Balmer (1885) fit part of the hydrogen spectrum. Rydberg generalized it:

\[\tilde{\nu} = R_H\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)\]

• \(R_H = 1.097 \times 10^7 \ \text{m}^{-1}\) (Rydberg constant)
• \(n_1 = 1,2,3,...\) and \(n_2 = n_1+1, n_1+2,...\)

• Fits the data beautifully, but offers no physics
• Why should integers govern atomic light?

Fit an integer number of de Broglie waves around the orbit:

\[2\pi r = n \lambda_e, \qquad \lambda_e = \frac{h}{m_e v}\]

Substituting gives the quantization condition:

\[m_e v r = \frac{n h}{2\pi} = n \hbar\]

• \(m_e v r\) is the angular momentum
• Bohr: angular momentum is quantized in units of \(\hbar\)

Electrostatic pull balances the centrifugal force:

\[\frac{e^2}{4\pi\varepsilon_0 r^2} = \frac{m_e v^2}{r}\]

Combined with \(m_e v r = n\hbar\), the allowed radii are:

\[r = n^2 a_0, \qquad n = 1,2,3,...\]

\[a_0 = \frac{4\pi \varepsilon_0 \hbar^2}{m_e e^2} \approx 0.529 \,\text{Å}\]

• The Bohr radius \(a_0\) sets the length scale of atoms

Total energy (kinetic plus Coulomb) at quantized radius:

\[E_n = -\frac{m_e e^4}{8 \varepsilon_0^2 h^2}\cdot\frac{1}{n^2}\]

The practical form for problem solving:

\[E_n = -13.6\,\frac{1}{n^2}\,\,\,[\text{eV}]\]

• Negative: electron is bound
• Ionization (\(n=1 \to \infty\)) costs exactly 13.6 eV
• Photon of a jump: \(\Delta E = h\nu\)

Photon energy of a transition, \(\tilde{\nu} = \nu/c\):

\[\tilde{\nu} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)\]

Now \(R_H\) is derived, not fitted:

\[R_H = \frac{m_e e^4}{8 \varepsilon_0^2 c h^3}\]

• Bohr’s model explains the empirical Rydberg formula
• The mysterious integers are quantum numbers

For one-electron ions (\(He^+\), \(Li^{2+}\)), add nuclear charge \(Z\):

\[E_n = -13.6\,\frac{Z^2}{n^2}\,\,\,[\text{eV}]\]

• \(Z=1\) for \(H\), \(Z=2\) for \(He^+\), \(Z=3\) for \(Li^{2+}\)
• Energies scale as \(Z^2\): \(He^+\) ionization is 54.4 eV
• Orbits shrink as \(r_n = n^2 a_0 / Z\)