de Broglie relation-duel nature of particle
de Broglie relation-duel nature of particle

In 1924 de Broglie’s relation pointed out that just as a light electron also has both particle and wave nature. According to de Broglie, this dual nature — wave and particle — should not be confined to radiations alone but should also be extended to matter. He suggested that electrons travel in waves, analogous to light waves. His idea could be fitted to drive the same relation that Bohr theory arrived at from his particle treatment of electrons.

Particle and wave nature of radiation

de Broglie proposed a relation between momentum and wavelength of a particle in motion. He considered the light of frequency ν, the energy of an electron is given by, E = hν = h (c/λ), where λ = wavelength, c = velocity of light, and h = Plank constant = 6.627× 10⁻²⁷. Again from the famous mass energy conservation relation, E = m c². The momentum of the photon (p) = mv = mc

Combining these two relations, we have, mc = h/λ or, p = h/λ. Therefore, λ = h/p. de Broglie extended this relationship to the dynamics of a particle and proposed that a wavelength λ is associated with a moving particle and is related to its momentum as, λ = h/p = h/mv, where m is the total mass of the particle and v is its velocity.

de Broglie waves in the Bohr model

According to Bohr’s model angular momentum of n-th orbital of moving electrons, mvr = n (h/2π), where m = mass of an electron and n = principle quantum number = 1, 2, 3, 4, …, and r = radius of the orbital of an atom.

Again according to de Broglie relation, λ = h/mv or, mv = h/λ, where λ = wavelength of the moving electron. Combining these two relations, we have, 2πr = n λ. Thus a standing produces a stationary pattern, its profile being fixed within the space allowed to it. It does not travel beyond the allowed space.

Wave-particle duality experiment

de Broglie’s suggestion of matter waves and its confirmation by Davisson and Germer’s electron diffraction spectrum experiment conclusively proves that electrons are not is not an ordinary particle. From the evidence by the experiment of determination of mass and e/m electron has particle nature. Electron diffraction experiment by Davison and Gramer’s given the evidence of the wave nature of the electron.

Kinetic energy and de Broglie wavelength

If the particle is an electron and if it is subjected to the potential difference V so as to acquires a velocity v, then, kinetic energy = Ve = ½ mv², where e is the charge of an electron. Again from the de Broglie relation, λ = h/mv. From these two energy relations, we have, λ = h/√(2mVe).
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