A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions
A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions
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DOI：
10.5488/cmp.25.33203
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发表时间：
2021-11
影响因子：
0.6
通讯作者：
W. N. Mathews;M. A. Esrick;Z. Teoh;J. Freericks
中科院分区：
文献类型：
--
作者：
W. N. Mathews;M. A. Esrick;Z. Teoh;J. Freericks
合流超几何方程，又称库默方程，是物理、化学和工程中最重要的微分方程之一。它的两个幂级数解是库默函数M（a，B，z），通常被称为第一类合流超几何函数，和M <$z1-bM（1+a-b，2-B，z），其中a和B是微分方程中出现的参数。第三个函数，Tricomi函数，U（a，B，z），有时被称为第二类合流超几何函数，也是通常使用的合流超几何方程的解。与通常的程序相反，在寻找合流超几何方程的两个线性无关解时，必须考虑所有这三个函数（以及更多）。当a、B和a - B是整数时，存在这样的情况，其中这些函数之一未被定义，或者其中两个函数不是线性无关的，或者微分方程的线性无关解之一不同于这三个函数。这些特殊情况中的许多情况正好对应于解决物理问题所需的情况。这导致了关于如何处理合流超几何方程的严重混乱，尽管有权威的参考文献，如NIST数学函数数字图书馆。在这里，我们仔细地描述了所有不同的情况下，必须考虑和显式公式是什么两个线性无关的解决方案的合流超几何方程。适当地解决合流超几何方程的程序总结在一个方便的表格中。作为一个例子，我们用这些解来研究类氢原子的束缚态，纠正了教科书中的标准处理。我们还简要地考虑了截止库仑势。我们希望本指南将帮助物理学家正确解决涉及合流超几何微分方程的问题。
The confluent hypergeometric equation, also known as Kummer's equation, is one of the most important differential equations in physics, chemistry, and engineering. Its two power series solutions are the Kummer function, M(a,b,z), often referred to as the confluent hypergeometric function of the first kind, and M ≡ z1-bM(1+a-b, 2-b,z), where a and b are parameters that appear in the differential equation. A third function, the Tricomi function, U(a,b,z), sometimes referred to as the confluent hypergeometric function of the second kind, is also a solution of the confluent hypergeometric equation that is routinely used. Contrary to common procedure, all three of these functions (and more) must be considered in a search for the two linearly independent solutions of the confluent hypergeometric equation. There are situations, when a, b, and a - b are integers, where one of these functions is not defined, or two of the functions are not linearly independent, or one of the linearly independent solutions of the differential equation is different from these three functions. Many of these special cases correspond precisely to cases needed to solve problems in physics. This leads to significant confusion about how to work with confluent hypergeometric equations, in spite of authoritative references such as the NIST Digital Library of Mathematical Functions. Here, we carefully describe all of the different cases one has to consider and what the explicit formulas are for the two linearly independent solutions of the confluent hypergeometric equation. The procedure to properly solve the confluent hypergeometric equation is summarized in a convenient table. As an example, we use these solutions to study the bound states of the hydrogenic atom, correcting the standard treatment in textbooks. We also briefly consider the cutoff Coulomb potential. We hope that this guide will aid physicists to properly solve problems that involve the confluent hypergeometric differential equation.