de Brogli wavelength

• Interference, diffraction and polarization establishes the wave nature of light and can be explained on the basis of wave theory
• photoelectric effect and compton effect are explained on the basis of quantum theory of light which establishes that quanta behaves like corpescules.
• Louis de Brogli proposed that idea of dual nature i.e., wave particle duality should be extanded to all micro particlesi.e., wave and corpescular nature should be associated with each particle.
• According to him material particles might have dual nature same as that of light.
• He suggested that moving particle whatever is its nature , has wave properties associated with it.
• Wavelength λ associated with any particle of momentum p is given by
  λ=h/p=h/mv
  where h is the Plank's constant.
• From Plank's theory of radiation , energy of photon is given by
  E=hν = hc/λ
  therefore λ=hc/E
  From special theory of relativity
  E=mc²
  mass of photon is m=hν/c²
  momentum of photon is
  mc=p=hν/c = h/λ
  therefore p=h/λ
  or, λ=h/p
• Thus what is true for energy packet (photon) is also true for material particle. Thus, for particle of mass m moving with velocity v , we have p=mv and de Brogli wavelength associated with it is
  λ=h/mv
  where m is the relativistic mass of the particle
• If m or v is large the de Brogli wavelength associated with a material particle would be small.
• The de Brogli wave associated with a material particle or photon of any charge associated with it can also be calculated. Thus we know that
  K=mv²/2
  therefore mv=√(2mK)
  λ=h/√(2mK)
• If a charged particle carrying a charge q is accelerated through a potential difference V volts, then kinetic energy K=qV
  Therefore be Brogli wavelength of charge particle for charge q and accelerated through a potential difference of V volts is given by
  λ=h/√(2mqV)