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。

Probability for a Range of Radius
Hydrogen Ground State

氢原子基态的半径范围的概率

Finding the probability that the electron in the hydrogen ground state will be found in the range r=b to r=c requires the integration of the radial probability density.

求电子在氢原子基态中处于r=b到r=c范围内的概率，需要对径向概率密度进行积分。

This requires integration by parts. The form of the solution is

这需要分部积分。解的形式是
For a range from b = a0 to c= a0
the probability is x 10^.
Radial behavior of ground state
基态的径向行为
Most probable radius
最可能半径
Expectation value for radius
半径的期望值
在 b 从 a0 到 c=a0 的范围内，概率是 x 10^ 。
中文译文中的待填/计算数值依次对应：1：b 2：c 3：pb 4：pp。实际数值以上方原输入框为准。
 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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