Hydrogen Energy Levels Calculator

The Hydrogen Energy Levels Calculator computes the quantized energy levels of the hydrogen atom and the properties of photons emitted or absorbed during transitions between levels. The hydrogen atom, with its single proton and single electron, is the simplest atomic system and the cornerstone of atomic physics.

Niels Bohr's 1913 model introduced the revolutionary idea that electrons occupy discrete energy levels given by $$E_n = -\frac{13.6 \text{ eV}}{n^2}$$ where n = 1, 2, 3, ... is the principal quantum number. The ground state (n = 1) has E₁ = –13.6 eV, meaning 13.6 eV of energy is needed to ionize the atom. As n increases, the energy levels converge toward zero (the ionization threshold), with ever-decreasing spacing.

When an electron transitions from a higher level nᵢ to a lower level n_f, it emits a photon with energy equal to the difference: $$\Delta E = 13.6 \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right) \text{ eV}$$ The photon wavelength follows from λ = hc/ΔE. Conversely, absorption of a photon with exactly this energy promotes the electron upward.

The spectral series of hydrogen are named after their discoverers: the Lyman series (transitions to n = 1, ultraviolet), Balmer series (to n = 2, visible), Paschen series (to n = 3, infrared), and Brackett and Pfund series (to n = 4 and 5, far infrared). The Balmer series is especially important historically because its visible lines (H-alpha at 656 nm, H-beta at 486 nm, etc.) were the first atomic spectral lines to be systematically studied.

Although Bohr's model has been superseded by the full quantum mechanical treatment (Schrödinger equation for hydrogen), the energy level formula remains exactly correct for hydrogen. The calculator is essential for spectroscopy, astrophysics (stellar classification relies on hydrogen lines), laser physics, and quantum chemistry courses.

Beyond hydrogen, the formula generalizes to hydrogen-like ions (He⁺, Li²⁺, etc.) by multiplying 13.6 eV by Z², where Z is the nuclear charge. This extension makes the model useful for understanding the spectra of highly ionized atoms in stellar atmospheres and fusion plasmas.