De-Broglie noted that according to the theory of relativity, the role played by momentum 'P' and the energy 'E' of the particle is done by angular frequency 'w' and the propagation vector 'k' of a wave. According to Einstein relation
E = mc2
Where 'm' is the mass of the particle and 'c' = is the velocity of light and according to quantum theory
E = hu
hu = mc2
(Since c = ul)
(Since mc = p)
This relation ship tells us that if there is a particle of mass 'm' moving with a velocity if manifests itself in the form of a wave, its wavelength would be h/p. This is de Broglie wave equation.
De-Broglie's hypothesis attributing a dual particle-wave character to matter appears very strange at first sight. Since it is very much against the direct evidence of one's senses at the macroscopic level. But at the level of the atom, the behavior of matter has been already found to be unconventional. It was such evidence of non-classical behavior, which gave scope for De Broglie's proposal. The precise form of the hypothesis was determined however by the speculative postulate that at the most fundamental level, matter and radiation which form the basic constituents of the physical should be similar in nature. Confirmation that nature does exhibit such an aesthetically pleasing symmetry between matter and radiation came from the experiments by Davison, Germer, and G.P. Thompson. These experiments demonstrated the existence of 'De Broglie Waves' associated with electrons in a very direct fashion.