A high-order accuracy method for numerical solving of the time-dependent Schrödinger equation

1999 ◽  
Vol 123 (1-3) ◽  
pp. 1-6 ◽  
Author(s):  
I.V. Puzynin ◽  
A.V. Selin ◽  
S.I. Vinitsky
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
High Order ◽  
Time Dependent ◽  
High Order Accuracy ◽  
Order Accuracy

EXPONENTIAL PROPAGATORS (INTEGRATORS) FOR THE TIME-DEPENDENT SCHRÖDINGER EQUATION

2013 ◽  
Vol 12 (06) ◽  
pp. 1340001 ◽  
Author(s):  
ANDRÉ D. BANDRAUK ◽  
HUIZHONG LU
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Time Derivative ◽  
Quantum Statistical Mechanics ◽  
Time Dependent ◽  
High Order Accuracy ◽  
Parabolic Partial Differential Equation ◽  
Imaginary Time ◽  
Time Step ◽  
Order Accuracy

The time-dependent Schrödinger Equation (TDSE) is a parabolic partial differential equation (PDE) comparable to a diffusion equation but with imaginary time. Due to its first order time derivative, exponential integrators or propagators are natural methods to describe evolution in time of the TDSE, both for time-independent and time-dependent potentials. Two splitting methods based on Fer and/or Magnus expansions allow for developing unitary factorizations of exponentials with different accuracies in the time step △t. The unitary factorization of exponentials to high order accuracy depends on commutators of kinetic energy operators with potentials. Fourth-order accuracy propagators can involve negative or complex time steps, or real time steps only but with gradients of potentials, i.e. forces. Extending the propagators of TDSE's to imaginary time allows to also apply these methods to classical many-body dynamics, and quantum statistical mechanics of molecular systems.

scholarly journals High-order symplectic FDTD scheme for solving a time-dependent Schrödinger equation

2013 ◽  
Vol 184 (3) ◽  
pp. 480-492 ◽  
Author(s):  
Jing Shen ◽  
Wei E.I. Sha ◽  
Zhixiang Huang ◽  
Mingsheng Chen ◽  
Xianliang Wu
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
High Order ◽  
Time Dependent
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