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.
Frequently Asked Questions
What is a hydrogen-like (hydrogenic) atom?
A hydrogen-like atom is any system with exactly one electron orbiting a nucleus of charge +Ze — this includes 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 formula for its energy levels.
What is the formula for hydrogen-like atom energy levels?
E_n = -13.6057 eV × Z²/n², where Z is the atomic number (nuclear charge) and n is the principal quantum number (1, 2, 3, ...). The energy is negative because the electron is bound to the nucleus; n=1 is the ground state.
How do I find the wavelength of a spectral line?
Compute the transition energy ΔE = |E_final − E_initial| in eV, then convert to wavelength with λ = 1239.84 eV·nm / ΔE (from λ = hc/E). For hydrogen's n=3 to n=2 transition (Balmer-alpha), this gives about 656 nm, the well-known red line in the hydrogen spectrum.
Why is the energy level negative, and what does that mean physically?
The negative sign shows the electron is bound: energy would need to be added to free it. The ground state (n=1) has the most negative energy, and levels approach zero as n increases toward ionization. The magnitude of E_1 is the atom's ionization energy — 13.6057 eV for hydrogen.
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Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
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| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
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