Ren G B, Rorison J M
Department of Electrical and Electronic Engineering, University of Bristol, BS8 1TR, United Kingdom.
Phys Rev E Stat Nonlin Soft Matter Phys. 2004 Mar;69(3 Pt 2):036705. doi: 10.1103/PhysRevE.69.036705. Epub 2004 Mar 31.
We present a very efficient scheme to calculate the eigenvalue problem of the time-independent Schrödinger equation. The eigenvalue problem can be solved via an initial-value procedure of the time-dependent Schrödinger equation. First, the time evolution of the wave function is calculated by the finite-difference time-domain method. Then the eigenenergies of the electron system can be obtained through a fast Fourier transformation along the time axis of the wave function after some point. The computing effort for this scheme is roughly proportional to the total grid points involved in the structure and it is suitable for large scale quantum systems. We have applied this approach to the three-dimensional GaN quantum dot system involving one million grid points. It takes only 7 h to calculate the confined energies and the wave functions on a standard 2-GHz Pentium 4 computer. The proposed approach can be implemented in a parallel computer system to study more complex systems.
我们提出了一种非常有效的方案来计算与时间无关的薛定谔方程的本征值问题。该本征值问题可以通过含时薛定谔方程的初值过程来求解。首先，利用时域有限差分法计算波函数的时间演化。然后，在经过某一点后，通过沿波函数时间轴进行快速傅里叶变换，可以得到电子系统的本征能量。该方案的计算量大致与结构中涉及的总网格点数成正比，适用于大规模量子系统。我们已将此方法应用于包含一百万个网格点的三维氮化镓量子点系统。在一台标准的2-GHz奔腾4计算机上计算受限能量和波函数仅需7小时。所提出的方法可以在并行计算机系统中实现，以研究更复杂的系统。
