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Numerically Absorbing Boundary Conditions for Quantum Evolution Equations

Summary: Numerically Absorbing Boundary Conditions for
Quantum Evolution Equations
Anton Arnold
Transparent boundary conditions for the transient Schrodinger equation on a domain can be derived
explicitly under the assumption that the given potential V is constant outside of this domain. In 1D these
boundary conditions are non-local in time (of memory type). For the Crank{Nicolson nite di erence
scheme, discrete transparent boundary conditions are derived, and the resulting scheme is proved to be
unconditionally stable. A numerical example illustrates the superiority of discrete transparent boundary
conditions over existing ad-hoc discretizations of the di erential transparent boundary conditions.
As an application of these boundary conditions to the modeling of quantum devices, a transient 1D
scattering model for mixed quantum states is presented.
Key words: Schrodinger equation, transparent boundary conditions, absorbing boundary conditions, nite
di erences, discrete transparent boundary conditions, quantum device contacts.
The formulation and implementation of physically reasonable and mathematically well-posed boundary con-
ditions (BC) is one of the big open problems and challenges for transient simulations of semiconductor devices
through quantum mechanical models. This paper is concerned with the construction and discretization of
absorbing boundary conditions (ABC) for the Schrodinger equation (SE)
ih t = ;h2


Source: Arnold, Anton - Institut für Analysis und Scientific Computing, Technische Universität Wien


Collections: Mathematics