A universal Laplace-transform approach to solving Schrödinger equations for all known solvable models
A universal Laplace-transform approach to solving Schrödinger equations for all known solvable models
复制标题
用于求解所有已知可解模型的薛定谔方程的通用拉普拉斯变换方法
DOI：
10.1088/0143-0807/35/1/015006
复制
发表时间：
2013
影响因子：
0.7
通讯作者：
Jyhpyng Wang
中科院分区：
文献类型：
--
作者：
Gin;Jyhpyng Wang
薛定谔方程的可解模型是量子系统的重要模型，因为它们是真实的量子系统的理想近似，并且从可解模型的精确解可以获得对真实的量子系统的许多了解。在本文中，我们表明，一个通用的拉普拉斯变换方案可以用来解决薛定谔方程的封闭形式的所有已知的可解模型。这项工作演示了如何将拉普拉斯变换应用于具有非常数系数的微分方程，这在除了量子力学之外的许多物理分支中都很有用。阐明了拉普拉斯变换相对于幂展开方法的优点及其与超对称形不变势和量子正则变换方法的联系，后者也给出了可解模型的封闭形式解。
Solvable models of the Schrödinger equation are important models of quantum systems because they are idealistic approximations of real quantum systems and much insight into real quantum systems can be gained from the exact solutions of the solvable models. In this paper we show that a universal Laplace transform scheme can be used to solve the Schrödinger equations in closed form for all known solvable models. The work demonstrates how to apply the Laplace transform to differential equations with non-constant coefficients, which is useful in many branches of physics in addition to quantum mechanics. The advantages of the Laplace transform over the power expansion method and its connection with the methods of supersymmetry shape-invariant potentials and quantum canonical transformation, which also give closed-form solutions for solvable models, are elucidated.