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arXiv:physics/0011069 · Journal of the Physical Society of Japan

Finite Element Approach for Simulating Quantum Electron Dynamics in a Magnetic Field

Naoki Watanabe, Masaru Tsukada

Published Sep 1, 2000• 11 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1143/jpsj.69.2962

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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.
A fast and stable method is formulated to compute the time evolution of a wave function by numerically solving the time-dependent Schrodinger equation. This method is a real-space-real-time evolution method implemented by several computational techniques such as Suzuki's exponential product, Cayley's form, the finite differential method, and an operator named adhesive operator. This method conserves the norm of the wave function, manages periodic conditions and adaptive mesh refinement technique, and is suitable for vector- and parallel-type supercomputers. Applying this method to some simple electron dynamics, we confirmed the efficiency and accuracy of the method for simulating fast time-dependent quantum phenomena.
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.
The particle method has no restriction on the particle arrangement, where the calculation is performed. The purpose is, by using this feature, to develop the method to analyze the dynamics of the electronic state. In order to describe the time evolution of the electronic state, we have developed a new method to solve the time-dependent wave equation, using the Bohmian that is very compatible with the particle method. In this form, there is also numerical instability in the region, where the value of the wave function is very small. We have applied our technique to the wave packet dynamics on harmonic potential and the analysis of interference by double slits as a simple system with analytical solution and easy to compare accuracy, and realized that both results are in good agreement.
In this work, we present a quantum-computing framework for time-domain simulation of Maxwell's equations. By defining the Riemann-Silberstein vector, Maxwell's curl equations reduce to a single Schrödinger-like evolution equation. The solution requires computing the exponential of the curl (Hamiltonian). On quantum hardware, matrix exponentials correspond to sequences of unitary gates. Therefore decomposing the Hamiltonian using the Trotter-Suzuki decomposition it is possible to simulate electrodynamics. To validate the method, we simulate the propagation of transverse-magnetic (TM) waves in a two-dimensional free-space domain. Results show accurate field propagation highlighting the potential advantage of quantum computing for solving electromagnetic problems.
In order to understand dynamics in the Heisenberg picture, a study is made of ordinary single particle quantum mechanics where the methods used can be compared with solving the Schroedinger equation numerically. The methods are generalized to field theory. Both analytic and numerical strategies are developed for solving the Heisenberg equations on a time lattice. The time development of an initial gaussian wave function is considered in various potentials in order to study both quantum tunnelling phenomena and the quantum roll. 9 refs., 1 fig. (LEW)