含义Meaning
在半径 r 到 r + dr 的球壳内找到电子的概率。1s 电子最可能出现在玻尔半径处，而不是核上（尽管 ψ² 在核处最大）。
Probability of finding the electron in a shell between r and r + dr. The 1s electron is most likely at the Bohr radius, not at the nucleus (where ψ² is largest).
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(P(r)\) | 径向分布函数radial distribution function | m⁻¹ |
| \(R_{nl}\) | 径向波函数radial wavefunction | m⁻³ᐟ² |
| \(a_0\) | 玻尔半径 52.9 pmBohr radius 52.9 pm | m |
推导Derivation
概率 = \(|\psi|^2\) × 体积；半径 r、厚 dr 的球壳体积 \(4\pi r^2dr\)。
Probability = \(|\psi|^2\) × volume; a shell has volume \(4\pi r^2dr\).
对角度积分，球谐函数已归一化，得 \(P(r)\,dr = r^2R^2\,dr\)。
Integrating over angles (normalised \(Y\)) leaves \(r^2R^2dr\).
1s：\(P = 4(Z/a_0)^3r^2e^{-2Zr/a_0}\)，\(dP/dr = 0\) ⇒ \(r = a_0/Z\)。
For 1s, \(dP/dr = 0\) at \(r = a_0/Z\).
2s 有一个径向节点（r = 2a₀）；2p 无径向节点但离核更远。2s 的内层小峰体现“穿透效应”，使 2s 能量低于 2p（多电子原子）。
2s has a radial node at 2a₀; its inner lobe (penetration) lowers 2s below 2p in many-electron atoms.
适用条件与局限Assumptions & limitations
严格适用于单电子体系；多电子原子形状类似但由有效核电荷修正。
Exact for one-electron systems; similar shapes with effective Z otherwise.