含义Meaning
玻尔（1913）假设电子在定态圆轨道上运动、角动量量子化 \(m_evr = n\hbar\)，得到氢原子能级，完美解释了氢光谱。
Bohr (1913) assumed stationary circular orbits with quantised angular momentum \(m_evr = n\hbar\), deriving hydrogen’s levels and explaining its spectrum.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(n\) | 主量子数principal quantum number | 1 |
| \(Z\) | 核电荷数nuclear charge | 1 |
| \(a_0\) | 玻尔半径 52.9 pmBohr radius | m |
推导Derivation
库仑力提供向心力：\(\frac{Ze^2}{4\pi\varepsilon_0r^2} = \frac{m_ev^2}{r}\)。
Coulomb force provides the centripetal force.
角动量量子化 \(m_evr = n\hbar\)，联立解出 \(r_n = \frac{4\pi\varepsilon_0\hbar^2n^2}{m_eZe^2} = n^2a_0/Z\)。
With \(m_evr = n\hbar\), \(r_n = n^2a_0/Z\).
总能量 \(E = \tfrac12m_ev^2 - \frac{Ze^2}{4\pi\varepsilon_0r} = -\frac{Ze^2}{8\pi\varepsilon_0r}\)，代入 r_n 得 E_n。
Total energy \(-Ze^2/(8\pi\varepsilon_0r)\) gives E_n.
适用条件与局限Assumptions & limitations
只适用于单电子体系（H、He⁺、Li²⁺）；不能解释多电子原子、谱线强度与精细结构，“轨道”概念被量子力学的轨道（波函数）取代。
One-electron systems only; fails for intensities, fine structure and many-electron atoms. Orbits are replaced by orbitals in quantum mechanics.
结果与薛定谔方程的能级相同（属巧合式的正确）。
The energies coincide with the Schrödinger result.
例题Worked example
氢原子电离能 = E_∞ − E₁ = 13.6 eV = 1312 kJ/mol；He⁺ 为 54.4 eV。
Hydrogen ionisation energy 13.6 eV (1312 kJ/mol); He⁺ 54.4 eV.