Energy Levels in Hydrogen
When sunlight passes through a prism, you see a continuous rainbow of colours. But when you heat hydrogen gas until it glows and pass that light through a prism, you see something strikingly different: a handful of sharp, coloured lines against a black background. This is the hydrogen emission spectrum, and for decades in the 19th century it was a complete mystery. Why does hydrogen emit only those specific colours — and nothing in between? The answer lies in the idea that electrons inside an atom cannot have just any energy. They are restricted to a set of discrete energy levels, like the rungs of a ladder rather than a smooth ramp. When an electron jumps from a higher rung to a lower one, it releases a photon whose energy exactly equals the gap between the two levels. Because the gaps are fixed, the photon energies — and therefore the colours of light — are fixed too. This is why the hydrogen spectrum is a fingerprint: every element has its own unique set of energy levels, producing its own unique set of spectral lines. The mathematical framework that predicts these energy levels was first worked out by Niels Bohr in 1913. Although the full picture requires quantum mechanics, Bohr's simple model correctly predicts the energies of hydrogen's levels and explains every line in its visible spectrum. Understanding hydrogen's energy levels is the gateway to modern atomic physics, chemistry's periodic table, lasers, and even how stars are classified.
- Quantisation — only certain orbits are allowed
- Energy formula — each level has a precise energy
- Transitions — jumping between levels produces photons
- Series — groupings of spectral lines
- Ionisation energy — kicking the electron out completely
Quantisation — only certain orbits are allowed
Bohr proposed that an electron can only orbit the nucleus in specific circular orbits where its angular momentum is a whole-number multiple of ħ (h-bar, Planck's constant divided by 2π). These allowed orbits correspond to the principal quantum numbers n = 1, 2, 3, … The orbit with n = 1, closest to the nucleus, is called the ground state. Higher values of n correspond to larger orbits with more energy.
Energy formula — each level has a precise energy
The energy of the nth level is given by Eₙ = −13.6 eV / n². The negative sign is crucial: it means the electron is bound to the nucleus (you would have to add energy to free it). At n = 1, the energy is −13.6 eV — the most tightly bound state. At n = 2 it is −3.4 eV, at n = 3 it is −1.51 eV, and so on. As n → ∞, the energy approaches 0 eV, which represents a free (ionised) electron.
Transitions — jumping between levels produces photons
When an electron drops from level nᵢ (initial, higher) to level nf (final, lower), it emits a photon. The photon's energy equals the difference: ΔE = Eᵢ − Ef = 13.6 eV × (1/nf² − 1/nᵢ²). This energy determines the photon's frequency via E = hf, and therefore its colour. Conversely, if a photon of exactly the right energy hits the atom, the electron absorbs it and jumps up — producing an absorption line.
Series — groupings of spectral lines
Transitions that land on the same final level form a named series. Those ending on nf = 1 form the Lyman series (ultraviolet). Those ending on nf = 2 form the Balmer series — these are the four visible lines: red at 656 nm, cyan at 486 nm, violet at 434 nm, and deep violet at 410 nm. Transitions ending on nf = 3 form the Paschen series (infrared). The Balmer series is why hydrogen glows pink-red in discharge tubes, and why nebulae glow red in astrophotographs.
Ionisation energy — kicking the electron out completely
If you supply 13.6 eV or more to a ground-state hydrogen atom, the electron escapes entirely — the atom is ionised. This value, 13.6 eV, is hydrogen's ionisation energy. It corresponds to moving the electron from n = 1 all the way to n = ∞. This is why high-energy UV photons from young stars can ionise the hydrogen gas surrounding them, creating glowing clouds called H II regions.
The Key Equation
- E_n — Energy of level n (eV (electron-volts)): Total mechanical energy of the electron in orbit n. Negative because the electron is bound.
- 13.6 \text{ eV} — Hydrogen ground-state energy (magnitude) (eV): The ionisation energy of hydrogen; equals m_e e⁴ / (8ε₀²h²) from first principles.
- n — Principal quantum number (dimensionless (integer)): Positive integer n = 1, 2, 3, … labelling each allowed energy level. n = 1 is the ground state.
Worked Example
Problem. An electron in a hydrogen atom falls from the n = 3 level to the n = 2 level. (a) Calculate the energy of the emitted photon in eV. (b) Calculate the wavelength of this photon and identify its colour.
Answer. The emitted photon has energy 1.89 eV and wavelength approximately 657 nm — red light (the Hα Balmer line).
Origins · 1 minNiels Bohr and how this was discovered
Bohr proposed the quantised atomic model in 1913, introducing the idea that electrons occupy discrete energy levels and that radiation is emitted or absorbed only when electrons transition between these levels. His formula correctly predicted the wavelengths of every line in the hydrogen spectrum, earning him the Nobel Prize in Physics in 1922. Bohr was also an excellent footballer (soccer player) in Denmark. His brother Harald represented Denmark in the 1908 Olympics as a professional-level player, but Niels was reportedly the better goalkeeper in informal games.
Related Concepts
- The Photoelectric Effect — Both concepts rely on photons having quantised energy E = hf; the photoelectric effect showed that light energy comes in discrete packets, directly supporting the quantum ideas used in Bohr's model.
- Quantum Mechanical Model of the Atom — The full quantum mechanical model replaces Bohr's circular orbits with probability orbitals but preserves the same energy level formula for hydrogen, extending the picture to multi-electron atoms.
- The Electromagnetic Spectrum — The photons emitted in hydrogen transitions fall in specific regions — UV (Lyman), visible (Balmer), infrared (Paschen) — connecting energy levels directly to regions of the EM spectrum.