Bessel Functions

贝塞尔函数

One of the varieties of special functions which are encountered in the solution of physical problems is the class of functions called Bessel functions. They are solutions to a very important differential equation, the Bessel equation:

在解决物理问题时遇到的特殊函数之一是被称为贝塞尔函数的函数类。它们是解一个非常重要微分方程——贝塞尔方程的解：

The solutions to this equation are in the form of infinite series which are called Bessel funtions of the first kind. The expression for the sum is

该方程的解形式为无穷级数，称为第一类贝塞尔函数。该级数的表达式为

Values for the Bessel functions can be found in most collections of mathematical tables. Bessel functions are encountered in physical situations where there is cylindrical symmetry. This occurs in problems involving electric fields, vibrations, heat conduction, optical diffraction and others.

贝塞尔函数的数值可以在大多数数学表集中找到。贝塞尔函数出现在具有圆柱对称性的物理问题中。这发生在涉及电场、振动、热传导和光学衍射等问题中。

For the specific application to a spherical potential well in quantum mechanics, another form called spherical bessel functions appears.

在量子力学中，针对球形势阱的具体应用，另一种形式称为球贝塞尔函数出现。

Index

Reference
Kreyszig
Sec 4.5

索引参考Kreyszig第4.5节

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Spherical Bessel Functions

球贝塞尔函数

A specific class of special functions called spherical Bessel functions arises in problems of spherical symmetry like the spherical potential well in quantum mechanics. The first three forms are

在具有球对称性的问题中，如量子力学中的球形势阱，会出现一类特殊的特殊函数，称为球贝塞尔函数。前三种形式为

From the first form, the higher order forms can be generated from the relationship

从第一种形式出发，可以通过该关系生成更高阶的形式。

It is sometimes useful to have the limiting cases for these functions for very large or very small distances:

在非常大的或非常小的距离情况下，了解这些函数的极限情况有时也是有用的：

Index

Reference
Krane
Ch 2

索引参考 Krane 第2章

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