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Quasi Monte-Carlo methods for the TDSE (Time dependent Schrödinger equation)

Gowri Suryanarayana, Dirk Nuyens, Ronald F. A. Cools

Published Nov 23, 2015• 0 citations
OpenAlex

Abstract

We approximate the solution for the time dependent Schrödinger equation (TDSE) in two steps. We first use a pseudo-spectral collocation method that uses samples of the functions on rank-1 lattice points. We then get a system of ordinary differential equations in time, which we solve approximately by stepping in time using the Strang splitting method.

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We approximate the solution for the time dependent Schrödinger equation (TDSE) in two steps. We first use a pseudo-spectral collocation method that uses samples of the functions on rank-1 lattice points. We then get a system of ordinary differential equations in time, which we solve approximately by stepping in time using the Strang splitting method.
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.
The time-dependent Schrödinger equation (TDSE) is a wave equation that is used to understand the behavior of small particles such as electrons or nuclei. The equation must be "solved" in the sense that a function must be found that can be plugged into the equation and that leaves the left-hand side equal to the right-hand side. The TDSE can be understood by first studying the classical wave equation that is used to "solve" the behavior of a guitar string or a water wave. The TDSE is a bit different in that it equates a double space derivative with a single time derivative (the classical equation equates two double derivatives); it also implicitly includes the symbol i , which stands for the square root of -1. These differences give rise to different sorts of solutions, including complex solutions with real and imaginary parts. In the case of the TDSE, the solution or "wave" does not directly show the motion of the particle. Instead, it shows where the particle is likely to be found. More specifically, it is a probability wave for particle location. To obtain the precise probability for the particle to be found at a specific position x at a specific time t , the wave function Ψ ( x , t ) must be squared via a special procedure called taking the square modulus. The wave contains information about the frequency, wavelength, position, and momentum of the particle. In the following chapters, we will be showing how to find the wave solution in different circumstances (different potentials), and how to extract the information about momentum, etc., once the wave function is known.
We solve the solution of the Time-Dependent Schrodinger Equation (TDSE) using the higher-order Trotter-Suzuki Decomposition. We use The Gaussian function as a stationary state in a linear potential system. The TDSE solution using Baker - Campbell - Hausdorff was used to validate the results and to measure the accuracy of the Trotter - Suzuki decomposition. So that the difference between the Baker - Campbell - Hausdorff result and the Suzuki - Trotter decomposition is considered an error. The error of the TDSE solution by the Trotter - Suzuki second-order decomposition was lower than the first order. Meanwhile, the error of the TDSE solution by second-order hybrid will be lower than the second-order Trotter - Suzuki decomposition when the value of dx=0.1 and 0.05 with dt < 0.0001. The error comparison of these three methods is only valid when time t<1.
Abstract In this paper, we study the Schrödinger equation with a Gaussian random potential (SE-GP) and develop an efficient numerical method to approximate the expectation of physical observables. The unboundedness of Gaussian random variables poses significant difficulties in both sampling and error analysis. Under time-splitting discretizations of SE-GP, we establish the regularity of the semi-discrete solution in the random space. Then, we introduce a non-standard weighted Sobolev space with properly chosen weight functions, and obtain a randomly shifted lattice-based quasi-Monte Carlo (QMC) quadrature rule for efficient sampling. This approach leads to a QMC time-splitting (QMC-TS) scheme for solving the SE-GP. We prove that the proposed QMC-TS method achieves a dimension-independent convergence rate that is almost linear with respect to the number of QMC samples. Numerical experiments illustrate the sharpness of the error estimate.