Write the de-Broglie hypothesis and derive the equation for the de-Broglie wavelength.

(N/A) If radiation has a dual (wave-particle) nature,why should particles of matter not also exhibit wave-like character?
Based on this,the scientist Louis Victor de-Broglie proposed the following hypothesis:
Moving particles of matter should display wave-like properties under suitable conditions.
Nature is symmetric,and the two basic physical entities—matter and energy—must have a symmetrical character.
If radiation shows a dual nature,then matter must also possess a dual nature.
de-Broglie showed that if the wavelength of a particle is $\lambda$ and its momentum is $p$,then:
$\lambda = \frac{h}{p} = \frac{h}{mv}$
where $m$ is the mass of the particle,$v$ is the speed of the particle,and $h$ is Planck's constant.
From the de-Broglie equation,the dual nature of matter can be easily observed.
The left-hand side of the equation represents the wavelength $\lambda$,whereas the right-hand side of the equation contains the momentum $p$,which is associated with a particle.
This equation is the hypothesis for particles of matter. It is also true for a photon:
For a photon,$E = pc = h\nu$. Since $\nu = \frac{c}{\lambda}$,we have $pc = \frac{hc}{\lambda}$,which gives $p = \frac{h}{\lambda}$ or $\lambda = \frac{h}{p}$.
Thus,the de-Broglie wavelength of a photon is associated with the wavelength of an electromagnetic wave. Therefore,a photon of radiation possesses both quantum energy and momentum.