Quantized Energy Levels

$$E_n = -\frac{13.6\ \text{eV}}{n^2}$$

• Energy is negative for bound states because we set $E=0$ at infinite separation (free electron + proton).
• $n=\infty$ is the ionization limit: the electron is no longer bound and the level energy is exactly $0\ \text{eV}$.
• Levels get closer together as $n$ increases — this is why spectral series converge to a limit.
• For hydrogen, the Bohr energy formula matches the exact Schrödinger solution because hydrogen is a pure $1/r$ Coulomb potential.
• The energies are exact. The circular orbits are not.

The Bohr Radius

$$r_n = n^2\, a_0, \quad a_0 = 0.0529\ \text{nm}$$

• The orbit radius grows as $n^2$: the $n = 3$ orbit is $9\times$ larger than $n = 1$.
• The Bohr atom view on the Explore tab uses a compressed scale for visibility — labeled "not to scale."
• Real electrons occupy probability clouds (orbitals), not circular paths. Bohr's picture gives correct energies, not correct shapes.

Photon Emission & Absorption

$$E_\gamma = h\nu = \frac{hc}{\lambda} = 13.6\ \text{eV}\left(\frac{1}{n_{\text{lower}}^2} - \frac{1}{n_{\text{upper}}^2}\right)$$

• Emission: electron drops from $n_{\text{upper}}$ to $n_{\text{lower}}$; photon carries away exactly $\Delta E$.
• Absorption: photon with exactly $\Delta E$ is absorbed; electron jumps up. Photons with wrong energy pass through.
• Emission and absorption occur at the same $\Delta E$ values: bright lines vs. dark dips at identical wavelengths.

The Rydberg Formula

$$\frac{1}{\lambda} = R_\infty \left(\frac{1}{n_{\text{lower}}^2} - \frac{1}{n_{\text{upper}}^2}\right), \quad R_\infty = 1.097 \times 10^7\ \text{m}^{-1}$$

• Discovered empirically by Rydberg (1888) — 25 years before Bohr explained it.
• As $n_{\text{upper}} \to \infty$, we reach the series limit: the shortest wavelength in a series and the ionization boundary for that lower level.
• The Balmer series limit (364.6 nm) is where Balmer lines pile up and merge into the continuum.

Why $1/n^2$ Appears

$$E_n \propto -\frac{1}{n^2}$$

Bound states in a $1/r$ potential produce energy eigenvalues proportional to $-1/n^2$. This scaling determines all hydrogen spectral structure.

Spectral Series of Hydrogen

$$\text{Lyman:}\ n_{\text{lower}} = 1\ (\text{UV}), \quad \text{Balmer:}\ n_{\text{lower}} = 2\ (\text{visible}), \quad \text{Paschen:}\ n_{\text{lower}} = 3\ (\text{IR})$$

• Each series is named for $n_{\text{lower}}$ — the level the electron falls to (emission) or jumps from (absorption).
• Only the Balmer series falls primarily in visible light — explaining why hydrogen glows red ($\text{H}\alpha$ at 656 nm).
• The Lyman series is even stronger but falls in the UV, invisible to our eyes.

Why Other Elements Have Different Lines

$$\text{Each element} \longrightarrow \text{unique energy levels} \longrightarrow \text{unique spectral fingerprint}$$

• Multi-electron atoms have more complex energy-level structures due to electron-electron interactions.
• Each element produces a unique line pattern — a "spectral fingerprint."
• Astronomers identify elements in distant stars by matching observed lines against laboratory reference spectra.