**Wave –Particle Duality, Dual Nature of Electron:**

Bohr’s and
Sommerfield proposed that angular momentum in closed orbit is “Quantized” but
they could not derived mathematically.

Derivation of
this fact was made by Loius de Broglie in 1924 assuming the dual nature of electron
i.e. particle as well as wave nature of electron. This proposal gave birth of a
new theory known as __“ Wave Mechanical theory of matter”__

**Loius de
Broglie** was suggested that a particle in motion also behaves like a wave. The wave
length associated with the moving particle is given by the following equation,

Where

*h* =
Planck’s constant,

*m* =
mass of the moving particle,

*v *= velocity
of the particle.

This above equation
(1) is known as **de Broglie equation.**
De Broglie equation was derived on the basis of Einstein’s equation, *E = mc ^{2}* and Planck’s equation

*E = hv.*From both of these relation,

Where *p* is the momentum of the moving particle.

**(1)** *The wave length associated with a particle in motion is inversely
proportional to its momentum.*

**(2)** *As the mass of the moving particle increases, the momentum also
increases; the wave length of the matter wave (associated
with the matter in motion) decreases.*

**(3)** *For the particles of finite size, i.e., having appreciable mass, the
momentum is very high, is very small and it
can be said that macroscopic bodies in motion do not possess matter waves.*

**(4)** *When different particles move with the same velocity, the wave length
of the matter wave is inversely proportional to the mass of the particle.*

**Related Questions:**

### (1) Show that circumference of the Bohr orbit for hydrogen atom is an integral multiple of the de Broglie wavelength associated with the electron revolving about that orbit.

**Solved Questions:**

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