Question

Explain de broglie equation related to the dual behaviour of matter

Answer 1

Answer:

Bohr's model failed to explain many concepts related to the spectrum of different atoms and splitting of spectral lines in magnetic and electric field. To overcome the shortcomings of Bohr’s atomic model, attempts were made to develop a more general model for atoms. One of those developments which contributed significantly in the formulation was the analysis of dual behavior of matter. De Broglie proposed that as light exhibits both wave like and particle like properties, matter too exhibit wave like and particle like properties. This nature was described as dual behavior of matter. On the basis of his observations, de Broglie derived a relationship between wavelength and momentum of matter. This relationship is known as de Broglie relationship.

Considering the particle nature, Einstein equation is given as,

E= mc2 —- (1)

Where,

E= energy

m= mass

c = speed of light

Considering the wave nature, the Plank’s equation is given as,

E = hν      ——– (2)

Where,

E= energy

h = Plank’s constant

ν = frequency

 

From (1) and (2),

mc2 = hν ——– (3)

Frequency, ν can be expressed in terms of wavelength, λ as,

ν = cλ

For a general particle, c can be replaced with the velocity of object, v. Hence, equation (3) can be given as,

mv2 = hvλ

⇒λ = hmv

The above equation is known as de Broglie relationship and the wavelength, λ  is known as de Broglie wavelength. Diffraction of electron beams explains the de Broglie relationship as diffraction is the property of waves. Electron microscope is a common instrument illustrating this fact. Thus, every object in motion has a wavelike character. Due to a large mass, the wavelengths associated with ordinary objects are so short that their wave properties cannot be detected. On the other hand, the wavelengths associated with electrons and other subatomic particles can be detected experimentally.

Answer 2

Answer:

The de Broglie equation:-

The de Broglie equation is one of the equations that is commonly used to define the wave properties of matter. It basically describes the wave nature of the electron.

Louis de Broglie in his thesis suggested that any moving particle, whether microscopic or macroscopic will be associated with a wave character. It was called ‘Matter Waves’. He further proposed a relation between the velocity and momentum of a particle with the wavelength, if the particle had to behave as a wave.

--->This means that photons, electrons possess both momentum as well as wavelength .

De Broglie, gave the following relation between wavelength (λ) and momentum (p) of a material particle.

λ = h/mv

= h/p

Here λ is wavelength , p is the momentum

Important Note-

1. de Broglie's prediction was confirmed experimentally when it was found that an electron beam undergoes diffraction, a phenomenon characteristic of waves.
2. Every object in motion has a wave character. The wavelengths associated with ordinary objects are so short (because of their large masses) that their wave properties cannot be detected.