8.4: 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—for instance, via interaction with a photon—then the electron can make a transition to another stationary state with different quantum numbers. (See Chapter [s13].)

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 Equation ([e9.55]), the energy of the electron will change by

$${\mit\Delta} E = E_0\left(\frac{1}{n_f^{\,2}}-\frac{1}{n_i^{\,2}}\right).$$ If ${\mit\Delta} E$ is negative then we would expect the electron to emit a photon of frequency $\nu=- {\mit\Delta}E/h$. [See Equation ([ee3.15]).] 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 wavelengths of the photons emitted by a hydrogen atom as its electron makes transitions between different energy levels are

$$\label{e9.77} \frac{1}{\lambda} = R\left(\frac{1}{n_f^{\,2}}-\frac{1}{n_i^{\,2}}\right),$$ where $$R = \frac{-E_0}{h\,c} =\frac{m_e\,e^{\,4}}{(4\pi)^3\,\epsilon_0^{\,2}\,\hbar^{\,3}\,c} = 1.097\times 10^7\,{\rm m^{-1}}.$$ Here, it is assumed that $n_f<n_i$. Note that the emission spectrum of hydrogen is quantized: that is, a hydrogen atom can only emit photons with certain fixed set of wavelengths. Likewise, a hydrogen atom can only absorb photons that have the same fixed set of wavelengths. This set of wavelengths constitutes the characteristic emission/absorption spectrum of the hydrogen atom, and can be observed as “spectral lines” using a spectroscope.

Equation ([e9.77]) 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 ultraviolet 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 infrared band—this set of lines is called the Paschen series, and so on.

Contributors and Attributions

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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