## Dual Nature – Particle and Wave Duality

Light shows many phenomena like diffraction, interference, black body radiation, photoelectric effect etc.

The diffraction and interference can be explained by considering nature of light as a wave nature whereas black body radiation and photoelectric effect can only be explained by considering nature of light as particle nature.

Thus, Light is said to have dual nature.

Louis de Broglie, in 1924 extended the idea of photons to material particles such as electron and he proposed that matter also has a dual character-as wave and as a particle.

#### de Broglie Equation:

The wavelength of the wave associated with any material particle can be calculated as follows:

In case of wave, the energy of photon is given by Planck’s equation i.e.

E = hν

whereas, If a photon is assumed to have particle nature then, according to Einstein Energy equation, Energy of photon will be

E = mc^{2}

Where m = mass of photon and C = Velocity of light.

from the above two equations,

hν = mc^{2}

but we know that ν = c/λ

=> hc/λ = mc^{2}

or λ = h/mc

The above equation is applicable to the material particle if the mass and velocity of the photon is replaced by the mass and velocity of the material particle. Thus for any material particle like electron,

λ = h/mv

or λ = h/p

where p = mv is the linear momentum of the particle.

#### Angular momentum from de Broglie Equation:

According to Bohr’s model of an atom, the electron revolves around the nucleus in circular orbits and as per de Broglie concept, the electron has wave character also.

If the wave associated with the electron is in phase than the circumference of the circular orbit must be equal to an integral multiple of wave length.

So,

2π r = n λ

Where n = 0, 1, 2, 3 … and r is the radius of the orbit.

However, as per de Broglie equation,

λ = h/mv

so, 2π r = nh/mv

on rearranging, we get,

mvr = nh/2π

which is Bohr’s postulate of angular momentum, where ‘n’ is the principal quantum number.

** “ Thus, the number of waves an electron makes in a particular Bohr orbit in one complete revolution is equal to the principal quantum number of the orbit ”.**

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