# 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 -