Hydrogen Atom Energy Levels Calculator

Hydrogen has an infinite number of discrete energy levels, labeled by the principal quantum number n = 1, 2, 3, … and so on up to infinity. At n = ∞, the electron is completely free from the nucleus (ionization). In practice, the most chemically and spectroscopically relevant levels are n = 1 through n = 7.

The energy of an electron in level n for a hydrogen-like ion is given by Eₙ = −(Z² × R_H) / n², where Z is the atomic number, n is the principal quantum number, and R_H ≈ 2.18 × 10⁻¹⁸ J (or 13.6 eV for hydrogen). The negative sign indicates that the electron is bound to the nucleus.

The ionization energy is the energy required to remove the electron from the ground state (n = 1) to infinity (n = ∞). For hydrogen (Z = 1), this equals |E₁| = 13.6 eV. For hydrogen-like ions, ionization energy = Z² × 13.6 eV.

The Rydberg equation, 1/λ = R × Z² × (1/n₁² − 1/n₂²), predicts the wavelength of light emitted or absorbed during electron transitions. It is directly derived from the energy level formula: the photon energy |ΔE| = h × c / λ, linking the change in energy levels to observable spectral lines.

Hydrogen's spectral series are named by the final level (n_final) of the electron transition: Lyman series (n_final = 1, UV), Balmer series (n_final = 2, visible), Paschen series (n_final = 3, infrared), Brackett series (n_final = 4), and Pfund series (n_final = 5). Each series corresponds to a specific range of emitted wavelengths.

The Bohr model is exact for hydrogen and hydrogen-like ions — species with only one electron, such as He⁺, Li²⁺, and Be³⁺. For multi-electron atoms, electron–electron repulsion makes the formula inaccurate, and more advanced quantum mechanical models are required.

When an electron drops from a higher level (n_initial) to a lower level (n_final), it releases energy in the form of a photon. The photon's energy equals |ΔE| = |E_final − E_initial|, its frequency ν = |ΔE| / h, and its wavelength λ = h × c / |ΔE|. If n_final > n_initial, the electron absorbs a photon instead.

The ground state energy of the hydrogen atom (n = 1, Z = 1) is approximately −13.6 eV or −2.18 × 10⁻¹⁸ J. This is the lowest possible energy state of the electron. All other energy levels are higher (less negative), and the electron requires energy input to move to them.