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

Explanation

The waves associated with the moving particles are called waves or de-Broglie waves.
de-Broglie equated he energy equation of Planck (wave) and Einstein (particle).
For a wave of frequency v, the energy associated with each photon is given by Plank.s relation.
E = hv
where h is Planck.s constant. ...(1)
According to Einstein.s mass energy relation, a mass m is equivalent to energy
`E=mc^(2)` ...(2)
where c is the velocity of light.
if `hv=mc^(2)`
`:. (hc)/(lambda)=mc^(2)` (or) `lambda=(h)/(mc)` ...(3)
(since `v=(c)/(lambda)`)
For a particle moving with a velocity v, If c=v from equation (3)
`lambda=(h)/(mc)=(h)/(p)` ...(4)
Where p=mv the momentum of the particle. These hypothetical matter waves will have appreciable wavelength only for very light particle.