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# 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.

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