Solution of time-independent Schrödinger equation by the imaginary time propagation method

2007 ◽  
Vol 221 (1) ◽  
pp. 148-157 ◽  
Author(s):  
L. Lehtovaara ◽  
J. Toivanen ◽  
J. Eloranta
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Imaginary Time ◽  
Time Propagation

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.

A Refined Speed Limit for the Imaginary-Time Schrödinger Equation

2020 ◽  
Vol 27 (02) ◽  
pp. 2050010
Author(s):  
Jie Sun ◽  
Songfeng Lu
Keyword(s):  
Schrödinger Equation ◽  
Quantum Dynamics ◽  
Schrodinger Equation ◽  
Time Dependent ◽  
Uncertainty Relations ◽  
Speed Limit ◽  
Full Time ◽  
Imaginary Time ◽  
Full Solution ◽  
The One

Recently, Kieu proposed a new class of time-energy uncertainty relations for time-dependent Hamiltonians, which is not only formal but also useful for actually evaluating the speed limit of quantum dynamics. Inspired by this work, Okuyama and Ohzeki obtained a similar speed limit for the imaginary-time Schrödinger equation. In this paper, we refine the latter one to make it be further like that of Kieu formally. As in the work of Kieu, only the initial states and the Hamiltonians, but neither the instantaneous eigenstates nor the full time-dependent wave like functions, which would demand a full solution for a time-dependent system, are required for our optimized speed limit. It turns out to be more helpful for estimating the speed limit of an actual quantum annealing driven by the imaginary-time Schrödinger equation. For certain case, the refined speed limit given here becomes the only useful tool to do this estimation, because the one given by Okuyama and Ohzeki cannot do the same job.

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