As the Newtonian classical approach to deal with problems involving particles at atomic level failed, the concept of wave mechanics was introduced by Louis Debroglie in 1924 and has been successfully applied to atomic and molecular structures.

Louis Debroglie, a French scientist extended the idea of dual nature of radiation to matter.

According to Debroglie, a materialistic particle posessess both wave and particle characteristics.

It is to be known that dual nature of light was explained by combining Planck's expression for energy of photon

E = h𝜈 --------------------(1)

Einsteins Mass energy relation

E = mc² ------------------(2)

h𝜈 = mc²------------------(3)

as we know,

𝜈 = C/ƛ -------------------(4)

From (3) & (4)

ƛ = h/P --------------------(5)

Just in analogy with equation 5, Debroglie proposed concept of matter waves according to which a material particles of mass M moving with a velocity 'v' should have an associated wave length ƛ called Debroglie's wavelength given by

ƛ = h/mv = h/P ----------(6)

'm' - according to theory of relativity is not an invariable quantity as in classical physics.

m = m₀/√ 1-(v²/c²) ------ (7)

If particles like electrons are accelerated to various velocities, we can produce waves of various wavelengths. The higher the velocity of electron, the smaller the debroglie's wavelength.

If electron is accelerated by applying potential V, then work done on electron is eV.

mv²/2 = eV ----------------(8)

mv = P = √ 2meV --------(9)

substituting (9) in (6), we have

ƛ = h/√ 2meV ------------(10)