Editing Wavenumber (section)

== In spectroscopy ==
In [[spectroscopy]], "wavenumber" <math>\tilde{\nu}</math> (in [[Reciprocal meter|reciprocal centimeters]], cm<sup>−1</sup>) refers to a temporal frequency (in hertz) which has been divided by the [[speed of light in vacuum]] (usually in centimeters per second, cm⋅s<sup>−1</sup>):
: <math> \tilde{\nu} = \frac{\nu}{c} = \frac{\omega}{2\pi c}. </math>
The historical reason for using this spectroscopic wavenumber rather than frequency is that it is a convenient unit when studying atomic spectra by counting fringes per cm with an [[interferometer]] : the spectroscopic wavenumber is the reciprocal of the wavelength of light in vacuum:
: <math>\lambda_{\rm vac} = \frac{1}{\tilde \nu},</math>
which remains essentially the same in air, and so the spectroscopic wavenumber is directly related to the angles of light scattered from [[diffraction grating]]s and the distance between fringes in [[interferometer]]s, when those instruments are operated in air or vacuum. Such wavenumbers were first used in the calculations of [[Johannes Rydberg]] in the 1880s. The [[Rydberg–Ritz combination principle]] of 1908 was also formulated in terms of wavenumbers. A few years later spectral lines could be understood in [[Quantum mechanics|quantum theory]] as differences between energy levels, energy being proportional to wavenumber, or frequency. However, spectroscopic data kept being tabulated in terms of spectroscopic wavenumber rather than frequency or energy.

For example, the spectroscopic wavenumbers of the [[Hydrogen spectral series|emission spectrum of atomic hydrogen]] are given by the [[Rydberg formula]]:
: <math> \tilde{\nu} = R\left(\frac{1}{{n_\text{f}}^2} - \frac{1}{{n_\text{i}}^2}\right), </math>
where ''R'' is the [[Rydberg constant]], and ''n''<sub>i</sub> and ''n''<sub>f</sub> are the [[principal quantum number]]s of the initial and final levels respectively (''n''<sub>i</sub> is greater than ''n''<sub>f</sub> for emission).

A spectroscopic wavenumber can be converted into [[photon energy|energy per photon]] ''E'' by [[Planck's relation]]:
: <math>E = hc\tilde{\nu}.</math>
It can also be converted into wavelength of light:
: <math>\lambda = \frac{1}{n \tilde \nu},</math>
where ''n'' is the [[refractive index]] of the [[optical medium|medium]]. Note that the wavelength of light changes as it passes through different media, however, the spectroscopic wavenumber (i.e., frequency) remains constant.

Often spatial frequencies are stated by some authors "in wavenumbers",<ref>See for example,
* {{cite journal |last1=Fiechtner |first1=G.
 |year=2001
 |title=Absorption and the dimensionless overlap integral for two-photon excitation
 |journal=[[Journal of Quantitative Spectroscopy and Radiative Transfer]]
 |volume=68 |issue=5 |pages=543–557
 |doi=10.1016/S0022-4073(00)00044-3
 |bibcode = 2001JQSRT..68..543F |url=https://zenodo.org/record/1259655
}}
* {{cite patent
 |invent1=Ray, James C.
 |invent2=Asari, Logan R.
 |pubdate=1991-09-10
 |title=Method and apparatus for spectroscopic comparison of compositions
 |country=US 
 |number=5046846
}}
* {{cite journal
 |year=2005
 |title=Boson Peaks and Glass Formation
 |journal=[[Science (journal)|Science]]
 |volume=308 |issue=5726 |pages=1221
 |doi=10.1126/science.308.5726.1221a
 |s2cid=220096687
}}</ref> incorrectly transferring the name of the quantity to the CGS unit cm<sup>−1</sup> itself.<ref>
{{cite book
 |last=Hollas
 |first=J. Michael
 |date=2004
 |title=Modern spectroscopy
 |url=https://books.google.com/books?id=lVyXQZkcKKkC&pg=PR22
 |publisher=John Wiley & Sons
 |page=xxii
 |isbn=978-0470844151
}}</ref>