De Broglie Equation and Wavelength

De Broglie’s hypothesis is one of the most fundamental theories that give a direction to quantum mechanics from classical physics. It describes the dual nature of matter, i.e., matter can behave like both particles and waves. The phenomenon of a beam of light with diffraction, just like a wave, is explained by this theory.

In 1924, a French physicist named Louis de Broglie gave a proposal about the wave nature of the particle. It was observed that the electron, which we usually think of as a particle, may, in some circumstances, behave like a wave. 

De Broglie’s theory

If a particle acts as a wave, it should have a wavelength and a frequency. According to De Broglie’s theory, a free particle with rest mass ‘m’, which is moving with a non-relativistic speed of ‘v’, should have a wavelength related to its momentum p=mv, the same way as a photon, as expressed by the equation:

hp=hmv 

Where, λ =De Broglie wavelength of a particle

             h=Planck’s constant

             p=Particle’s momentum

             m=Particle’s mass

             v=Particle’s speed

Derivation of De Broglie equation

According to Einstein’s equation –

E= mc²           …. (i)

Where E is energy, m is mass, and c is the speed of light (3 x 10⁸  ms-1)

Plank’s equation tells us that quantum energy (E) is directly proportional to the frequency of radiation (n). 

E=hv    =    h c/ λ       … (ii)

Where E = Energy, h = Plank’s constant, v = frequency= c/ λ, λ = wavelength and c = velocity of light.

The energy emitted from a moving object is proportional to the frequency of waves and inversely proportional to the wavelength. The emission of electromagnetic radiation (EMR) occurs due to the acceleration of charged particles in an atom. In simpler terms, when an atom is accelerated, it radiates a photon, which is a fundamental quantum particle that carries energy away in the form of EM waves.

 Putting value of E=mc2 from equation (i) in equation (ii)

                                 mc2 = hc/ λ             

 (c can be replaced by velocity v for a general particle)

                                 mv2 = hv/ λ     

                                 λ   = h/mv          … (iii)

The equation (iii) is known as the De Broglie relationship or equation.   

The formula for De Broglie wavelength

There are a few explanations for the fact that the wavelength is exposed in experiments with Broglie particles. However, not all of these definitions can be mathematically represented, nor do they provide a practical, justified formula. 

Particle waves:

If particles are stimulated by other particles during testing or particle collisions with measuring instruments, internal waves may occur in the particles. It can be electric waves or waves associated with solid particle interactions, gravity in the gravitational force model of solid interactions, etc. With Lorentz modification, the wavelength of internal oscillations can be translated into wavelengths by an external viewer. Statistics provide de Broglie’s wavelength formula and Broglie’s wavelength distribution speed:

cB = λB/TB = c2/v

Here, TB = Period of oscillation

Atoms having electrons:

The movement of electrons in atoms occurs around the nuclei of the atom. In the larger model, the electrons have the shape of a disk-shaped cloud. This is the result of action equal to the force of magnitude arising:

1) the electron attraction to the nucleus due to strong gravitational force and attraction of Coulomb by the charge of the electron and nucleus,

2) the expulsion of the charged electron, the matter from which it is derived, and

3) the transit of the electron matter from the nucleus due to rotation, defined by the central force.

If we assume that the electron atom in the atom comprises the n of the Broglie wave, then in the case of a circular cycle with radius r and an angular force of electron L, we will find the following:

2πr = nλB 

L = rp = nh/2π

λB = h/p

Conclusion

Although de Broglie was credited with his hypothesis, he had no proof of it. In 1927, Clinton J. Davisson and Lester H. Germer sprayed electron particles into crystal nickel, and what they saw was an electron-like separation of waves against crystals. That same year, English physicist George P. Thomson fired electrons toward a thin metal plate giving him the same results as Davisson and Germer.