Wavelength, wavenumber, frequency and photon energy are five ways of writing the same physical quantity, linked by the Planck-Einstein relation and the speed of light.

Constants

All four are exact under the 2019 SI definitions, so the conversions carry no measurement uncertainty of their own. The Planck constant h is 6.62607015 × 10⁻³⁴ J·s, the speed of light c is 299,792,458 m/s, and the elementary charge e is 1.602176634 × 10⁻¹⁹ C. The product used throughout is hc = 1239.8419843 eV·nm, which is where the familiar shortcut E in eV = 1239.84 / λ in nm comes from.

On the wavenumber symbol

ṽ is the spectroscopic wavenumber, 1/λ in reciprocal centimetres, which is the unit line positions are tabulated in. It is not the angular wavenumber k = 2π/λ used in wave physics, and the two differ by a factor of 2π. Line lists such as HITRAN use the spectroscopic form.

Vacuum, not air

Every conversion here is for propagation in vacuum. In a medium the frequency and the photon energy are unchanged but the wavelength shortens by the refractive index, and so the wavenumber rises. The refractive index of air is about 1.00027 in the near infrared at ambient conditions, so an air wavelength runs roughly 0.03 % below the vacuum value: at 1550 nm that is about 0.4 nm, which is far wider than an absorption line. Spectroscopic databases quote vacuum wavenumbers while some laser datasheets quote air wavelengths, and mixing the two will put you on the wrong side of a line. A precise conversion needs the Edlén or Ciddor equation with the actual temperature, pressure and humidity.

Band names

The band boundaries on the scale follow ISO 20473: ultraviolet below 400 nm, visible from 400 to 780 nm, IR-A from 780 to 1400 nm, IR-B from 1400 to 3000 nm and IR-C from 3 µm to 1 mm. Other conventions draw the near, mid and far infrared boundaries in different places, so check which one a datasheet means before comparing.