, among others, in mesoscopic and condensed matter systems. In this research, the dynamics of the quantum propagator for the ring of oscillators was analyzed anew. White noise analysis was applied to derive the quantum mechanical propagator for a ring of four harmonically coupled oscillators. The process was done after performing four successive coordinate transformations obtaining four separated Lagrangian of a one-dimensional harmonic oscillator. Then, the individual propagator was evaluated via white noise path integration where the full propagator is expressed as the product of the individual propagators. In particular, the frequencies of the first two propagators correspond to degenerate normal mode frequencies, while the other two correspond to non-degenerate normal mode frequencies. The full propagator was expressed in its symmetric form to extract the energy spectrum and the wave function.

Decoherence decay time and thermal distributions for quadratic quantum Lagrangians

Quantum propagator for a general quadratic Lagrangian is obtained using position and momentum operators in Heisenberg picture and properties of propagators. For a quantum system governed by a general quadratic Lagrangian containing external forces, decoherence decay time, thermal distribution functions, and thermal Wigner function are obtained in high temperature and weak coupling regime.