Hydrogen Atom Energy Levels Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/8/2026
The energy of an electron level in the hydrogen atom is calculated with E(n) = −13.6 ÷ n² electron-volts. For the ground state (n=1), this energy is exactly −13.6 eV — hydrogen's ionization energy, one of the most famous values in all of atomic physics.
Explanation
The Bohr model of the hydrogen atom, proposed in 1913, predicts that the electron can only occupy discrete orbits around the nucleus, each associated with a precise, quantized energy given by an integer n called the principal quantum number. The energy of each level, E(n)=−13.6/n² eV, is always NEGATIVE: this sign reflects the fact that the electron is BOUND to the nucleus by electrostatic attraction, with the zero reference corresponding to an electron fully removed from the atom (ionized) and at rest at infinite distance. The larger n is, the further out the electron's orbit from the nucleus, the more weakly it is bound, and the closer its energy gets to zero without ever reaching it — the ground state (n=1) is thus the most stable level, with the most negative energy (−13.6 eV), while successive excited states (n=2, n=3...) progressively approach the ionization threshold. This calculator directly complements our Rydberg formula calculator, which calculates the wavelength of the photon emitted or absorbed during a TRANSITION between two levels (the energy difference between them, converted to wavelength via E=hc/λ): here, it's the ABSOLUTE energy of each level taken individually that is calculated, the base quantity from which all possible transitions between any two levels are then derived. These quantized energy levels directly explain why hydrogen's light spectrum isn't continuous but made up of very precise discrete lines (the Lyman, Balmer, Paschen series...), each line corresponding to a transition between two well-defined energy levels and to a photon of an exact energy — exactly the kind of energy quantum already at play in the photoelectric effect.
Example: hydrogen's ground state
Inputs
Principal quantum number: n=1 (ground state).
Calculation
E(1) = −13.605693122994 ÷ 1² = −13.605693122994 eV.
Result
The energy of hydrogen's ground state is about −13.61 eV — exactly the energy (in positive terms) needed to fully ionize a hydrogen atom in its ground state.
Frequently asked questions
Why is the energy of each level negative?
Because the universally adopted convention in atomic physics sets the zero energy level at the state where the electron is fully freed from the nucleus (ionized) and at rest at infinite distance. Since the electron is BOUND within the atom (energy must be supplied to remove it), its energy in the atom is necessarily below this zero reference level, hence the systematic negative sign of all bound levels.
What does the −13.6 eV energy actually represent?
It is precisely the ionization energy of the hydrogen atom in its ground state: the amount of energy that must be supplied to a hydrogen atom at rest to fully remove its electron and free it from the nucleus. It is one of the most measured and most fundamental quantities in all of atomic physics, serving as a comparison point for electron binding energy in more complex atoms.
Does this formula apply to atoms other than hydrogen?
Not directly: this precise formula only applies to hydrogen and to so-called 'hydrogen-like' ions, which have only one electron (such as He⁺ or Li²⁺), where it generalizes by simply multiplying by the square of the atomic number Z. For a multi-electron atom, interactions between electrons considerably complicate the calculation, and this simple Bohr model formula is no longer enough to precisely describe the actual energy levels.