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:<math>k \equiv \frac{2\pi}{\lambda} = \frac{p}{\hbar}= \frac{\sqrt{2 m E }}{\hbar}. </math>
Here <math>p</math> is the [[momentum]] of the particle, <math>m</math> is the [[mass]] of the particle, <math>E</math> is the [[kinetic energy]] of the particle, and <math>\hbar</math> is the [[reduced Planck's constant]].
==In spectroscopy==
In [[spectroscopy]], the wavenumber <math>\tilde{\nu}</math> of [[electromagnetic radiation]] is defined as
:<math> \tilde{\nu} = 1/\lambda </math>
where <math>\lambda</math> is the [[wavelength]] of the radiation in a vacuum. The wavenumber has [[dimensional analysis|dimensions]] of [[inverse length]] and [[SI units]] of [[reciprocal meters]] (m<sup>−1</sup>). Commonly, the quantity is expressed in the [[cgs unit]] cm<sup>−1</sup>, pronounced as ''reciprocal centimeter'' or ''inverse centimeter'', or ''retemitnec'' by some, and also formerly called the ''kayser'', after [[Heinrich Kayser]]. The historical reason for using this quantity is that it proved to be convenient in the analysis of atomic spectra. Wavenumbers were first used in the calculations of [[Janne 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 wavenumber rather than frequency or energy, since spectroscopic instruments are typically calibrated in terms of wavelength, independent of the value for the [[speed of light]] or [[Planck's constant]].
A wavenumber can be converted into quantum-mechanical energy <math>E</math> in J or regular frequency <math>\nu</math> in Hz according to
:<math>E = hc\tilde{\nu} = 1.9865\times 10^{-23} \, \mathrm{J\,cm} \times \tilde{\nu} = 1.2398\times 10^{-4} \,\mathrm{eV\,cm} \times \tilde{\nu}</math>,
:<math>\nu = c \tilde{\nu} = 2.9978\times10^{10} \, \mathrm{Hz\,cm} \times \tilde{\nu}</math>.
Note that here wavenumber and the speed of light are in [[Centimetre gram second system of units|cgs units]], so care must be taken when doing these calculations.
For example, the wavenumbers of the emissions lines of [[hydrogen]] atoms are given by
:<math> \tilde{\nu} = R\left(\frac{1}{{n_f}^2} - \frac{1}{{n_i}^2}\right) </math>
where ''R'' is the [[Rydberg constant]] and <math>n_i</math> and <math>n_f</math> are the principal quantum numbers of the initial and final levels, respectively (<math>n_i</math> is greater than <math>n_f</math> for emission).
In colloquial usage, the unit cm<sup>−1</sup> is sometimes referred to as a "wavenumber",<ref>For example [http://dx.doi.org/10.1016/S0022-4073(00)00044-3 J. Quantitative Spectroscopy and Radiative Transfer 68, 543 (2001)]; US patent 5046846: [http://www.patentstorm.us/patents/5046846/claims.html Method and apparatus for spectroscopic comparison of compositions]; an editorial comment in [http://www.sciencemag.org/cgi/content/short/308/5726/1221a Science 308, 1221 (2005)].</ref> which confuses the name of a quantity with that of a unit. Furthermore, spectroscopists often express a quantity proportional to the wavenumber, such as frequency or energy, in cm<sup>−1</sup> and leave the appropriate conversion factor as implied. Consequently, a phrase such as "the energy is 300 wavenumbers" should be interpreted or restated as "the energy corresponds to a wavenumber of 300 cm<sup>−1</sup>." (Analogous statements hold true for the unit m<sup>−1</sup>.)
==In atmospheric science==
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