Hydrogen Atom Energy Levels Calculator
Computes hydrogen electron energy levels, transition energies, emitted wavelengths, and photon frequencies from two quantum numbers. Use it when studying the Bohr model, spectral series, or atomic emission spectra.
About this calculator
The Bohr model assigns each hydrogen electron a discrete energy Eₙ = −13.6 eV / n², where n is the principal quantum number (n = 1, 2, 3, …). When an electron moves from level n₁ to n₂, the transition energy is ΔE = 13.6 × (1/n₂² − 1/n₁²) eV. A positive ΔE means the photon is absorbed; negative means it is emitted. The emitted wavelength follows the Rydberg formula: 1/λ = R_H × |1/n₂² − 1/n₁²|, where R_H = 1.097×10⁷ m⁻¹. Photon frequency is given by ν = 3.29×10¹⁵ × |1/n₂² − 1/n₁²| Hz. The familiar spectral series — Lyman (n₂=1), Balmer (n₂=2), Paschen (n₂=3) — all follow directly from these expressions, explaining hydrogen's characteristic line spectrum.
How to use
Find the wavelength of the first Balmer-series line (n₁ = 3 → n₂ = 2). Set n₁ = 3, n₂ = 2, calculation type = wavelength. The formula gives: 1/λ = 1.097×10⁷ × |1/4 − 1/9| = 1.097×10⁷ × (5/36) = 1.524×10⁶ m⁻¹. Therefore λ = 1 / 1.524×10⁶ ≈ 6.56×10⁻⁷ m = 656 nm — the red Hα line visible in hydrogen discharge tubes. The transition energy is ΔE = 13.6 × (1/4 − 1/9) = 13.6 × 0.1389 ≈ 1.89 eV.
Frequently asked questions
What are the hydrogen spectral series and what quantum numbers define them?
Each spectral series groups all transitions that share the same lower energy level n₂. The Lyman series (n₂ = 1) produces ultraviolet lines; the Balmer series (n₂ = 2) produces visible lines including the red Hα at 656 nm and blue Hβ at 486 nm; the Paschen series (n₂ = 3) falls in the near-infrared. Higher series (Brackett, Pfund) lie in the mid-infrared. Astronomers use these series to identify hydrogen in stellar atmospheres, and laboratory spectroscopists use them to calibrate spectrometers.
Why does the hydrogen atom only emit specific wavelengths of light?
Quantum mechanics restricts the electron to discrete, quantised energy levels described by integer quantum numbers n. A photon is emitted only when an electron drops from a higher level to a lower one, and the photon's energy exactly equals the energy difference ΔE = 13.6 × (1/n₂² − 1/n₁²) eV. Because the levels are discrete, the differences are discrete, producing a line spectrum rather than a continuous one. This quantisation was the key evidence that classical physics could not explain atomic stability or spectra.
How do I calculate the ionisation energy of hydrogen from the energy level formula?
Ionisation means removing the electron entirely, which corresponds to a transition from level n₁ to n = ∞. Substituting n₂ = ∞ into the transition-energy formula gives ΔE = 13.6 × (1/∞² − 1/n₁²) = 13.6 / n₁² eV. For the ground state n₁ = 1, the ionisation energy is exactly 13.6 eV (≈ 2.18×10⁻¹⁸ J). For an excited state n₁ = 2 it is only 3.4 eV, explaining why excited hydrogen is much easier to ionise. This principle is used in astrophysics to model photoionisation of interstellar gas.