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Journal of the Physical Society of Japan

Time Dependent Schrödinger Equations with Vector Potentials - Numerical Calculations and Visualizations -

Hiroshi Natori, Tomo Munehisa

Published Feb 15, 1997• 9 citations
Publisher

Abstract

We study algorithm to calculate the time dependent Schrödinger equation numerically and to visualize the results. We extend the product formula by the Trotter-Suzuki decomposition to cases of Hamiltonians with vector potentials. This method is unconditionally stable and it can be used for a time dependent Hamiltonian so that one can apply it to various problems without any difficulty. After examining numerical errors for time evolutions of one dimensional wave packets, we present the errors under a constant magnetic field. Also we discuss a cyclotron motion in quantum mechanics, where an advantage for the unconditional stability is demonstrated. In order to visualize the results we make animations of the time evolutions, where the phase of the wave function is represented by colors and its absolute value is shown by brightness. Here we suggest a new way to represent a complex number by colors.

DOI
10.1143/jpsj.66.351

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We study algorithm to calculate the time dependent Schrödinger equation numerically and to visualize the results. We extend the product formula by the Trotter-Suzuki decomposition to cases of Hamiltonians with vector potentials. This method is unconditionally stable and it can be used for a time dependent Hamiltonian so that one can apply it to various problems without any difficulty. After examining numerical errors for time evolutions of one dimensional wave packets, we present the errors under a constant magnetic field. Also we discuss a cyclotron motion in quantum mechanics, where an advantage for the unconditional stability is demonstrated. In order to visualize the results we make animations of the time evolutions, where the phase of the wave function is represented by colors and its absolute value is shown by brightness. Here we suggest a new way to represent a complex number by colors.
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— The fundamental possibility of a numerical solution of the time-dependent Schrödinger equation for a wave function defined on a non-uniform coordinate grid by the spectral method using the fast Fourier transform algorithm is discussed. The method is based on reducing the non-uniform coordinate grid to a uniform one by a non-linear transformation of coordinates and approximating the obtained evolution operator using the Lie–Trotter–Suzuki product formula. Algorithms for the numerical solution of the first and second orders are developed.
A fast and stable numerical method is formulated to compute the time evolution of a wave function in a magnetic field by solving the time-dependent Schroedinger equation. This computational method is based on the finite element method in real space to improved accuracy without any increase of computational cost. This method is also based on Suzuki's exponential product theory to afford an efficient way to manage the TD-Schroedinger equation with a vector potential. Applying this method to some simple electron dynamics, we have confirmed its efficiency and accuracy.
Numerical calculations on two-dimensional spin-½ XXZ Heisenberg models are presented. The computational technique entails solving the time-dependent Schrödinger equation. The time propagation of wave functions is performed employing Trotter-Suzuki product formulas. The algorithms are numerically stable and are used to study models involving ≈10^7 states. With this approach both the thermodynamic properties as well as the many-particle dynamics can be obtained. Densities of states and zero-temperature dynamic magnetic structure factors are presented. For the isotropic Heisenberg antiferromagnet thermodynamic properties are computed for lattices up to 26 sites.
Numerical calculations on two-dimensional spin-1/2 XXZ Heisenberg models are presented. The computational technique entails solving the time-dependent Schr\"odinger equation. The time propagation of wave functions is performed employing Trotter-Suzuki product formulas. The algorithms are numerically stable and are used to study models involving \ensuremath{\approxeq}107{10}^{7}107 states. With this approach both the thermodynamic properties as well as the many-particle dynamics can be obtained. Densities of states and zero-temperature dynamic magnetic structure factors are presented. For the isotropic Heisenberg antiferromagnet thermodynamic properties are computed for lattices up to 26 sites.
We develop methodology for performing time dependent quantum mechanical calculations by representing the wave function as a sum of Gaussian wave packets (GWP) each characterized by a set of parameters such as width, position, momentum, and phase. The problem of computing the time evolution of the wave function is thus reduced to that of finding the time evolution of the parameters in the Gaussians. This parameter motion is determined by minimizing the error made by replacing the exact wave function in the time dependent Schrödinger equation with its Gaussian representation approximant. This leads to first order differential equations for the time dependence of the parameters, and those describing the packet position and the momentum of each packet have some resemblance with the classical equations of motion. The paper studies numerically the strategy needed to achieve the best GWP representation of time dependent processes. The issues discussed are the representation of the initial wave function, the numerical stability and the solution of the differential equations giving the evolution of the parameters, and the analysis of the final wave function. Extensive comparisons are made with an approximate method which assumes that the Gaussians are independent and their width is smaller than the length scale over which the potential changes. This approximation greatly simplifies the calculations and has the advantage of a greater resemblance to classical mechanics, thus being more intuitive. We find, however, that its range of applications is limited to problems involving localized degrees of freedom that participate in the dynamic process for a very short time. Finally we give particular attention to the notion that the GWP representation of the wave function reduces the dynamics of one quantum degree of freedom to that of a set of pseudoparticles (each represented by one packet) moving according to a pseudoclassical (i.e., classical-like) mechanics whose ‘‘phase space’’ is described by a position and momentum as well as a complex phase and width.