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Derive de Brogue equation and give its significance.

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1. Louis de Brogue extended the concept of dual nature of light to all forms of matter. To quantify this relation, he derived an equation for the wavelength of a matter wave.

2. He combined the following two equations of energy of which one represents wave character (hu) and the other represents the particle nature (mc2).

Planck’s quantum hypothesis: 

E = hv ….(1) 

Einsteins mass – energy relationship: 

E = mc2 …(2) 

From (1) and (2) 

hv = mc2

hc/λ = mc2

∴ λ = h/mc…(3) 

The equation (3) represents the wavelength of photons whose momentum is given by mc. (Photons have zero rest mass).

3. For a particle of matter with mass m and moving with a velocity y, the equation (3) can be written as 

λ = h/mc  …(4)

4. This is valid only when the particle travels at speed much less than the speed of Light.

5. This equation implies that a moving particle can be considered as a wave and a wave can exhibit the properties of a particle (i.e momentum).

6. Significance of de Brogue equation: For a particle with high linear momentum, the wavelength will be too small and cannot be observed. For a microscopic particle such as an electron, the mass is 9.1 x 10-31 kg. 

Hence the wavelength is much larger than the size of atom and it becomes significant.

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