What are matter waves? Derive an expression for the de Broglie wave length.

Explanation

Waves associated with material particles in motion are called matter waves or de Broglie waves. The energy of a photon is given by `E = hv` where h is the Planck.s constant and `v` is  the frequency of radiation. If photon is regarded as a particle of mass m, then its energy given by `E = mc^(2)`where c is the velocity of electromagnetic radiation in free space
`E = hv = mc^(2)` ....(1)
i.e., `h c/ lambda = mc^(2) or h/lambda = mc , " " lambda = h/(mc)=h/p` ....(2)
where `p=mc` momentum of the photon. de Broglie suggested that equation (2) is applicable to both photons and other material particles. Thus, if a particle of mass m moves with a velocity v, then the de Broglie wave length associated with it is
`lambda = h/p = h/(mv)` ....(3)
The above equation is known as de Broglie wave equation.