Bilingual edition: English is preserved and Chinese follows each unit. Terminology uses the confirmed v20260916 glossary; automated semantic review remains traceable.
中英双语版：英文原文完整保留，中文紧随对应单元；术语采用 v20260916 确认表，自动语义审校结果可追溯。

Radial Equation Separation

径向方程分离

Substituting the form

代入该形式

into the hydrogen Schrodinger equation allows all the partial derivatives to be expressed as ordinary derivatives. Direct substitution and a bit of rearranging gives

将变量代入氢原子的薛定谔方程可以使所有偏导数表示为普通导数。直接代入并稍作整理可得

Note that the first two terms contain only variable r, and the remaining terms contain the angular variables. Since the equation must hold for all values of these variables, the collection of radial terms must be equal to a constant. This separates out the radial equation. The sum of the angular terms must therefore be equal to the negative of that constant. Also, the terms in the two angles must separately equal a constant since they must hold for all angle values. This separates the two angular parts into the colatitude and azimuthal equations.

注意，前两项仅包含变量r，其余项包含角变量。由于该方程必须对所有这些变量的值成立，因此径向项的集合必须等于一个常数。这将径向方程分离出来。因此，角项的总和必须等于该常数的负数。此外，两个角度中的项必须各自等于一个常数，因为它们必须对所有角度值成立。这将两个角部分分别分离为极角和方位角方程。
 Index

Schrodinger equation concepts

Hydrogen concepts
索引 希尔伯特方程概念 氢原子概念
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The Radial Equation

径向方程

Upon separation of the Schrodinger equation for the hydrogen atom, the radial equation is:

在分离氢原子的薛定谔方程后，径向方程为：

In order to separate the equations, the radial part is set equal to a constant, and the form of the constant on the right above reflects the nature of the solution of the colatitude equation which yields the orbital quantum number.

为了分离方程，将径向部分设为常数，右边的常数形式反映了球面角坐标方程的解的性质，该解决定了轨道量子数。

Solution of these equations under the constraints placed on the wavefunction leads to series solutions in the form of polynomials called the associated Laguerre functions. In order to fit the physical boundary conditions, these solutions contain a parameter n which can take only positive integer values; this parameter is called the principal quantum number. The form of the radial solutions is

解这些方程并在波函数所受的约束下求解，得到一系列以多项式形式表示的解，称为关联拉格朗日函数。为了满足物理边界条件，这些解中包含一个参数n，其只能取正整数值；这个参数称为主量子数。径向解的形式为

where Ln,l is the associated Laguerre function. The first few radial wavefunctions R are shown as part of the hydrogen wavefunctions.

其中 $ L_{n,l} $ 是关联的拉格朗日函数。氢原子波函数中展示了前几个径向波函数 $ R $。
 Index

Schrodinger equation concepts

Hydrogen concepts
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The Principal Quantum Number

主量子数

The principal quantum number or total quantum number n arises from the solution of the radial part of the Schrodinger equation for the hydrogen atom. The bound state energies of the electron in the hydrogen atom are given by

主量子数或总量子数n来源于氢原子的径向薛定谔方程的解。氢原子中电子的束缚态能量由

In the notation of the periodic table, the main shells of electrons are labled K(n=1),L(n=2),M(n=3), etc. based on the principal quantum number.

在元素周期表的表示法中，电子的主壳层被标记为K(n=1)，L(n=2)，M(n=3)等，这是基于主量子数。
Quantum numbers for atomic electrons
原子电子的量子数
 Index

Schrodinger equation concepts

Hydrogen concepts
索引 希尔伯特方程概念 氢原子概念
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