Derivation of de-Broglie Relationship

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The relationship may be derived by combining the mass-energy relationship proposed by Max Planck and Einstein. According to Planck, photon of light having energy E is associated with a wave of frequency n as:

E = hn .....(1)

According to Einstein, mass and energy are related as:

E = mc² .....(2)

where c is the velocity of light.

Combining the above two relations in eqs. (1) and (2), it is seen:

hn = mc²

The equation is valid for a photon. De-Broglie suggested that on substituting the mass of the particle m and its velocity v in place of velocity of light c, the equation can also be applied to material particles.

Thus, the wavelength of material particles, l is:

This equation is known as de-Broglie's equation.

where p stands for the momentum (mv) of the particle. Since h is constant, l 1/Momentum.

It means that the wavelength of a particle in motion is inversely proportional to its momentum.

The concept of dual nature of matter is significant only for microscopic bodies. For larger bodies the wavelengths of the associated waves are very small and cannot be measured by any of the available methods. Therefore, practically these larger bodies are said to have no wavelengths.