What is a hydrogen-like atom?

Any atom or ion with exactly one electron: H, He⁺, Li²⁺, Be³⁺, etc. The Bohr model is exact for these systems.

Why do energy levels scale with Z²?

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.

What is the Rydberg constant?

The fundamental constant R∞ = 13.6 eV that sets the energy scale for hydrogen-like atoms.

What is the Balmer series?

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.

Does the Bohr model work for multi-electron atoms?

No — it is only accurate for one-electron systems. Multi-electron atoms require quantum-mechanical treatments.

What is the reduced mass correction?

The Bohr model assumes an infinitely heavy nucleus. The reduced mass correction (< 0.1% for hydrogen) accounts for the finite nuclear mass.

Hydrogen-Like Atom Calculator

Calculate energy levels, transition wavelengths, orbital radii, and spectral series for hydrogen-like atoms using the Bohr model.

Atomic Number (Z)

Lower Energy Level (n₁)

Upper Energy Level (n₂)

Calculation Type

Reduced Mass Correction

Energy Level n=1

-13.6000 eV

E_n = −13.6 × Z²/n² eV

Energy Level n=2

-3.4000 eV

Energy of the upper level

Photon Energy

10.2000 eV

Energy of absorbed/emitted photon = |E₂ − E₁|

Wavelength

121.57 nm

λ = 1240 / E (eV·nm) — Ultraviolet

Frequency

2.468e+15 Hz

f = c / λ

Orbital Radius (n=1)

52.9 pm

r_n = a₀ × n²/Z (Bohr radius = 52.9 pm)

Ionization Energy

13.6000 eV

Energy to remove electron from n=1

Orbital Speed (n₁)

2.188e+6 m/s

v_n = 2.188×10⁶ × Z/n

Spectral Region

UV (10 nm)VisibleIR (1500 nm)

Series	Lower Level	Spectral Range	Transitions
Lyman	n = 1	UV (91–122 nm)	n→1
Balmer	n = 2	Visible (365–656 nm)	n→2
Paschen	n = 3	Near IR (820–1875 nm)	n→3
Brackett	n = 4	IR (1458–4051 nm)	n→4
Pfund	n = 5	Far IR (2279–7458 nm)	n→5

Planning notes, formulas, and examples

About the Hydrogen-Like Atom Calculator

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.

When This Page Helps

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.

How to Use the Inputs

Enter the atomic number Z (1 for hydrogen, 2 for He⁺, etc.).
Enter the lower principal quantum number n₁.
Enter the upper principal quantum number n₂.
Select the calculation type: transition energy or orbital properties.
Optionally apply a reduced-mass correction for hydrogen or deuterium.
Review energy levels, photon energy, wavelength, orbital radius, and speed.
Check the spectral series table for context.

Formula used

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)

Example Calculation

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.

Tips & Best Practices

Check that all inputs use the same scale and assumptions before trusting the result.
Compare the answer with the worked example or a rough estimate to catch entry mistakes.

What The Model Captures

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.

Spectral-Series Checks

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.

Where Accuracy Runs Out

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.

Sources & Methodology

Last updated: January 15, 2025

Frequently Asked Questions

Any atom or ion with exactly one electron: H, He⁺, Li²⁺, Be³⁺, etc. The Bohr model is exact for these systems.