Explore quantized states for hydrogen with confidence now. Use transitions to model emission spectra easily. Check energies, wavelengths, and radii in one place fast.
Enter quantum numbers and settings. Results appear above this form after submit.
Sample values for hydrogen (Z=1) using reduced mass correction.
| n | Eₙ (eV) | rₙ (m) | Example transition | λ (nm) |
|---|---|---|---|---|
| 1 | -13.60 | 5.29e-11 | 2 → 1 (Lyman) | 121.6 |
| 2 | -3.40 | 2.12e-10 | 3 → 2 (Balmer) | 656.3 |
| 3 | -1.51 | 4.76e-10 | 4 → 3 (Paschen) | 1875 |
Values are typical reference numbers and may differ slightly by constants and corrections.
Energy of a hydrogen-like atom in the Bohr model is:
En = -Ry · (μ/me) · Z² / n²
Ry ≈ 13.6057 eV is the Rydberg energy. μ is the reduced mass.
Photon energy for a transition nᵢ → n𝒇:
ΔE = Ry · (μ/me) · Z² · (1/n𝒇² − 1/nᵢ²)
Wavelength from Planck relation:
λ = hc / |ΔE|
Radius:
rn = a0 · (me/μ) · n² / Z
Speed estimate:
vn = αc · Z / n
Hydrogen is the reference system for atomic structure because its spectrum is dominated by clean, quantized transitions. Each principal quantum number n defines a discrete bound energy En, so emission and absorption occur only at specific photon energies.
In the Bohr model, the ground state energy is about −13.6 eV for Z=1. The n=2 level is near −3.40 eV, and n=3 is near −1.51 eV. The spacing decreases with n, which is why higher lines crowd together near the series limit.
The calculator identifies common series by the lower level. Lyman lines end at n=1 and fall in the ultraviolet, including the famous 2→1 line near 121.6 nm. Balmer lines end at n=2 and include the 3→2 line near 656.3 nm in the visible red.
Spectroscopy often uses wavenumber in cm⁻¹. This tool computes wavenumber from λ and also from the Rydberg form to provide a consistency check. If inputs are valid, both values closely agree, with small differences only from rounding.
The Bohr radius a0 sets the size scale. Orbital radius grows as n² and shrinks as 1/Z. For hydrogen, r1 ≈ 5.29×10⁻¹¹ m and r2 ≈ 2.12×10⁻¹⁰ m. For a hydrogen-like ion with Z=2, radii are roughly half for the same n.
The electron speed estimate uses vn = αc·Z/n, giving v≈2.19×10⁶ m/s for hydrogen at n=1. This is a useful scale for trends, while detailed spectra also include fine structure, Lamb shift, and hyperfine effects.
Real nuclei are not infinitely heavy. Using the reduced mass μ slightly increases accuracy by scaling energies by μ/me. For hydrogen, μ/me is about 0.99946, shifting predicted lines by a fraction of a percent—small but measurable in precision work.
Start by choosing Z and the transition nᵢ→n𝒇 that matches the spectral line. Use emission when nᵢ>n𝒇. Record ΔE, λ, f, and cm⁻¹, then compare with lab or reference spectra. Export results to CSV for reports and to PDF for quick sharing.
Negative En indicates a bound electron. Zero energy is defined at infinite separation. A more negative value means the electron is more tightly bound to the nucleus.
Use nᵢ greater than n𝒇 for emission, because the electron drops to a lower level and releases a photon. Use nᵢ less than n𝒇 for absorption, because the electron gains energy from a photon.
Energies scale with Z² for hydrogen-like atoms. Larger Z increases transition energies, which reduces wavelength because λ = hc/|ΔE|.
Series names group transitions by the lower level. Lyman ends at n=1, Balmer ends at n=2, Paschen ends at n=3, and so on. This helps classify spectral regions.
Enable it for better accuracy, especially when comparing with measured spectra. It accounts for nucleus motion by replacing the electron mass with the electron–nucleus reduced mass.
Yes. Set Z=2 for He⁺ and Z=3 for Li²⁺. The formulas are for hydrogen-like ions with one electron. Multi-electron atoms need additional screening and quantum corrections.
Differences can come from rounding, alternative constants, or higher-order physics such as fine structure and relativistic corrections. This calculator is based on the Bohr hydrogen-like model with optional reduced mass.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.