### Particle and wave nature of an electron

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 arrived at from his particle treatment of electrons.

#### Momentum and wavelength of a particle

de Broglie relation |

- de Broglie proposed a relation between momentum and wavelength of a particle in motion. He considered the light of frequency Î½, the energy is given by

E = hÎ½ = h (c/Î»)

- Where Î» = wavelength, c = velocity of light, and h = Plank constant = 6.627× 10⁻²⁷.

- Again from the famous mass-energy equivalence relation from Einstien,

E = m c²

The momentum of the photon is

p = mv = mc

The momentum of the photon is

p = mv = mc

- Combining these two relations, we have

mc = h/Î»

or, p = h/Î»

∴ Î» = h/p

or, p = h/Î»

∴ Î» = h/p

- de Broglie extended this relationship to the dynamics of a particle and proposed that a wavelength Î» is associated with a moving particle and is related to its momentum as

Î» = h/p = h/mv

- Where m is the total mass of the particle and v is its velocity.

### Bohr's model and de Broglie relation

- Bohr's model for the angular momentum of n-th orbital of moving electrons

mvr = n (h/2Ï€)

or, mv = n (h/2Ï€m)

or, mv = n (h/2Ï€m)

- where m = mass of an electron, n = 1, 2, 3, 4, ..., and r = radius of the orbital of an atom.

- Again according to de Broglie relation

Î» = h/mv

or, mv = h/Î»

or, mv = h/Î»

- where Î» = wavelength of the moving electron.

- Combining these two relations

nh/2Ï€r = h/Î»

or, 2Ï€r = n Î»

or, 2Ï€r = n Î»

- 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.

#### Electron diffraction experiment

- de Broglie's suggestion of matter waves and its confirmation by Davisson and Germer's electron diffraction 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.

### The kinetic energy of an electron

- 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 of an electron = Ve = ½ mv²

where e is the charge of an electron.

where e is the charge of an electron.

Again from the de Broglie relation

Î» = h/mv

Î» = h/mv

From these two relations

Î» = h/√(2mVe)

Î» = h/√(2mVe)