Wave Number

Wavenumber is a crucial topic in physics. Wavenumber is the reciprocal of the wavelength of a wave. It is specially used to calculate the number of cycles per unit distance. Wavenumber has a magnitude with no direction. 

A Wave number definition states that “A wave's spatial frequency, measured in cycles per unit distance or radians per unit distance”. The Wavenumber is the magnitude of the wave vector in multidimensional systems. Wavenumber is a scalar quantity.

Wavenumber definition in spectroscopy and chemistry fields is defined as the number of wavelengths per unit distance.

angular wavenumber is a quantity in theoretical physics that is defined as the number of radians per unit distance. 

Wavenumber Formula in Spectroscopy 

Wavenumber \[\bar{v}\] is a term used in spectroscopy to describe a frequency that has been separated by the speed of light in a vacuum. The formula of Wave number in spectroscopy and chemistry fields is given as,

 \[\bar{v}=\frac{1}{\lambda }=\frac{\omega }{2\pi c}=\frac{v}{c}\]

Where \[\bar{v}\] is the spectroscopy wavenumber.

λ is the wavelength sometimes called a spectroscopic wavenumber which is equal to the spatial frequency.

\[\omega =2\pi v\] is the angular frequency.

• By using Planck’s relation spectroscopic wavenumber can be converted into energy per photon as follows,

\[E=hc\bar{v}\]

Where E is the energy per photon,

h is the reduced Planck’s constant = 6.62607004 x 10^-34m²kg/s

c is the speed of light

\[\bar{v}\] is the Spectroscopic Wavenumber

• Spectroscopic wavenumber can be converted to the wavelength of light as follows,

λ=\[\frac{1}{n\bar{v}}\]   

Where λ  is the wavelength,

n is the refractive index of the medium,

\[\bar{v}\] is the Spectroscopic Wavenumber

The SI unit of measurement of Spectroscopic Wavenumber is reciprocal of a meter that is m^-1. The CGS unit of measurement of Spectroscopic Wave number is reciprocal of a centimetre that is cm^-1.

Wavenumber Formula for Wave Equations

The formula for angular wave number in theoretical physics is given by 

K = \[\frac{2\pi }{\lambda }=\frac{2\pi v}{v_p}=\frac{\omega }{v_p}\]

Where k is the angular wave number,

λ is the wavelength,

\[\omega =2\pi v\] is the angular frequency,

V_p is the phase velocity

For the special case of an electromagnetic wave propagating at the speed of light in a vacuum, the Wave equation k is given by

k = \[2\pi \frac{E}{hc}\]

Where k is the angular wave number,

E is the energy of the wave,

h is the reduced Planck’s constant = 6.62607004 × 10^-34 m² kg / s

c is the speed of light

For the special case of a matter-wave in the non-relativistic approximation that is in the case of a free particle (a particle that has no potential energy), the Wave equation k is given by

k = \[\frac{2\pi }{\lambda }=2\pi \frac{p}{h}=2\pi \frac{\sqrt{2mE}}{h}\]

Where k is the angular wave number,

λ is the wavelength,

p is the momentum of the particle,

h is the reduced Planck’s constant = 6.62607004 × 10^-34 m² kg / s

m is the mass of the particle

E is the kinetic energy of the particle

Applications of Wavenumber

• A wavenumber is used to calculate the spatial frequency.

• Other than spatial frequency, wavenumber can be used to define other quantities such as optics and wave scatterings in physics.

• Wavenumbers and wave vectors are crucial in X-ray diffraction, neutron diffraction, electron diffraction, and elementary particle physics.

• The canonical momentum of quantum mechanical waves is the wavenumber multiplied by the reduced Planck's constant.

• Group velocity can be defined by using a wavenumber.

Problems on Wave Number

In this section, we will learn how to find wavenumber by using the different formulas which we discussed till now. Since there are different types of formulas to calculate the wavenumber depending on the various associated terms. Let us solve some problems to understand the concepts of wavenumber more clearly.

Calculate the Angular Wavenumber if the Wavelength of the LightWave is 600 Nanometers. Also, Calculate the Spectroscopic Wavenumber.

Ans: Here we are given only the wavelength of the light wave, so we will use the angular wavenumber which is directly related to the wavelength.

The angular wavenumber formula is given by

k = \[\frac{2\pi }{\lambda }\]

Here is the wavelength of the light wave is given as 600 nanometers which is equal to 600 x 10^-9m.

Now substituting the values we will get the angular wavenumber as follows:

k = \[\frac{2\pi }{600\times 10^{-9}}\]

k = 10.48 x 10⁶m^-1

To calculate the spectroscopic wavenumber use the formula which describes the relation between spectroscopic wavenumber and wavelength of the light wave.

The spectroscopic wavenumber is given as follows:

\[\bar{v}\]= \[\frac{1}{\lambda }\]

Here λ is the wavelength of the light wave is given as 600 nanometers which is equal to 600 × 10^-9m. Now substituting the values we will get the spectroscopic wavenumber as follows:

Now substituting the values we will get the spectroscopic wavenumber as follows:

\[\bar{v}\]= \[\frac{1}{600\times 10^{-9}}\]

\[\bar{v}\]= 1.67 x 10⁶m^-1

Conclusion

In atomic, molecular, and nuclear spectroscopy, a wavenumber is defined as the true frequency divided by the speed of light, and which is equal to the number of waves in a unit distance. In optics and wave scattering physics, such as X-ray diffraction, neutron diffraction, electron diffraction, and elementary particle physics, wavenumbers and wave vectors are frequently used.