Calculate energy levels, transition wavelengths, orbital radii, and spectral series for hydrogen-like atoms using the Bohr model.
The hydrogen-like atom — an atom or ion with just one electron — is the only atomic system that can be solved exactly in quantum mechanics. The Bohr model, while simplified, correctly predicts the energy levels, orbital radii, and spectral lines for hydrogen (Z=1), He⁺ (Z=2), Li²⁺ (Z=3), and higher hydrogen-like ions.
This Hydrogen-Like Atom Calculator uses the Bohr model to compute energy levels, transition energies, emitted or absorbed photon wavelengths, orbital radii, and orbital speeds for any hydrogen-like species. Simply enter the atomic number Z and two principal quantum numbers to see the transition properties, or explore individual orbital characteristics.
The calculator also identifies the spectral region of each transition and provides a reference table of the hydrogen spectral series such as Lyman, Balmer, Paschen, Brackett, and Pfund. It is useful for spectroscopy homework, introductory quantum mechanics, and quick checks on one-electron ions before moving to more complete quantum treatments.
Use this calculator to move quickly between energy levels, wavelengths, and orbital sizes without manually reworking the Bohr-model constants for each ion or transition. It is especially handy when you want to check a hydrogen line or one-electron ion before moving on to a more detailed quantum treatment. It also keeps the spectral-series context in view when you are comparing several candidate transitions. That makes it easier to sanity-check a line assignment or compare several candidate ions side by side.
Energy Level: Eₙ = −13.6 × Z² / n² (eV) Photon Energy: ΔE = |Eₙ₂ − Eₙ₁| Wavelength: λ = 1240 / ΔE (nm, when ΔE in eV) Orbital Radius: rₙ = a₀ × n² / Z (a₀ = 52.9 pm) Orbital Speed: vₙ = 2.188 × 10⁶ × Z / n (m/s)
Result: ΔE = 10.2 eV, λ = 121.5 nm (UV, Lyman-α)
The transition from n=1 to n=2 in hydrogen emits/absorbs a 121.5 nm ultraviolet photon — the Lyman-alpha line.
For one-electron atoms and ions, the Bohr model gives the correct structure of the energy levels and the familiar 1/n² dependence. That makes it a practical teaching model for hydrogen, He+, Li2+, and similar ions where electron-electron repulsion is absent.
If you are identifying a line, first confirm the lower energy level. Transitions ending at n = 1 belong to the Lyman series in the ultraviolet, n = 2 gives the Balmer series in the visible and near ultraviolet, and higher terminal levels move into the infrared. Matching the final level is often the fastest way to sanity-check a wavelength.
The Bohr model does not include fine structure, relativistic corrections, spin-orbit coupling, or Lamb shifts. It is excellent for order-of-magnitude reasoning and basic spectroscopy, but not for precision atomic data or multi-electron atoms.
Any atom or ion with exactly one electron: H, He⁺, Li²⁺, Be³⁺, etc. The Bohr model is exact for these systems.
A larger nuclear charge pulls the electron into a tighter orbit and increases the binding energy. In the Bohr model, that scaling appears as Z² in the energy expression.
The fundamental constant R∞ = 13.6 eV that sets the energy scale for hydrogen-like atoms.
It is the family of transitions that end at n = 2. In hydrogen, several Balmer lines fall in the visible range, including H-alpha near 656 nm.
No — it is only accurate for one-electron systems. Multi-electron atoms require quantum-mechanical treatments.
The Bohr model assumes an infinitely heavy nucleus. The reduced mass correction (< 0.1% for hydrogen) accounts for the finite nuclear mass.