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If you heat hydrogen gas with an electrical discharge, it doesn't emit a smooth rainbow of colors like a hot filament. Instead, it emits a distinct magenta glow made of just a few pinpoint wavelengths.
Why do atoms produce specific, sharp lines of color instead of continuous bands of light?

Quantized Energy Levels

In 1913, Danish physicist Niels Bohr resolved this mystery by proposing that electrons in atoms are confined to specific, stable orbits called quantized energy levels.
For hydrogen, the energy of an electron at principal quantum number nn (n=1,2,3,…n = 1, 2, 3, \dots) is fixed by the relation: En=βˆ’2.18Γ—10βˆ’18Β Jn2=βˆ’13.6Β eVn2E_n = -\frac{2.18 \times 10^{-18}\text{ J}}{n^2} = -\frac{13.6\text{ eV}}{n^2}
The energy values are negative because the electron is bound to the nucleus, with 0Β eV0\text{ eV} defined as a completely free, ionized electron at infinity.
What happens when an electron leaps from one energy rung to another?

Absorption and Emission Transitions

By the law of conservation of energy, an electron cannot change states without exchanging energy with light. The energy of the involved photon must exactly match the difference between the two states: Ξ”E=∣Efβˆ’Ei∣\Delta E = |E_f - E_i|
In absorption, an electron absorbs a photon to jump to a higher level. In emission, an excited electron drops to a lower level, releasing that exact energy as an outgoing photon.
We connect the photon's energy Ξ”E\Delta E to its frequency ff and wavelength Ξ»\lambda using the Planck-Einstein relation: Ξ”E=hf=hcΞ»\Delta E = hf = \frac{hc}{\lambda}
Here, h=6.626Γ—10βˆ’34Β Jβ‹…sh = 6.626 \times 10^{-34}\text{ J}\cdot\text{s} is Planck's constant, and c=3.00Γ—108Β m/sc = 3.00 \times 10^8\text{ m/s} is the speed of light in vacuum.