The Rydberg equation (sometimes called the Balmer equation) is an analytical
equation for determining the wavelength of a photon emitted or absorbed
when an electron changes energy levels in a hydrogen atom.

In 1884, Johannes Balmer, a Swiss high school teacher, first
determined that the prominent emission lines of hydrogen
gas had wavelengths which followed the simple equation

1/λ = R × (1/4 - 1/n^{2})

where λ is the wavelength, *R* is a constant
(now called Rydberg's constant) equal to 1.0967758(±7) × 10^{7}
meter^{-1}, and
*n* is an integer greater than two. The optical emission (and
absorption) lines of hydrogen have since been known as the
*Balmer series*.

In 1900, the Swede Johannes Rydberg derived this equation independently,
but generalized it to cover *all* transitions in the hydrogen atom, rather
than just the Balmer series. It is

1/λ = R × (1/n_{1}^{2} - 1/n_{2}^{2})

where *n*_{1} is the lower energy level, and *n*_{2}
is the higher.

Suppose an electron is in the second electronic level
(n_{1}=2). In order to boost that electron to the third electronic
level (n_{2}=3), the atom must absorb a photon with a wavelength, λ,
of exactly

1/λ = R × (1/2^{2} - 1/3^{2})

or about 6563 Å. If the electron *falls* from level 3 to
level 2, it will emit a photon with that wavelength. You
can also use this equation to figure out what photon energy is required to
ionize the atom by setting n_{2} to infinity. Thus
a level 1 (ground state) electron can be ejected by a 911 Å photon, a
level two electron
by a 3645 Å photon, and so on. In this case, however, you do not need
a photon with *exactly* that wavelength, just one that is that wavelength
*or shorter* (shorter wavelengths = higher energy). The extra energy
goes into kinetic energy of the atom and electron.

Transition series are given names depending upon what their lowest and highest
levels are. The first six are: *Lyman* (n_{1}=1),
*Balmer* (n_{1}=2), *Paschen* (n_{1}=3),
*Brackett* (n_{1}=4), *Pfund* (n_{1}=5), and *Humphreys* (n_{1}=6).
Furthermore, these are each divided into additional groups denoted with
Greek letters.
For example, Lyman *alpha* is the transition between levels 1 and 2,
Lyman *beta* is the transition between levels 1 and 3, and so on. (One
particularly important line in astronomy is the Balmer alpha line at
6563 Å, often called "H-alpha." It is frequently observed in emission
line nebulae.)
The Lyman lines are all in the ultraviolet, the Balmer in the optical,
and the Paschen, Brackett, Pfund, and Humphreys are all in the infrared regions of
the electromagnetic spectrum. Each is named after the scientist who
discovered the series.

Rydberg's formula (at the time) supported the Bohr model of the atom, at
least for the simple hydrogen atom with its single electron. However, the
equation can be modified for other elements ionized to the point that they
contain only one electron (singly-ionized helium, doubly-ionized lithium,
etc), simply by multiplying the Rydberg constant by the square of the
nuclear charge (the atomic number). However, the equation was
not perfect. In particular, it could not explain phenomena like the
Zeeman and Stark effects. Eventually,
the Bohr model was scrapped in favor of the quantum mechanical model of
Erwin Schrodinger.

The name *Pfund* makes me giggle.