Bohr Orbit Radius

📐 计算公式

玻尔轨道半径 r_n = a₀ × n²，其中 a₀ = 0.529 × 10⁻¹⁰ m（0.529 Å，氢原子基态半径）、n 为主量子数（正整数）；n = 1 时 r = 0.529 Å、n = 2 时 2.12 Å、n = 3 时 4.76 Å；对应能级 E_n = −13.6 ÷ n² eV（基态 −13.6 eV、n = 2 为 −3.4 eV），n 越大能级越靠近电离极限 0。

Compute the radius of the n-th hydrogen orbit (an integer-square multiple of the Bohr radius), suitable for atomic-structure models and quantum-physics teaching demos.

主量子数 n

Formula & Standard References

Item	Formula / Note	Source / Standard
Bohr radius	a₀ = 0.529 Å	Bohr 1913
n-th orbit	r_n = n²a₀/Z	Bohr model
Fine-structure constant	α ≈ 1/137	CODATA 2018
Hydrogen ground state	r₁ = 0.529 Å	Quantum mechanics
Model limits	Exact only for hydrogen-like	Modern physics
Reduced mass	Use reduced mass for precision	QM correction

📌 计算说明

a₀ ≈ 0.529 Å。
n=1 基态半径 0.529 Å。

📐 Physical Constants

Constants used on this page and their authoritative sources, for verification and citation

Constant	Value	Unit	Source
Bohr radius a₀	5.29177210903 × 10⁻¹¹	m	CODATA 2018 (rel. std. unc. 1.5×10⁻¹⁰)
Reduced Planck constant ħ = h/2π	1.054571817 × 10⁻³⁴	J·s	Derived from exact h and π
Electron rest mass m_e	9.1093837015 × 10⁻³¹	kg	CODATA 2018 (rel. std. unc. 3.0×10⁻¹⁰)
Elementary charge e	1.602176634 × 10⁻¹⁹	C	SI defining constant (exact)

Note: a₀ = 4πε₀ħ²/(m_e e²), derived from ħ, m_e, e and ε₀; orbital radii follow r_n = n²·a₀/Z.

📚 深度解析：玻尔轨道半径

💡 常见使用场景

氢原子各能级轨道半径。
原子尺度估算。
里德伯常数关联验证。

基态 n=1

r=a_0·n²=5.29e-11×1=5.29e-11 m=0.529Å（玻尔半径）。

n=2

r=5.29e-11×4=2.12e-10 m；半径按 n² 迅速增大。

❓ 常见问题（FAQ）

a_0 是经验值吗？: 否，a_0=4πε_0ħ²/(m_e e²)=5.29e-11 m，由基本常量导出。
玻尔模型适用范围？: 仅对类氢单电子近似好；多电子需用薛定谔/哈特里-福克。

玻尔轨道半径计算器