de Broglie hypothesis

• According to de-Broglie, a wave is always associated with every moving particle. This wave is known as de-Broglie wave or matter wave.
• It means a matter wave have particle nature as well as wave nature i.e., dual nature.
• From quantum theory of radiation, energy of photon is
  E = hν,   where ν = frequency of incident photon
• From Einstein’s theory of relativity E = √(m₀²c⁴ + p²c²)
• If m₀ = 0, then E = pc,  where p = momentum and c = velocity of light
• Now E = hν and E = pc

∴     hν = pc     or     p = hν/c

∵      c = νλ      or     λ = c/ν

∴      p = h/λ     or     λ = h/p

This is known as de-Broglie wavelength

• This wavelength is always associated with a photon
• Since momentum is the characteristic of particles, and the wavelength is characteristic of wave.
• Hence a wave is always associated with a particle.

Conclusion

• The wavelength of a particle is independent of the charge or the nature of the particle.
• The E.M. waves are produced only due to the charged particle. Hence the matter waves are not E.M. in nature.

∵     λ = h/p    and   p = mv

∴     λ ∝ 1/v    and   λ ∝ 1/m,      v = velocity of particle

• Thus
• Faster is the particle, smaller is its wavelength.
• Heavier is the particle, smaller is its wavelength.
• If v is comparable to c, then m = m₀/√(1 − v²/c²)

For any massive particle    λ = h/√2mE            where E is kinetic energy

• If a matter particle at absolute temperature T is in thermal equilibrium, then

           E = 3/2 kT      where   k = Boltzmann’s constant and T is absolute temperature

∵     λ = h/√2mE

∴   λ = h/√3mkT

• According to Bohr’s postulate, an electron can revolve around the nucleus only in those orbits whose angular momentum is an integral multiple of h/2π.

∴     mvr = nh/2π

• Since a wave is associated with a moving particle, so 2πr = nλ
• Here 2πr is the circumference of the stable orbit
• From de-Broglie hypothesis,  λ = h/mv

Now 2πr = nλ  and  λ = h/mv

∴     2πr = nh/mv

or   mvr = nh/2π

• Which is the Bohr’s hypothesis