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International Journal of Quantum Chemistry

Time‐dependent schrödinger equation with Markovian outgoing wave boundary conditions: Applications to quantum tunneling dynamics and photoionization

Chia‐Chun Chou, Róbert E. Wyatt

Published Feb 13, 2012• 10 citations
Publisher

Abstract

Abstract Markovian outgoing wave boundary conditions are introduced as an approximate method to reduce the size of the computational grid for time integration of the time‐dependent Schrödinger equation. The ratio and polynomial methods developed as open boundary conditions are applied to the wave function at the boundaries of the computational grid. This computational method is used to study the wave packet dynamics for a metastable well, a double well, and strong‐field ionization of a model atom. Accurate results demonstrate that this method can significantly reduce the number of grid points required in a dynamical calculation for quantum dynamical problems. © 2012 Wiley Periodicals, Inc.

DOI
10.1002/qua.24005

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Abstract Markovian outgoing wave boundary conditions are introduced as an approximate method to reduce the size of the computational grid for time integration of the time‐dependent Schrödinger equation. The ratio and polynomial methods developed as open boundary conditions are applied to the wave function at the boundaries of the computational grid. This computational method is used to study the wave packet dynamics for a metastable well, a double well, and strong‐field ionization of a model atom. Accurate results demonstrate that this method can significantly reduce the number of grid points required in a dynamical calculation for quantum dynamical problems. © 2012 Wiley Periodicals, Inc.
The scattered wave formalism is developed for a quantum subsystem interacting with the external environment through open boundaries. The total wave function is divided into incident and scattered components and Markovian outgoing wave boundary conditions are applied to the scattered wave function. This formalism significantly reduces the computational effort relative to other methods which rely on Green functions and memory kernels. The method is applied to one-dimensional barrier scattering and to a three-dimensional model for the field effect transistor.
The scattered wave formalism developed for a quantum subsystem interacting with reservoirs through open boundaries is applied to one- or two-dimensional barrier scattering and quantum transistors. The total wave function is divided into incident and scattered components. Markovian outgoing wave boundary conditions are imposed on the scattered or total wave function by either the ratio or polynomial methods. For barrier scattering problems, accurate time-dependent transmission probabilities are obtained through the integration of the modified time-dependent Schrödinger equations for the scattered wave function. For quantum transistors, the time-dependent transport is studied for a quantum wave packet propagating through the conduction channel of a field effect transistor. This study shows that the scattered wave formalism significantly reduces computational effort relative to other open boundary methods and demonstrates wide applications to quantum dynamical processes.
The open-system boundary conditions for the one-dimensional Schr\"odinger equation are derived by dividing the unbounded domain into a finite system and two semi-infinite reservoirs. The resulting boundary conditions on the system are non-Markovian, as they contain a convolution over the history of the system. Thus, time-irreversibility arises in a pure-state problem. The propagator which appears in the boundary condition is derived for a simple discrete model. The correctness of the boundary conditions is verified and the usefulness of the discrete model is demonstrated by a numerical calculation of the time-evolution of a wave packet.
Abstract A new approach based upon the Taylor series method is proposed for propagating solutions of the time‐dependent Schrödinger equation. Replacing the spatial derivative of the wave function with finite difference formulas, we derive a recursive formula for the evaluation of Taylor coefficients. The automatic differentiation technique is used to recursively calculate the required Taylor coefficients. We also develop an implicit scheme for the recursive evaluation of these coefficients. We then advance the solution in time using a Taylor series expansion. Excellent computational results are obtained when this method is applied to a one‐dimensional reflectionless potential and a two‐dimensional barrier transmission problem. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010
Abstract Two methods for the numerical integration of the time‐dependent Schrödinger equation with given initial conditions (initial wave packet) are presented. The first method (method A) is based on the Schrödinger representation of the quantum‐dynamical system while the second one (method B) is based upon the intermediate representation. In both cases the quantum dynamical equation is transformed into a system of Hamilton‐Jacobi type equations of motion as occurring in multi particle classical dynamics, i.e. standard molecular dynamics techniques can be applied for the integration. The dynamics of a minimum uncertainty Gaussian wave packet in a strongly anharmonic oscillator is taken as an example.
A method for the solution of the time-dependent Schrödinger equation in two dimensions is presented which is based on the hydrodynamic analogy to quantum mechanics. The continuum introduced in that analogy is approximated by a finite number of particles, whose trajectories are computed. The procedure is applied here to the dynamics of a Gaussian wave packet on a two-dimensional quadratic potential surface containing a saddle point. Comparison with the known analytical solution indicates that the method is capable of accurate results with substantially less computer time required than in methods previously employed. This particular check problem was chosen because of its close relationship to the wave-packet dynamics of chemical reaction rates.