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# Wave Particle Duality

Visible light or commonly called as light is a part of electromagnetic radiation that is visible through the human eye. It ranges from 380 nano meters to 740 nm wavelength located in between the invisible infrared and the ultraviolet region. The nature of light was a mystery from a long time.

Initially light considered as waves washing around like water waves as well as considered as particles shooting around like tiny bullets. How anyone can decide the nature of light at any instant that was a big puzzle for scientist. Until the 20th century, most of the scientist considered light as either particle or wave.

Many scientists have compared light with waves to explain the phenomenon of reflection, refraction, diffraction etc. However, in 1905 Albert Einstein suggested the dual nature and character of light as matter and radiation. Only the particle nature of light and many by the wave nature of light could not explain many experiments.

Therefore, Albert combined these two different natures of light into one in order to understand all experiments. Light is a kind of radiation and finally it was concluded that all radiations behaves like waves as well as like matter.

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## Particle Wave Duality

The matter waves are distinctly different from electromagnetic waves. The speed of these waves is not same as that of light. These waves cannot radiate in the empty space. They are certainly not emitted by the parities under consideration, they are simply associated with it.

In 1932, the Francis physicist Luis De Broglie purposed the duel nature of light. Some experiments like interference and diffraction proves the wave nature of light. At the same time the particle nature light spots appearing singly in the fluorescent detector as they arrived.

While the photoelectric effect proves the particle nature of light and the Davisson Germer experiment showed the wave-nature of matter. Hence two sided nature of the light is called as the Wave-Particle Duality.

Initially there were some observations like diffraction and interference of light radiation can be explained if light is assumed electromagnetic radiation like waves, but these electromagnetic radiations cause photoelectric effect, which can only be explained based on particle (matter) theory of light.

In order to explain this entire phenomenon, Louis De Broglie explained that every particle or material moving with certain velocity can be considered to have a wave associated with it, for example electron microscope works on the principle of wave nature of electrons. Further it has been said that wavelength is associated with a moving particle (matter) is inversely proportional to the momentum, which tells us the fact that as kinetic energy of a particle (matter) increases the frequency of wave also increases and hence this concludes that phenomenon of duality in nature of light.

## Wave Particle Duality of Matter

The wave particle duality of light was a big mystery for scientist. The light produce energy which traverses through space just like as the ripples spreading across the surface of a still pond after being disturbed by a dropped rock, it proves the wave like nature of light.

While light can be considered as a steady stream of particles just like tiny droplets of water sprayed from a garden hose nozzle. There are various experiments which can be proves the wave nature of light. At the same time some experiments favors the particle nature.

## Wave Particle Duality of Light

The refraction of light based on the concept of the wave nature and proves that the velocity of light in any substance is inversely proportion to its refractive index. When a beam of light passes through from one medium to another one which having different refractive indices, the beam changes its direction and the phenomenon is termed as refraction.

At the same time refraction can also explained by using corpuscular theory, which proves the particle nature of light.

The refraction of light is just like the reflection of light through a smooth, specular surface, just like a mirror.

The reflection of light far more favor the particle nature of light, in which particles get bounces away after striking on a smooth in opposite direction. Since the light particles are very tiny and in a huge number present in a propagating light beam, hence after impacting the mirror, the particles bounce from different points and the light beam is reversed upon reflection to produce a reversed image.

No doubt the reflection of light proves the wave as well as particle nature of light, at the same time the particle nature also proves the scattering the light from the rough surface. Another experiment which proves the duality of light is diffraction pattern of light. When a light beam crossed through a narrow slit , it spreads and becomes wider.

The diffraction through the edge of an object proves the wave nature of light.
The English physicist named Thomas Young proves the wave nature of light by using an experiment in which a screen containing a single, narrow slit is used to produce a coherent light beam which contains waves that propagate in phase from ordinary sunlight.

At the same time the photoelectric effect proves the particles nature of light .

## De Broglie Wavelength Equation

The dual nature of light get certified by French scientist Louis-Victor de Broglie who proposed that all matter and radiation have properties which are resemble to both a particle and a wave.

De Broglie used the Max Planck's lead for the wave nature of light and related it to the Einstein’s famous formula relating mass and energy to include Planck's constant.

E = mc2 = h$\nu$
Where

E = Energy of a particle,
m = mass of particle
c = Speed of light
h = Planck's constant
$\nu$ = Frequency of light

The relation between the frequency of a wave to the energy and mass of a particle, became fundamental in the development of a new field that used to explain both the wave-like and particle-like nature of light. Hence, De Broglie equation proves that at times light behaves as a particle, and at other times as a wave.

This duality of light can be used to describe almost all of the known properties of light like reflection, interference, refraction , photoelectric effect and diffraction.

The wavelength of light can be estimated by using the De Broglie equation.

E = mc2 = h$\nu$
E = mc2 = $\frac{hc}{\lambda}$
Hence, mc = $\frac{h}{\lambda}$
Or
p = $\frac{h}{\lambda}$
Where;
$\lambda$ = wavelength of light
p = momentum of light

## De Broglie Wavelength Formula

Since De Broglie proved the dual nature of light, so the De Broglie equation is able to explain different properties of wave as well as the particles. The de Broglie wave length equations give a relation between the wavelength λ to the momentum p, and frequency "υ" to the total energy E of a particle

$\lambda$ = $\frac{h}{p}$
$\nu$ = $\frac{E}{h}$

The de Broglie wave length equations can be equivalently written as

p = ħκ
E = ħω

Where

ħ = h/2π = Reduced Planck's constant or Dirac's constant
κ = $2\pi / \lambda$ (angular wave number)
$\omega$ = $2\pi \nu$ (angular frequency)

These relations are also known as Planck-Einstein relation because they have both concepts of Planck as well as Einstein.

By using the relativistic mass formula from special relativity

Since
m = $\Upsilon m_{0}$
Hence

$\lambda = \frac{h}{\gamma m_0 v}$ = $\lambda = \frac{h}{m_0 v}\sqrt{1 - \frac{v^2}{c^2}}$

f = $\frac{\gamma m_0 c^2}{h}$ = $\frac{m_0 c^2}{h\sqrt{1 - \frac{v^2}{c^2}}}$

Where

m0 = particle's rest mass
v = particle's velocity,
$\Upsilon$ = Lorentz factor
c = speed of light in vacuum

## De Broglie Wavelength of an Electron

If light can be considered as a particle or as a wave, is the same possible for an electron?

The De Broglie equation was hold well for light weight objects; it means it must be applicable on electrons also. The Bohr model explained the wave nature of electron and considered as standing waves.

At the same time Davisson & Germer and George Thomson observed the diffraction of electrons. In an atom electron is moving in a circular orbit of radius r around a nucleus.
Just like light, electron can be considered as particle as well as wave in an atom. If the wave is to remain continually in phase, the circumference of the circular orbit must be integral multiples of wavelength Î» , that is

2$\pi$ r = n$\lambda$

2
$\pi$r = $\frac{nh}{mc}$

Hence,

Angular momentum = mcr = $\frac{nh}{2\pi}$

This is the last Bohr;s postulate, it states if the circumference is bigger or smaller than the value of $\frac{nh}{2\pi}$, the wave will no longer remain in phase.

Hence, according to De Broglie concept

Electron revolves around the nucleus in certain circular orbit in the form of wave. If the radius of these orbit is r and circumference is 2$\pi$r , than the relation between wavelength and radius of orbit is

2$\pi$r = $\frac{nh}{mc}$
2$\pi$r = $\frac{nh}{p}$
(where p = angular momentum = mv)

Since,
$\lambda$ = $\frac{h}{p}$

So,
2$\pi$r = n$\lambda$

Here $\lambda$ is the wavelength of electron and this equation termed as De Broglie wave length equation for an electron. Electron can move only in those orbits where the angular momentum of electron is multiple integral of $\frac{h}{2\pi}$. So

mcr = $\frac{nh}{2\pi}$

Here, n is the whole number, which gives the number of orbit associated with the wave length. It proves that the wave concept of De Broglie was same as the Bohr's hypothesis.

## De Broglie Wavelength of Electrons

The De Broglie equation is applicable for electron also. Electron possesses kinetic and potential energy sum of which is equals to the total energy of electron. According to Einstein concept energy is related to mass and velocity of particle by mc2. Hence,
$E = mc^{2} = KE + m_{0}c^{2}$

Or,

$E = \sqrt{p^{2}c^{2}+m_{0}^{2}c^{4}}$

The momentum of particle as zero rest mass p = $\frac{E}{c}$

For a electron; E = hc… = $\frac{hc}{\lambda}$

So p = $\frac{hc}{c\lambda}$ = $\frac{h}{\lambda}$

This momentum-wavelength relationship of electron can be related to De Broglie wavelength equation for electron.

$\lambda$ = $\frac{h}{p}$

The electron diffraction supports the wave nature of electrons as suggested in the De Broglie hypothesis and Bohr's energy levels of electron in an atom favor the wave nature.

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