The Hydrogen Spectrum: Spectral Series

From energy levels to spectral lines. Bohr's third postulate (Section 1.4) states that a photon is emitted whenever an electron jumps from a higher-energy orbit $n_i$ to a lower-energy orbit $n_f$ ($n_i>n_f$), with the photon's energy exactly equal to the energy released:

Deriving the Rydberg formula. Substituting the energy-level formula $E_n=-13.6\,\text{eV}/n^2$ (hydrogen, Section 1.5) for both $n_i$ and $n_f$:

Dividing through by $hc$ and writing $R \equiv \dfrac{13.6\ \text{eV}}{hc}$ (a fixed combination of constants) gives the celebrated Rydberg formula:

Evaluating the constant $R$ numerically gives $R\approx1.097\times10^7\ \text{m}^{-1}$ -- matching, to remarkable precision, the value spectroscopists had already measured empirically decades before Bohr's model existed, without knowing WHY hydrogen's lines followed this pattern. That match was one of the single strongest early confirmations that Bohr's quantised picture of the atom was physically correct.

The named spectral series. Every choice of the LOWER level $n_f$ generates its own family, or series, of spectral lines, one line for every possible higher starting level $n_i=n_f+1,\,n_f+2,\,\ldots$; five such series, named after the spectroscopists who first identified them, are recognised for hydrogen (Table 1): the Lyman series ($n_f=1$, entirely in the ultraviolet), the Balmer series ($n_f=2$, the only series with lines in the visible region -- historically the first hydrogen series ever identified, since it could be seen directly), and the Paschen, Brackett and Pfund series ($n_f=3,4,5$ respectively), all lying in the infrared.

Series limits. Within any one series, as $n_i$ increases from $n_f+1$ toward $\infty$, the lines become progressively more closely spaced and their wavelengths shorten, approaching -- but never quite reaching -- a shortest possible wavelength called the series limit, obtained by setting $n_i\to\infty$ in the Rydberg formula: …