Explanation of the Line Spectrum of Hydrogen Using Bohr Theory

Bohr's theory does not just describe individual stationary states — it explains WHY hydrogen's emission spectrum shows exactly the lines it does. By Bohr's second postulate, radiation is emitted only when an electron falls from an outer orbit of higher principal quantum number ($n_i$, the initial state) to an inner orbit of lower principal quantum number ($n_f$, the final state), and by the third postulate the energy of that emitted radiation equals the energy difference between the two orbits, $\Delta E = E_i - E_f$. Substituting the orbit-energy expression $E_n = -R_H\left(\frac{1}{n^2}\right)$ for both $E_i$ and $E_f$ and simplifying gives $\Delta E = R_H\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)$, and putting in $R_H = 2.18\times10^{-18}$ J alongside $E=h\nu$ and $\bar\nu = \nu/c$ converts this into a wavenumber expression, $\bar\nu = 109677\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)\ \text{cm}^{-1}$ — which is exactly the empirical Rydberg equation from section 4.5.2, with $n_f = n_1$ and $n_i = n_2$. In other words, Bohr's theory doesn't just fit the Rydberg equation after the fact, it derives it from first principles, and in doing so gives the previously-empirical integers $n_1$ and $n_2$ real physical meaning: they are the principal quantum numbers of the final and initial concentric orbits of an electronic transition. Every one of the five named series (Lyman, Balm …