Formula and Method for Hydrogen-Like Atom Energy Levels
A hydrogen-like (or "hydrogenic") atom is any system with exactly one electron bound to a nucleus of charge +Ze — this covers neutral hydrogen (Z=1) as well as ions such as He+ (Z=2), Li2+ (Z=3), and Be3+ (Z=4). Because there is only one electron, the Bohr model gives an exact, closed-form expression for the allowed energy levels: E_n = -13.6057 eV × Z²/n², where n = 1, 2, 3, ... is the principal quantum number. This calculator also finds the transition energy and photon wavelength between any two levels n_i and n_f.
How the calculation works
Enter the atomic number Z along with an initial quantum number n_i and a final quantum number n_f. The tool evaluates E_n = -13.6057 eV × Z²/n² at both levels, then takes the transition energy ΔE = |E_f − E_i|. That energy is converted to a photon wavelength with λ = hc/ΔE, using hc = 1239.84 eV·nm, a standard constant in atomic and optical physics. If n_i is greater than n_f, the electron falls to a lower level and the atom emits a photon; if n_i is less than n_f, the electron is promoted to a higher level and the atom must absorb a photon of that exact energy.
Common mistakes
- Reading energy sign backwards: energy levels are negative because the electron is bound. E = -1.51 eV (n=3) is a higher (less tightly bound) level than E = -3.40 eV (n=2), even though -1.51 is the larger number.
- Using mass number instead of atomic number: Z is the nuclear charge (proton count) — hydrogen is Z=1, helium is Z=2 — not the isotope's mass number.
- Applying the formula to multi-electron neutral atoms: E_n = -13.6057 eV × Z²/n² is exact only for one-electron systems; electron-electron repulsion breaks the formula for neutral helium, lithium, and heavier atoms.
Real-world applications
- Astronomers identify hydrogen and helium in stars and nebulae by matching observed emission lines to Bohr-model transition wavelengths (e.g., the Balmer series in stellar spectra).
- The n=3 → n=2 hydrogen transition (Balmer-alpha) produces the well-known 656 nm red line seen in emission nebulae and hydrogen discharge lamps.
- Laser and plasma physics use the same formula for hydrogen-like ions (He+, Li2+) at higher photon energies, since heavier one-electron ions scale as Z².
- Introductory quantum mechanics courses use this model as the first exactly solvable atom, bridging classical orbits and quantum energy quantization.