量子流體力學-理論與計算(1/2)

Abstract

近來量子流體力學的研究在各個領域蓬勃發展, 在de Broglie-Bohm 方法中將複數的波函 數以極座標形式來表達, 再將其代入非線性薛丁格方程式之中, 就可以得到一組相對應的量子 流體力學方程式, 其中有類比於古典流體力學的密度、速度, 但亦有新的量子位能與壓力項出 現, 它們將是產生量子行為的因素, 量子位能有著特殊的形式, 它與密度之量值和變化量相關, 量子流體力學形式裡當波函數為零時, 即會造成有奇異的現象發生, 此即是在超流體中漩渦的 現象, 奇異的現象在數值計算上亦會造成若干的困難, 故建立一套解量子流體力學的數值方法 除要考量其精確性外, 對於奇異問題的處理更是一大重點。 本計畫目前所進行之工作主要分成兩個方向, 一為利用高解析流體守恆律算則來解量子流 體力學方程式, 利用量子流體力學方程式與古典流體力學方程式相似的特性, 便可將計算流體 力學的方法運用在計算量子力學的問題之中, 二為利用高精確度的徑向基底函數展開法來解非 線性薛丁格方程式與量子流體力學方程式, 徑向基底函數展開法是一種無網格且高精確的方法, 所以運用在計算當中非常地有效率。從現階段的結果中可以發現到各種流體的現象, 並且可以 發現一些量子流體特有之性質。

Recently, interest in the de Broglie-Bohm formulation of quantum mechanics has increased dramatically within various fields. In the de Broglie-Bohm approach, the complex wave function is expressed in polar form and substituted into the time-dependent linear or nonlinear Schrödinger equation results a set of hydrodynamic-like equations which describe the flow of the probability. This set of equations are very similar to those of classical hydrodynamics except that an additional quantum potential term is present. The quantum potential and its associated quantum force give rise to all quantum effects such as tunneling and interference. Despite its conceptually attractive features, there are major computational problems inherent to the de Broglie-Bohm approach when a direct numerical solution of the quantum hydrodynamics is attempted. In particular, the quantum potential possess some unique features, it does not respond to the intensity of the wave but rather depends upon its form. A desirable numerical method for solving the quantum hydrodynamics based on Schrödinger equation not only requires high accuracy but also the ability to handle discontinuities caused by the singularity of quantum potential. The current work has two independent directions: one is high resolution scheme, the other is radial basis function based scheme.