de Broglie’s Relationship

Introduction

Louis de Broglie was a French physicist. He stated that particles such as electrons could be analyzed as particles and waves and worked on both the particle-like and wave-like nature of matter to describe their properties. He observed the dual nature of matter. de Broglie created a relation between the momentum of matter and wavelength. Therefore, this relation is referred to as de Broglie’s relationship.

de Broglie’s Relationship

 Bohr’s atomic structure model failed to explain several concepts related to splitting spectral lines in the electric and magnetic fields. Several experiments were formulated to overcome the drawbacks of Bohr’s atomic structure model and to develop a more reliable and more accurate atomic structure model. de Broglie postulated a hypothesis regarding the dual nature of matter. Based on this, the highly accelerated microscopic particles reveal dual nature viz particles and wave nature. 

The Relationship Between de Broglie’s Wavelength and Kinetic Energy

According to de Broglie, the wavelength correlated with a particle of mass m,

moving with velocity v by the relationship, 

λ = h/mv

in which, 

h is interpreted as Planck’s constant.

Relation between kinetic energy and de Broglie’s wavelength

12mv2=K.E.

m2v2 = 2m K.E.

mv = 2m K.E.

= h2m K.E.

Significance of de Broglie’s Relationship

de Broglie stated that every object has a particle nature in motion. For example, a wave is observed when we consider a moving car or ball. Thus, de Broglie derived the wavelengths of electrons in a cricket ball.

• The wavelength of a cricket ball:

For instance, we are assuming that

the mass of balls = 150 g = 150 x 10 -3 kg

velocity = 35 m/s 

and h = 6.626 x 10-34 Js

Thus, settling the values in the equation 

λ = h/mv

λ = 6.626 × 10-34/(150 × 10-3 × 35)

By solving this equation, we get the wavelength of the ball.

 λ = 1.2621 x 10-34 m

The main concern that arises is whether the ball depicts a wave nature. The ball has a wavelength associated with it but is small. 

• The wavelength for an electron:

As, me = 9.1 x 10-31 kg, and ve = 218 x 106  m/s. Adding these values in the equation, we get

 λ = h/mv = 6.626 × 10-34/9.1 × 10-31 × 218 × 106

λ = 3.2 Å

Thus, the electrons have wave-particle duality. Similarly, every big object consists of microscopic objects like electrons, depicting a wave-particle nature and wave nature.

• Hypothesis:

Broglie’s equation: λ = h/p = h/mv, 

We assume that 

In case v = 0, λ = ∞, and

In case v = ∞, λ = 0

Derivation of de Broglie Relationship

de Broglie formulated the wavelength equation with the help of Einstein’s mass equation of energy relation and plank equation. 

 With the particle nature, the equation of Einstein is as:

E = mc2 ——(1)

in which, 

E is referred to the energy

m is referred to the mass

and c is referred to the speed of light

With the wave nature, the equation of Planck’s is as:

E = hν——(2)

in which, 

E is referred to the energy

h is referred to the Planck’s constant

and ν is referred to the frequency 

Adding Equation (1) and Equation (2), we get

mc2 = hν ——(3) 

The (v) frequency can be conveyed in terms of the (λ) wavelength as  

ν = c/λ

for general particles. When speed c is replaced by v, Equation (3) can be written as 

mv2 = hv/λ

λ = h/mv

Conclusion

The significance of de Broglie’s relationship is that this equation explains the dual nature of microscopic particles. The diffraction of electron beams explains de Broglie’s relationship as the diffraction in the wave property. The common instrument exemplifying this fact is an electronic microscope. Therefore, each object represents a wave-like character in motion. The wavelengths correlated with the standard objects are short due to a large mass. Thus, the wave properties cannot be detected; however, experiments can detect the wavelengths when they correlate with other subatomic particles and electrons.