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.

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. (676), the energy of the electron will change by

$${\mit\Delta} E = E_0\left(\frac{1}{n_f^{ 2}}-\frac{1}{n_i^{ 2}}\right).$$ (699)

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. (58)]. 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

$$\frac{1}{\lambda} = R\left(\frac{1}{n_f^{ 2}}-\frac{1}{n_i^{ 2}}\right),$$ (700)

where

$$R = \frac{-E_0}{h c} =\frac{m_e e^4}{(4\pi)^3 \epsilon_0^{ 2} \hbar^3 c} = 1.097\times 10^7 {\rm m^{-1}}.$$ (701)

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 wavelengths. Likewise, a hydrogen atom can only absorb photons which 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 (700) 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.