QUANTUM MECHANICS : WAVE NATURE, DE-BROGLIE WAVELENGTH, SOLVED PROBLEMS, MOCK TEST

To Be Discussed --

• Wave Nature of Particle
• de-Broglie hypothesis
• de-Broglie wavelength in terms of  - 
• Velocity
• Energy
• Temperature
• Potential difference

Wave Nature of Particle --

The successful explanation of the photoelectric and Compton effect established that EM radiation travels not in the form of a continuous stream of energy but in the form of a tiny packet of energy. These packets of energy called photons that behaves exactly like a material particle. On the other hand, a phenomenon like interference, diffraction, or X-rays could not be explained unless EM radiation is assumed to possess a wave character. When radiation interacts with matter it exhibits its particle character whereas when radiation interacts with radiation, it exhibits wave character.

When --

In 1925, a new concept was introduced by de-Broglie and then by Schrodinger. De-Broglie put forward his hypothesis of matter-wave. Schrodinger presented an equation that could satisfy these matter waves. De-Broglie extended the wave-particle dualism from radiation to all fundamental entities of Physics. The electrons, protons, atoms, and molecules, When in motion, should have some type of wave motion associated with them.

De-Broglie Hypothesis --

De-Broglie was led to this hypothesis from consideration based upon STR and Quantum theory.

The expression for de-Broglie wavelength for photon --

The energy of a photon can be written as :

If a photon possesses some mass, its energy according to the theory of relativity is :

Where m is the mass of the photon.

From equations (1) and (2), we get

where is the momentum of the photon.

De-Broglie wavelength in term of velocity - 

For a material particle - 

If a material particle of mass m moving with velocity v, then the momentum of a material particle - 

the wavelength associated with the material particle, 

This wavelength is called the de-Broglie wavelength.

De-Broglie wavelength in term of Energy - 

The kinetic energy is given by - 

De-Broglie wavelength in terms of energy will be 

De-Broglie wavelength in term of Temperature - 

From the Kinetic energy of gases, the average K.E of the material particle is given by -

As we know, 

De-Broglie wavelength in term of Potential Difference - 

For Non-Relativistic Particle - 

Let an electron having charge e and mass m accelerated through a potential V volt and attains a velocity v,

Also, K.E of the electron is given by

From (1) and (2), we get

The de-Broglie wavelength of electron -

For Relativistic Particle - 

In the Relativistic case, the relation between Energy E and momentum p is given by 

Also, we have, 

From (1), 

Remember all the formulae of wavelength in terms of velocity, Energy, Temperature, and potential difference - 

For Lecture-1 - Click Here

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