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。

What Causes Electron Energies to Depend Upon the Orbital Quantum Number?

什么导致电子能量依赖于轨道量子数?
From the Bohr model or the hydrogen Schrodinger equation, the solution for the electron energy levels gives:

This fits the hydrogen spectrum unless you take a high resolution look at fine structure or the structure produced by external magnetic fields (Zeeman effect), etc.

这符合氢谱线的情况,除非你用高分辨率观察精细结构或由外部磁场产生的结构(塞曼效应)等。

Hydrogen-like atoms such as lithium and sodium might be expected to exhibit similar energy levels. They consist of closed shells with a single electron outside. Envisioning a Bohr-type shell structure with just a single electron in the outer shell, the net charge inside that shell is just one net positive charge. This leads to the following expectation:

像锂和钠这样的氢原子可能表现出相似的能量级。它们由闭合壳层和一个外层电子组成。设想一个类似玻尔的壳层结构,仅有一个电子在最外层,那么该壳层内的净电荷仅为一个净正电荷。这导致以下预期:

However, when data from spectra are used to build energy level diagrams for these atoms, a strong orbital dependence of the energy is found for the electrons of low angular momentum as shown below.

然而,当用光谱数据构建这些原子的能级图时,发现低角动量电子的能级表现出显著的轨道依赖性,如下所示。

What is the origin of the orbital quantum number dependence?

轨道量子数的依赖性来源于何处?
从玻尔模型或氢原子的薛定谔方程来看,电子能级的解为:
Index

Schrodinger equation concepts

Hydrogen concepts

Rohlf
Ch 9
索引 希尔伯特方程概念 氢原子概念 Rohlf 第9章
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Origin of Orbital Quantum Number Dependence of Electron Energies

电子能量的轨道量子数依赖性的起源

When the wavefunctions for electrons with different orbital quantum numbers are examined, it is found that there is a different amount of penetration into the region occupied by the 1s electrons. This penetration of the shielding 1s electrons exposes them to more of the influence of the nucleus and causes them to be more tightly bound, lowering their associated energy states.

当考察具有不同轨道量子数的电子波函数时,发现它们对1s电子所处区域的穿透程度不同。这种穿透使屏蔽的1s电子更接近核的影响,从而更紧密地束缚,降低其相应的能量状态。
In the case of lithium, the 2s electron shows more penetration inside the first Bohr radius and is therefore lower than the 2p. In the case of sodium with two filled shells, the 3s electron penetrates the inner shielding shells more than the 3p and is significantly lower in energy.
Hydrogen radial probability distribution
氢原子径向概率分布
在锂的情况下,2s电子更深入第一玻尔半径内,因此其能量低于2p电子。在钠的情况下,具有两个填满的壳层,3s电子比3p电子更深入内层屏蔽壳层,其能量显著更低。
Index

Schrodinger equation concepts
索引:薛定谔方程概念
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