De Broglie Wavelength Formula With Solved Examples

De Broglie Formula

The De Broglie equation is an equation that relates a particle’s momentum to its wavelength, and the de Broglie wavelength is the wavelength calculated using this equation.

λ = h(m v) = h × momentum

The widely used De Broglie equation widely defines the wave characteristics of matter. For example, the wave nature of an electron is defined using this equation.

Electromagnetic radiation persists in both the properties of particles and waves. This dual nature was also found in electron particles.

In his thesis, Louis de Broglie proposed that each moving particle, whether microscopic or macroscopic, has a wave character. ‘Matter Waves’ was the title. De Broglie also stated that if a particle behaves like a wave, then a relationship exists between the particle’s velocity and momentum and its wavelength if the particle had to act like a wave.

Derivation:

Particles of very tiny mass travelling faster than the speed of light act like a particle and a wave.

According to Plank’s quantum theory, the energy of an electromagnetic wave is related to its wavelength or frequency. 

  E = hν 

       = hc/ λ …….(1)

E = mc2 was Einstein’s formula for relating particulate matter’s energy to its mass and velocity…….. (2)

De Broglie equated both of these relations for the particle travelling with velocity ‘v’ since the smaller particle shows dual nature and has the same energy as the larger particle.

E= hc / λ = mv2

Then, h / λ = mv

Or, λ = h / mv = h/ momentum

Where h is known as Plank’s constant.

Solved Examples

1. Calculate the Electron’s Wavelength if it travels at the speed of light.

The wavelength equation of De Broglie is as follows: λ=h/ mv 

6. The Planck’s constant and its value is 6.6260 x 10-34Js. 

The velocity v is 3 x 108ms-1, which is the speed of light in this case.

The electron’s mass is 9.1 x 10-31kg.

We can get this by substituting all of these values.

λ = (6.620 × 10-34) / 9.1×10-31 × 3×108

λ = 0.2424 × 10-11m

λ = 2.424 nm

1. De Broglie’s theory of the dual behaviour of matter led to the development of the electron microscope, which is commonly used to magnify images of biological molecules and other materials. Calculate the de Broglie wavelength associated with this electron if the electron’s velocity in this microscope is 2 × 106 ms-1.

λ = h/ mv

           = (6.63 × 10-34) /( 9.1 × 10-31×2 × 106)

           = 3.64 × 10-10m