Hydrogen-Like Atom Calculator — Bohr Energy Levels
Compute the energy, orbital radius, and emitted photon wavelength for any level of hydrogen-like ions (H, He⁺, Li²⁺, …) — enter atomic number Z, lower quantum number n, and upper level n′ for the transition.
Bound-state energy (negative means the electron is bound to the nucleus)
- 1
Z² (nuclear charge squared)
1² = 1 × 1 = 1 - 2
n² (principal quantum number squared)
2² = 2 × 2 = 4 - 3
Energy of level n
Eₙ = −13.6 × 1 ÷ 4 = -3.4000Negative energy confirms the electron is bound to the nucleus.
How does this calculator work?
Eₙ = −13.6 Z²/n² eV; orbital radius rₙ = a₀ n²/Z. Emission wavelength from n′→n: 1/λ = Z² R∞ (1/n² − 1/n′²). Works for any one-electron ion (H, He⁺, Li²⁺, …). For hydrogen n=2→n=3 gives the Hα red line at ~656 nm.
Formula
How this is calculated
The Bohr model predicts that an electron in a hydrogen-like ion of nuclear charge Z occupies circular orbits with energy Eₙ = −13.6 Z²/n² eV and radius rₙ = a₀ n²/Z, where a₀ = 0.0529 nm is the Bohr radius. Negative energy means the electron is bound; the ground state (n = 1) has the most negative energy. Larger Z pulls the electron closer and lowers all levels proportionally to Z².
When an electron falls from a higher level n′ to a lower level n, it emits a photon with energy ΔE = 13.6 Z² (1/n² − 1/n′²) eV. The Rydberg formula 1/λ = Z² R∞ (1/n² − 1/n′²), with R∞ = 1.0974 × 10⁷ m⁻¹, gives the photon wavelength. For hydrogen (Z=1, n=2, n′=3), this gives the Hα Balmer line at ~656 nm (red).
The Bohr model is exact for one-electron systems and works well for highly ionised atoms (He⁺, Li²⁺, …). It does not account for fine structure, relativistic corrections, spin, or multi-electron shielding.
Frequently asked questions
For hydrogen (Z=1, n=1), Eₙ = −13.6 eV. This equals the ionisation energy — the energy needed to completely remove the electron from a ground-state hydrogen atom.
The Hα line is the n′=3 → n=2 transition in hydrogen. Using Z=1: 1/λ = R∞ × (1/4 − 1/9) = 1.524 × 10⁶ m⁻¹, giving λ ≈ 656 nm (red visible light).
A larger nuclear charge Z attracts the electron more strongly, shrinking the orbital radius by 1/Z and deepening the energy well by Z². Both the Coulomb attraction and the resulting radius change contribute a factor of Z, giving Z² overall.
TG we-Calculate Editorial Team. (2026). Hydrogen-Like Atom Calculator — Bohr Energy Levels [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/hydrogen-like-atom-calculator
TG we-Calculate Editorial Team. "Hydrogen-Like Atom Calculator — Bohr Energy Levels." TG we-Calculate. 2026. https://we-calculate.com/calculator/hydrogen-like-atom-calculator.
TG we-Calculate Editorial Team, "Hydrogen-Like Atom Calculator — Bohr Energy Levels," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/hydrogen-like-atom-calculator
@misc{wecalculate_hydrogen_like_atom_calculator, title = {Hydrogen-Like Atom Calculator — Bohr Energy Levels}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/hydrogen-like-atom-calculator}}, year = {2026}, note = {TG we-Calculate} }
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