In an isolated atom, electrons can only occupy certain energy levels. These levels are quantised, meaning the electron’s energy can take only specific values. When an electron jumps from a higher level to a lower one, it emits a photon; when it absorbs a photon, it jumps to a higher level.
For hydrogen (single proton + electron), Niels Bohr showed that the allowed energies are given by
\$E_n = -\frac{13.6\ \text{eV}}{n^2}\$
where \$n = 1,2,3,\dots\$ is the principal quantum number. The negative sign indicates that the electron is bound; \$E_1 = -13.6\ \text{eV}\$ is the ground state.
| \$n\$ (level) | Energy \$E_n\$ (eV) |
|---|---|
| 1 | \$-13.6\$ |
| 2 | \$-3.4\$ |
| 3 | \$-1.51\$ |
| 4 | \$-0.85\$ |
When an electron drops from level \$ni\$ to a lower level \$nf\$, the emitted photon has energy
\$\Delta E = E{ni} - E{nf} = 13.6\ \text{eV}\left(\frac{1}{nf^2} - \frac{1}{ni^2}\right).\$
Using \$E = hf = \frac{hc}{\lambda}\$, the wavelength of the emitted light is
\$\frac{1}{\lambda} = RH\left(\frac{1}{nf^2} - \frac{1}{n_i^2}\right),\$
where \$R_H = 1.097\times10^7\ \text{m}^{-1}\$ is the Rydberg constant for hydrogen.
| Series | Final level \$n_f\$ | Typical wavelength range |
|---|---|---|
| Lyman | 1 | UV (\$<122\ \text{nm}\$) |
| Balmer | 2 | Visible (\$365-656\ \text{nm}\$) |
| Paschen | 3 | IR (\$>820\ \text{nm}\$) |
🌟 Understanding quantised energy levels helps explain the colours we see in gas‑discharge tubes and stars! 🌟