What is De Broglie Wavelength of Electron & Its Derivation

Wavelength in physics can be defined as the distance from one crest to another crest is called wavelength, and it is denoted with λ. According to its definition, the wave repeating its characteristics after a time period. Before going to discuss this concept, we should know the basics of an electron and what it is actually? Electron is a sub particle in the atom, denoted by “e-”. This electron has a negative electrical charge. These electrons play an important role in transferring electricity into solid materials. According to the French scientist Louis de Broglie, even electrons also having the wave properties. In his thesis, he proved that all matters/particles have wave properties even electron also. De Broglie proposed an equation to describe the properties of any matter/particle. In this article will know the de Broglie wavelength of the electron, its equation, derivation, and de Broglie wavelength of an electron at 100 EV.

What is De Broglie Wavelength of Electron?

According to Louis de Broglie, all the particles hold the properties of a wave. They can show some wave-type properties. The same theory applies to the electron also as per his statement.

An electron wave has a wavelength λ and this wavelength dependent on the momentum of the electron. Momentum (p) of the electron is expressed in terms of the mass of the electron (m) and the velocity of the electron (v).

∴Momentum of the electron (p) = m * v

Then the wavelength λ is

∴ Wavelength λ = h/p

Here h is the Planck’s constant and its value is 6.62607015×10-34 J.S

The formula for λ is known as the de Broglie wavelength of the electron. By analyzing this we can say that slowly moving electrons are having the large wavelength and fast-moving electrons are having a short or minimum wavelength.

De Broglie Wavelength of Electron Derivation

The derivation of De Broglie Wavelength of an Electron states the relation between matter and energy. To derivate the de Broglie wavelength of an electron equation, let’s take the energy equation which is

E = m.c²

Here m = mass

E = energy

C = speed of light

And Planck’s theory also states that the energy of a quantum is related to its frequency along with plank’s constant.

E = h.v

∴ Equating the two energy equations to get the de Broglie wavelength equation.

 m.c²  =  h.v

Any real particles can’t travel with the speed of light. So, replace the velocity (v) by the speed of the light (c).

m.v² = h.v

Substitute the ‘v’ by v/ λ, then, m.v2 = h.v/ λ

∴ λ = h.v/m.v^2a

The above equation indicates the de Broglie wavelength of an electron.

For example, we can find the de Broglie wavelength of an electron at 100 EV is by substituting the Planck’s constant (h) value, the mass of the electron (m) and velocity of the electron (v) in the above equation. Then the de Broglie wavelength value is 1.227×10-10m.

Any particle or a matter has the wave type properties in this universe according to de Broglie. And they can have the wavelength. Those values can be known by the de Broglie wavelength equation. By considering the particle velocity and mass value along with Planck’s constant we can find out its wavelength. The particles which are having more mass value than the fewer particles have the least wavelength.