高斯相互作用束缚态Sinc基与谐振子基的比较
Comparing Sinc and Harmonic Oscillator Basis for Bound States of a Gaussian Interaction
摘要：
研究了用正弦配置基和谐振子基求解由径向薛定谔方程描述的高斯势束缚的两粒子系统。我们分析了束缚态波函数的性质，通过调查的基础状态的波函数的故障，并与故障的红外和紫外标度的两个基地。我们提出了一个修正的渐近红外区，长程尾的波函数。我们比较了两种基内计算得到的束缚态本征值和均方半径。从数值结果的趋势，我们确定的优点和缺点的两个基地。我们发现sinc基在精确计算深度和弱束缚态方面表现更好，而谐振子基更方便，因为基态波函数是正交的，并且在位置和动量空间中保持相同的数学结构。谐振子基的这些数学性质在同时使用位置和动量空间的问题中特别有利。在这项工作中所示的谐振子基础的主要缺点是大的基础空间尺寸需要同时获得准确的结果，深和弱束缚状态。sinc基的主要缺点可能是在多体应用中实现它的数值挑战。
英文摘要：
We investigate the use of the sinc collocation and harmonic oscillator bases for solving a two-particle system bound by a Gaussian potential described by the radial Schrödinger equation. We analyze the properties of the bound state wave functions by investigating where the basis-state wave functions break down and relate the breakdowns to the infrared and ultraviolet scales for both bases. We propose a correction for the asymptotic infrared region, the long range tails of the wave functions. We compare the calculated bound state eigenvalues and mean square radii obtained within the two bases. From the trends in the numerical results, we identify the advantages and disadvantages of the two bases. We find that the sinc basis performs better in our implementation for accurately computing both the deeply- and weakly-bound states whereas the harmonic oscillator basis is more convenient since the basis-state wave functions are orthogonal and maintain the same mathematical structure in both position and momentum space. These mathematical properties of the harmonic oscillator basis are especially advantageous in problems where one employs both position and momentum space. The main disadvantage of the harmonic oscillator basis as illustrated in this work is the large basis space size required to obtain accurate results simultaneously for deeply- and weakly-bound states. The main disadvantage of the sinc basis could be the numerical challenges for its implementation in a many-body application.
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