Wavenumber

Ordinary wavenumber is defined as the number of wave cycles divided by length; it is a physical quantity with dimension of reciprocal length, expressed in SI units of cycles per metre or reciprocal metre (m−1).

Angular wavenumber, defined as the wave phase divided by time, is a quantity with dimension of angle per length and SI units of radians per metre.

For quantum mechanical waves, the wavenumber multiplied by the reduced Planck constant is the canonical momentum.

For example, a wavenumber in inverse centimeters can be converted to a frequency expressed in the unit gigahertz by multiplying by 29.9792458 cm/ns (the speed of light, in centimeters per nanosecond);[5] conversely, an electromagnetic wave at 29.9792458 GHz has a wavelength of 1 cm in free space.

In theoretical physics, a wave number, defined as the number of radians per unit distance, sometimes called "angular wavenumber", is more often used:[6] When wavenumber is represented by the symbol ν, a frequency is still being represented, albeit indirectly.

For electromagnetic radiation in vacuum, wavenumber is directly proportional to frequency and to photon energy.

The imaginary part of the wavenumber expresses attenuation per unit distance and is useful in the study of exponentially decaying evanescent fields.

Wavelength, phase velocity, and skin depth have simple relationships to the components of the wavenumber: Here we assume that the wave is regular in the sense that the different quantities describing the wave such as the wavelength, frequency and thus the wavenumber are constants.

(in reciprocal centimeters, cm−1) 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−1): 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: which remains essentially the same in air, and so the spectroscopic wavenumber is directly related to the angles of light scattered from diffraction gratings and the distance between fringes in interferometers, when those instruments are operated in air or vacuum.

For example, the spectroscopic wavenumbers of the emission spectrum of atomic hydrogen are given by the Rydberg formula: where R is the Rydberg constant, and ni and nf are the principal quantum numbers of the initial and final levels respectively (ni is greater than nf for emission).

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",[11] incorrectly transferring the name of the quantity to the CGS unit cm−1 itself.