Derive de-Broglie’s equation.

Hint: As you know, de-Broglie’s equation is one of the equations used to define the wave properties of matter. It is actually describing the wave nature of the electron.

Complete step by step answer: We know that electromagnetic radiation exhibits the dual nature of a particle and wave. Microscopic particles like electrons also possess this type of dual nature.
Let us derive the de-Broglie equation:
Very low mass particle moving at speed less than that of light behaves like a particle and wave. De-broglie derived an expression relating the mass of such smaller particles and its wavelength.
Plank’s quantum theory relates the energy of an electromagnetic wave to its wavelength or frequency.
${{E = h\nu = }}\dfrac{{{\text{hc}}}}{{{\lambda }}}$ ………..(1)
Einstein related the energy of particle matter to its mass and velocity, as ${\text{E = m}}{{\text{c}}^{\text{2}}}$ ………(2)
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,
${\text{E = }}\dfrac{{{\text{hc}}}}{{{\lambda }}}{\text{ = m}}{{\text{v}}^{\text{2}}}{\text{:Then,}}\dfrac{{\text{h}}}{{{\lambda }}}{\text{ = mv}}$
 ${{\lambda = }}\dfrac{{\text{h}}}{{{\text{mv}}}}{\text{ = }}\dfrac{{\text{h}}}{{{\text{momentum}}}}$, where ‘h’ is the plank’s constant.

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.

Additional Information:
Particle and wave nature of matter, however, looked contradictory as it was not possible to prove the existence of both properties in any single experiment. This is because of the fact that every experiment is normally based on some principle and results related to the principle are only reflected in that experiment and not the other.

Note: You should notice that both the particle nature and the wave nature are necessary to understand or describe the matter completely. Hence, particles and wave nature of matter are actually ‘complementary’ to each other. It is not necessary for both to be present at the same time though. The significance of de-Broglie relation is that it is more useful to microscopic, fundamental particles like electrons.