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首页> 外文会议>Mathematical models and methods in applied sciences >Plenary Lecture 5 Variational Treatment of Screened Coulomb Potentials: The Yukawa Potential
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Plenary Lecture 5 Variational Treatment of Screened Coulomb Potentials: The Yukawa Potential

机译：全体会议第5讲：筛查库仑电势的变化处理：汤川电势

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The most fundamental equation of Theoretical Chemistry and of Atomic Physics is the Schroedinger equation for a hydrogen like system. Its solution can be found in any standard textbook on Atomic Physics, Quantum Chemistry and so on. A similar equation which is somewhat more complicated is the Schroedinger equation for a particle bound in what is known in the literature as screened Coulomb potential. The screening function that will be discussed is one which is solely dependent on the radial variable r and is known in the literature as the Yukawa potential. This potential arises naturally as the position space version of the solution of the Klein-Gordon equation for a static meson field. It was the deuteron problem which inspired the first solutions to the corresponding eigenvalue equation. It is commonly known in plasma physics as the "Debye-Hueckel" potential and represents the effect of the plasma sea on localized two-particle interactions. The Debye-Hueckel potential also approximates the Thomas-Fermi potential in the calculation of the energy levels of the impurity centers in doped semiconductors. Together with the Hulten and the exponential potentials the Yukawa potential plays an important role as a good test case in potential scattering studies also. In quantum chemistry the effect of the core electrons on the valence electrons can be modeled by means of a linear combination of Yukawa or similar potentials. Various approaches have been made to attempt to solve the eigenvalue problem associated to the corresponding Schroedinger equation having Yukawa or similar screened coulomb potentials. Quite a few of these use perturbational and variational techniques. There were also group theoretical approaches. Direct numerical integration of the corresponding Schroedinger equation were also employed and quite succesfully so. Regge trajectories were determined via this means or by utilizing continued fractions. There are of course plenty of other works related to Yukawa potential. The method that will be discussed during the talk is also based on variational treatment of the radial Schroedinger equation with Yukawa potential. It employs a Laguerre basis set extended by an extra function. A parameter used in this extra function and its relation with the energy of the system results in the utilization of an auto-coherent (or self-consistent) scheme. The proposed method does not only give energy values for the ground and the first few excited states consistently up to thirty digits but also gives threshold screening parameter values accurate to 15-20 decimal points.
机译：理论化学和原子物理学的最基本方程是类氢系统的Schroedinger方程。可以在有关原子物理学，量子化学等的任何标准教科书中找到其解决方案。一个类似的方程式稍微复杂一些，它是粒子束缚的Schroedinger方程，在文献中称为屏蔽库仑势。将要讨论的筛选函数是一种仅取决于径向变量r的函数，在文献中称为Yukawa势。作为静态介子场的Klein-Gordon方程解的位置空间版本，自然会产生这种潜力。氘核问题激发了相应特征值方程的第一个解决方案。在等离子体物理学中，通常将其称为“ Debye-Hueckel”势，它代表了等离子体海对局部两粒子相互作用的影响。在计算掺杂半导体中杂质中心的能级时，Debye-Hueckel势也近似于Thomas-Fermi势。与Hulten和指数势一起，Yukawa势在势散射研究中作为一个良好的测试案例也起着重要的作用。在量子化学中，可以通过Yukawa或类似电势的线性组合来模拟核心电子对价电子的影响。已经采取了各种方法来尝试解决与具有汤川或类似的被筛选的库仑势的相应的施罗丁格方程相关的特征值问题。其中许多使用微扰和变分技术。也有小组理论方法。相应的Schroedinger方程的直接数值积分也被采用，并且非常成功。 Regge轨迹是通过这种方式或利用连续分数来确定的。当然，还有许多与汤川潜力有关的其他作品。谈话中将要讨论的方法也是基于对具有Yukawa势的径向Schroedinger方程的变分处理。它采用了Laguerre基础集，并通过附加功能进行了扩展。在此额外功能中使用的参数及其与系统能量的关系导致使用自相关（或自洽）方案。所提出的方法不仅给出地面和前几个激发态的能量值，一直到三十位数，而且给出了精确到15-20个小数点的阈值筛选参数值。

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