Hydrogen Energy Levels Calculator

Hydrogen has one proton and one electron, making it the simplest atom and a foundational model in quantum physics. In the Bohr description, the electron may occupy only discrete stationary energy levels labeled by the positive principal quantum number n. The energy of a level is negative 13.6 electronvolts divided by n squared. Negative energy indicates that the electron is bound to the nucleus; zero corresponds to a free electron infinitely far away. As n increases, levels become less negative and crowd toward the ionization limit. An electron moving between two allowed levels must exchange a photon whose energy equals the magnitude of the level-energy difference. A transition from a higher principal quantum number to a lower one emits a photon. A transition from a lower number to a higher one absorbs a photon. Photon wavelength follows from energy through the Planck relation, while frequency equals energy divided by Planck's constant. This calculator uses a convenient energy-wavelength constant of 1239.841984 electronvolt nanometres and Planck's constant in electronvolt seconds. Hydrogen spectral lines form named series according to their final level. Transitions ending at n equals one are in the Lyman series and are primarily ultraviolet. Those ending at n equals two form the Balmer series, whose first lines include visible red, blue-green, and violet light. Transitions ending at n equals three form the infrared Paschen series. The familiar Balmer-alpha line comes from n equals three to n equals two and has a wavelength near 656.4 nanometres. Lyman-alpha comes from two to one near 121.6 nanometres. The calculation represents ideal, isolated hydrogen without fine structure. Real observed lines can split or shift because of electron spin, relativistic corrections, nuclear motion, electric and magnetic fields, collisions, pressure broadening, and Doppler motion. The reduced mass of the electron-proton system also makes the most precise Rydberg value slightly different from a simple infinite-nuclear-mass model. These effects matter in precision spectroscopy but are usually much smaller than the main Bohr-level separation. Enter whole-number levels because principal quantum numbers cannot be fractional or zero. Reversing the levels leaves photon energy, frequency, and wavelength unchanged but changes emission to absorption. Equal levels do not represent a transition and therefore produce no photon. The calculator is useful for physics coursework, spectroscopy estimates, and checking energy-unit conversions. It should not be applied directly to multi-electron atoms, whose electron interactions produce substantially different level structures.