Question

What is De Broglie's law for small particles like electrons moving at high speeds ?

How is their wavelength , energy calculated ?
please give formulas.

Answer 1

De- Broglie suggested that like electromagnetic radiation, matter (both charged and uncharged ) also possesses dual nature. according to him, like the photon has also momentum and wavelength.
     in a better way, we can say that any subatomic particle or comparable size of it has dual nature, the particle as well as wave.

this can be explained by combining of two theories,
        i) plank's quantum theory [ E = hυ ]
        ii) Einstien's energy equation. [ E = mc² ]

E = mc² = hυ 
   but we know, υ = c/λ  
here, υ is frequency 
         λ is the wavelength
        m is mass of the subatomic particle or comparable size of it
        h is plank's constant
  mc² = hc/λ
    λ = h/mc

hence, the formula of wavelength = h/mc
here, c is the speed of the particle.
    we know,
               momentum = mass * velocity of particle 
  hence, we can say that,
           wavelength = h/p 
we know, kinetic energy (K.E) = p²/2m
      $so, wavelength = \frac{h}{\sqrt{2m(K.E)}}$
 
* for a charged particle, 1/2mv² = qv  = p²/2m [ where q is charge and v is potential difference ]
      $P = \sqrt{2qVm}$
hence, 
   wavelength = h/√(2qVm)

* for  gaseous molecule,
           λ ( wavelength ) = h/mVrms 
here, Vrms is the rms speed of the gaseous molecule 
we also know,
         Vrms = √(3kT/m)
where k is Boltzmann's constant and T is the temperature.
then, wavelength = h/√(3mkT)

[ here i gave formulae of wavelength, you can calculate energy by using formula, E = hc/λ, after finding wavelength. ]


      

Answer 2

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ANSWER @@@

According to de Broglie’s hypothesis, a moving material particle sometimes acts as a wave and sometimes as a particle; or a wave is associated with moving material particle, which controls the particle in every respect. The wave associated with moving particle is called matter wave or de Broglie wave, λ = h/(mv)

Where, m and v are the mass and velocity of the particle and h is Planck’s constant

According to Planck’s quantum theory, the energy of a photon of a radiation of frequency ν and wavelength λ is

E = hν

According to Einstein’s mass-energy relation,

E = mc2

From (i) and (ii), we obtain

hν = mc2

=> m = hv/c2

Since each photon moves with the same velocity c, the momentum of photon,

p = Mass × Velocity

So, p = (hv/c2)× c = hv/c = h/λ

That is,

λ = h/p

This equation is equally applicable to both the photons of radiation and other material particles.


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