DE-BROGLIEWAVES

DE-BROGLIEWAVES | λ =h/mv | de-Broglie relation | DE-BROGLIE WAVELENGTH OF ELECTRON
de-Broglie relation


Radiation behave both as wave and particle. In 1924, Louis de-Broglie put forward a bold hypothesis that matter should also posses dual nature.

The following observations led him to the duality hypothesis for matter :
  • The whole energy in this universe is in the form of matter, and electromagnetic radiation.               
  • The nature loves symmetry. As the radiation has got that nature, matter should also posses dual nature.

Thus according to de-Broglie, a wave is associated with every moving particle. These wave are called de- Broglie waves or matter waves.

According to the quantum theory of radiation, energy of a photon is given by
               
E=hง            .......(1)

Further, the energy of a relativistic particle is given by

   



Since photon is a particle of zero rest mass, setting m0=0 in the above equation,
we have
           
  E=pc          .......(2)

From the (1) and (2), we have
              
 pc=hv

or                        p=hv/c = hv/vλ       (∵c=vλ)

or      p=h/λ

Therefore, the wavelength of the photon is given by
         
    λ=h/p          ........(3)

de- Broglie asserted that the equation (3) is completely a general formula and applies to photon as well as other moving moving particle. The momentum of a particle of mass m moving with velocity v is mv. Hence, de Broglie wavelength is given by
           
   λ =h/mv      ........(4)

This is called de-Broglie relation.
It connects the momentum, which is characteristic of the particle, with wavelength, which is characteristic of the wave. From this equation, the following conclusions can be drown:

  1. Lighter the particle, greater is its de-Broglie wavelength.
  2. The faster the particle moves, the smaller is its de- Broglie wavelength.
  3. The de-Broglie wavelength of a particle is independent of the charge or nature of the particle.
  4. The matter wave are not electromagnetic in nature. The electromagnetic waves are produced only by the accelerated charged particle.     

DE-BROGLIE WAVELENGTH OF ELECTRON

Consider that an electron of mass m and charge e is accelerated through a potential difference V. If E is the energy acquired by the particle, then
         
 E=eV

if v is the velocity of electron, then
          
E=1/2mv.v

     v=√(2E/m)

Now, de-Broglie wavelength of electron is given by
        
  λ=h/mv

or     λ=h/√{m(2E/m)}

or     λ=h/√(2mE)

Substituting the value of E (=eV), we get

        λ=h/√(2meV)

Setting m=9.1х10^-31 kg; e=1.6х10^-19 C and h=6.62х10^-34 Js, we get
      
 λ=[12.27/√V]х10^-10 m

or    λ=12.27/√V   À

For example, the de-Broglie wavelength of electron, when accelerated through a potential difference of 100 volt, will be
  
  λ=12.27/√100   ~1.227  À

Thus, the wavelength of de-Broglie waves associated with 100 eV electrons is of the order of the wavelength of  X-rays, but the two are quite different from each other in nature.