1.2: The hydrogen atom

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The simplest atom is hydrogen. It has just one proton and one electron (a few weird hydrogen atoms might have a neutron or two as well). Its line spectrum is also the simplest. Therefore explaining the origin of the lines in the hydrogen spectrum seems like a good place to start a quest to understand such things.

Balmer and Rydberg

Johann Jakob Balmer derived a formula to predict the locations of the visible lines in the hydrogen spectrum, which can be written \[ \frac{1}{\lambda} = 1.097\times10^7\left(\frac{1}{4} - \frac{1}{n^2}\right) \] where \(\lambda\) is wavelength in meters, and \(n\) is an integer greater than 2.

A few years later, Johannes Rydberg derived a more general formula which can be written \[ \frac{1}{\lambda} = \frac{R_H}{hc}\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right) \] where \(\lambda\) is wavelength in meters, \(R_H\) is the Rydberg constant \(2.179 872 171\times10^{-18}\) J, \(h\) is Planck's constant, \(c\) is the speed of light, and \(n_f\) and \(n_i\) are integers such that \(n_f < n_i\) for emission spectra and each \(n_f\) defines a particular series of lines. E.g., when \(n_f = 2\) the lines given by the Rydberg formula are the visible lines of the Balmer series.

Bohr

The above is all well and good, but WHY do these equations work? Niels Bohr derived a model of the hydrogen atom in which the electron was only allowed to occupy certain energy levels, i.e., the energy of the electron was quantized. Specifically, if the only allowed energies of the electron in a hydrogen atom are taken to be \[ E=-\frac{R_H}{n^2}\] where \(n=1, 2, 3, ...\infty\) and we note that the energy of the emitted photon is \(E=h\nu\) then \[\begin{aligned}
\Delta E & = & (E_f - E_i)\\
\Delta E & = & - R_H\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right) \\
h\nu & = & R_H\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right) \\
\frac{1}{\lambda} & = & \frac{R_H}{hc}\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)
\end{aligned}\]