# de Broglie wavelength relation

## de Broglie relation in chemistry

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 confine to radiations alone but should also be extended to matter.

Classical mechanics of particles, wave motion, and laws of energy had been the main tools for study and explanation of the different physical phenomena up to almost the advent of the twentieth century.

He suggested that electrons, protons, atoms could not be regarded simply corpuscles. The periodicity of wave motion must also be assigned to them.

de Broglie was led to this hypothesis from the consideration of quantum theory and the special theory of relativity. He also proposed a relation between momentum and wavelength of a particle in motion.

#### 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 this light is E.

$\therefore&space;E&space;=&space;h\upsilon&space;=&space;h\frac{c}{\lambda&space;}$

where λ = wavelength of light
c = velocity of light
h = Plank constant = 6.627× 10⁻²⁷

Theory of relativity from Einstien
E = m c².

Combining these two relations,

$mc&space;=&space;\frac{h}{\lambda&space;}$

$or,&space;P&space;=&space;\frac{h}{\lambda&space;}$

where p = mc = mv = momentum.

$\therefore&space;\lambda&space;=\frac{h}{p}$

de Broglie extended this relationship to the dynamics of a particle and proposed that a wavelength associated with a moving particle and momentum of the particle.

$\lambda&space;=&space;\frac{h}{p}=\frac{h}{mv}$

where m = total mass of the particle
v = velocity.

#### Bohr’s theory and de Broglie relationship

Angular momentum of moving electrons in Bohr’s model

$mvr=n\frac{h}{2\pi&space;}$

$mv=\frac{h}{2\pi&space;m}$

where m = mass of an electron,
n = principle quantum number = 1, 2, 3, 4, …
r = radius of the orbital of an atom.

According to de Broglie

$\lambda&space;=&space;\frac{h}{mv}$

$or,mv=\frac{h}{\lambda&space;}$

where λ = wavelength of the moving electron.

Combining these two relations

$\frac{nh}{2\pi&space;r}=\frac{h}{\lambda&space;}$

$2\pi&space;r=n\lambda$

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.

But the electron diffraction experiment by Davison and Gramer’s given the evidence of the wave nature of the electron.

#### How to find 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 it has two types of energy potential and kinetic energy.

Thus the energy of an electron

$=Ve=\frac{1}{2}mv^{2}$

where e is the charge of an electron.

Hence from the de Broglie relation

$\lambda=&space;\frac{h}{mv}$

$\therefore&space;\lambda&space;=\frac{h}{\sqrt{2mVe}}$