Explain de Broglie wave theory

Matter-wave or a de Broglie wave

Louis de Broglie, a French physicist, in 1924, suggested that all microscopic, as well as

macroscopic objects, possess dual character. The wave associated with the particle is

called a matter-wave or a de Broglie wave.

The wavelength of the wave associated with any material particle was calculated as:

In the case of a photon, if it is assumed to have a wave character, its energy is given by

$E=h\nu$

where ν is the frequency of the wave and h is Planck’s Constant.

If the photon is supposed to have particle character, its energy is given by

$E=m{c}^2$

where m is the mass of the photon and c is the velocity of light.
As the smaller particle exhibits dual nature, and energy being the same, de Broglie equated both these relations for the particle moving with velocity ‘$v$’ as,

$h\nu =m{v}^2$

$\frac{hc}{\lambda }=m{v}^2$

$\Rightarrow \lambda =\frac{h}{mv}$

This equation relating the momentum of a particle with its wavelength is the de-Broglie equation and the wavelength calculated using this relation is the de-Broglie wavelength.