De Broglie’s Relationship

De Broglie’s Relationship 

The popular Bohr’s Model was not successful in explaining concepts about the spectrum of atoms, splitting into spectral lines in an electric or magnetic field. Attempts were made to generate a common model for the atoms to overcome the failure in Bohr’s model. 

It was Louis de Broglie, a French physicist, who proposed the theory that said that similar to light, even electrons possess properties such as waves and particles. This was called dual behaviour. He rearranged the denotations of Plank-Einstein relation to make it applicable to all atoms. Here de Broglie found a relation between momentum and wavelength, known as de Broglie wavelength. It shows the wave nature shown by the electrons.

i.e. λ = h/mv (de Broglie equation)

Relation Between Wavelength and Momentum 

Let the energy of a photon be in terms of its frequency v.

So, E = hv -eq (1)

Where:

E= Energy

h= Planck’s constant (6.63 × 10-34 J s)

v= frequency

This theory of relativity provides a different aspect in terms of the velocity of light. Here, m is the relativistic mass of light. Relativistic mass is non-zero because it is travelling with a velocity of c. It can be zero only when it is at rest.

E = mc2 -eq (2)

Where:

E= Energy

m= mass

c= speed of light

On comparing, de Broglie disagreed that a particle having non-zero rest mass m and velocity v would have a wavelength given by:

λ = h/mv

Where h= Planck’s constant

λ= wavelength

m= mass of particle moving at velocity v

Since, mv=p

Where p= particle’s momentum

So now, the de Broglie relation is:

λ = h/p

Significance of De Broglie Equation

The de Broglie relationship can be explained using the diffraction of an electron beam; diffraction is considered as a property of waves. Thus, he states that every moving object will have a wave-like character. De Broglie proves that every object when in motion has a particle nature. But when we look we may not see that nature. Still, each of them has very minute momentum. 

Relation Between De Broglie Wavelength and Kinetic Energy

When a photon or an electron undergoes a potential difference V to gain a velocity v it produces two types of energy such as potential and kinetic energy. 

So, the energy of an electron, E = Ve = ½ mv2, where e is the electron charge. 

Again, λ = h/mv. 

From the two, wavelength (λ) = h/(√2mVe).

This is the relation between de Broglie wavelength and the kinetic energy of a photon or electron.