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The Rydberg formula

An electron in a given stationary state of a hydrogen atom, characterized by the quantum numbers $n$, $l$, and $m$, should, in principle, remain in that state indefinitely. In practice, if the state is slightly perturbed--e.g., by interacting with a photon--then the electron can make a transition to another stationary state with different quantum numbers (see Sect. 13).

Suppose that an electron in a hydrogen atom makes a transition from an initial state whose radial quantum number is $n_i$ to a final state whose radial quantum number is $n_f$. According to Eq. (655), the energy of the electron will change by

\begin{displaymath}
{\mit\Delta} E = E_0\left(\frac{1}{n_f^{\,2}}-\frac{1}{n_i^{\,2}}\right).
\end{displaymath} (678)

If ${\mit\Delta}E$ is negative then we would expect the electron to emit a photon of frequency $\nu=- {\mit\Delta}E/h$ [see Eq. (41)]. Likewise, if ${\mit\Delta}E$ is positive then the electron must absorb a photon of energy $\nu={\mit\Delta}E/h$. Given that $\lambda^{-1}=\nu/c$, the possible wave-lengths of the photons emitted by a hydrogen atom as its electron makes transitions between different energy levels are
\begin{displaymath}
\frac{1}{\lambda} = R\left(\frac{1}{n_f^{\,2}}-\frac{1}{n_i^{\,2}}\right),
\end{displaymath} (679)

where
\begin{displaymath}
R = \frac{-E_0}{h\,c} =\frac{m_e\,e^4}{(4\pi)^3\,\epsilon_0^{\,2}\,\hbar^3\,c} = 1.097\times 10^7\,{\rm m^{-1}}.
\end{displaymath} (680)

Here, it is assumed that $n_f<n_i$. Note that the emission spectrum of hydrogen is quantized: i.e., a hydrogen atom can only emit photons with certain fixed set of wave-lengths. Likewise, a hydrogen atom can only absorb photons which have the same fixed set of wave-lengths. This set of wave-lengths constitutes the characteristic emission/absorption spectrum of the hydrogen atom, and can be observed as ``spectral lines'' using a spectroscope.

Equation (679) is known as the Rydberg formula. Likewise, $R$ is called the Rydberg constant. The Rydberg formula was actually discovered empirically in the nineteenth century by spectroscopists, and was first explained theoretically by Bohr in 1913 using a primitive version of quantum mechanics. Transitions to the ground-state ($n_f=1$) give rise to spectral lines in the ultra-violet band--this set of lines is called the Lyman series. Transitions to the first excited state ($n_f=2$) give rise to spectral lines in the visible band--this set of lines is called the Balmer series. Transitions to the second excited state ($n_f=3$) give rise to spectral lines in the infra-red band--this set of lines is called the Paschen series, and so on.



Subsections
next up previous contents
Next: Problems Up: Central potentials Previous: The hydrogen atom   Contents
Richard Fitzpatrick 2006-12-12