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Waves are things that do not have a definite position we can not specify a definite position to waves they are spread across space, but they have an oscillating field. waves can be of many forms they only differ in what is actually oscillating examples of waves includes sound waves, electromagnetic waves, and water waves. in the case of sound waves the oscillating medium is the air molecules around, this is the reason why in space you can not talk. because there is no air in space. and in electromagnetic waves, the oscillating fields are electric and magnetic fields, and in the case of water waves, the water itself is an oscillating field.

## Properties of waves

Various properties of waves are listed below

Frequency of waves(f): Waves have an essential property called their frequency of oscillation it is the number of oscillations made per unit of time. i.e.

f=1/T

where f is frequency and T is time.

Wavelength(λ): Wavelength is the distance between two consecutive Crest or Trough as shown in the figure: and wavelength is also inversely proportional to the frequency of the wave, i.e.

f∝1/λ

and the product of frequency and wavelength gives the speed of the wave.

v=fλ

Particles are the things that have a definite position (at least in the common world in the quantum world that is not true ), and we can specify a definite position to particles that are the main distinction between particles and waves.

Examples of particles or matter may include any rigid body like a ball or sugar etc. but as the developments in physics were made the particle and waves definitions have changed and why we defined the above quantities to come to this theory of wave matter  ( particle ) duality.

## Properties of Particles (Matter)

Matter as the name suggests should contain mass and occupy space.

• The particles are rigid objects having definite mass and occupy the space.
• Unlike the waves which are distributed, around the space, the particles are localized, and they occupy a definite position.
• The particles have intrinsic energy that is their mass, given by E=mc2
• Every matter is composed of the elementary particles in two groups, the first are fermions and the second are bosons.
• Fermions are mass-providing particles and bosons are the force-carrying particles.
• When the matter particle interacts with another matter particle it transfers both energies as well as momentum to the colliding particle. During the collision the conservation laws both ( momentum conservation and energy conservation are followed).
• Matter also affects the curvature of space-time, by curving the space-time, the more the mass of the object the more it bends the space-time.
• The matter particles are also, affected by the gravitational field.

Example: What is the energy associated with the particle having a mass of 1kg?

Given Mass = 1kg

since we know the mass-energy formula,

E = mc2

E = 1×(3×108)2

E = 9×1016 joule

## De Broglie Hypothesis

At the start of 20 century the developments in physics and other sciences were in the boom quantum mechanics have come into existence and the Maxwells Equations confirmed the wave nature of light, there were long-run arguments over the nature of light some scientist has treated it as a wave and some have considered it as particles.

Because all the phenomena that occur with light couldn’t be explained by any one of the pictures of light some phenomena could only be explained if the light was treated as waves like diffraction, reflection, interference, etc. and some could be explained only with particle nature of light like black body radiation and photoelectric effects.

So they came up with the idea of the dual nature of light and accepted the double nature of light.

Around the same time De Broglie a well-known physicist at that time came up with the idea of matter waves, can matter too can have wave-like properties?

And gave his hypothesis known as De Broglie’s hypothesis and gave this formula for matter waves.

According to De Broglie’s Hypothesis

The wavelength of a particle with mass m is given by

λ=h/p

where p is the momentum of the particle.

or

λ=h/mv

where,

m = mass of the particle

h = is Planck’s constant with value 6.64×10-34 js

v = velocity of the particle

λ = wavelength associated with that particle.

But as we can see if the velocity of the particle is very small then the associated wavelength is negligible .That is why in our everyday life we cannot observe the wavelengths of everyday objects.

## Solved Examples on Dual Nature of Light

Example 1: What is the wavelength of a particle moving with a velocity of 664 m/s with a mass of 1kg?

Solution:

Mass of the object 1kg,

Velocity of object 664 m/s.

λ = ?

we know the wavelength of a particle is given by,

λ = h/mv

then

λ = 6.64×10-34/664

λ =10-36

which is very small and not detectable.

Example 2: What is the frequency of particles having a mass of 2 kg and moving with a velocity of 10 m/s?

Solution:

Mass of the object 2kg,

Velocity of object 10 m/s.

f = ?

we know the wavelength of a particle is given by,

λ = h/mv

then

λ = 0.332×10-34

since we know that fλ = v

so f = 10/3.32×10-34

f = 3.1×1034 s-1

Example 3: What is the velocity of an object having a wavelength of 0.994×10-34 and a mass of 6.64 kg?

Solution:

Mass of the object 6.64kg,

v = ?

we know the wavelength of a particle is given by,

λ = h/mv

then

v = h/ λm

v = 1.1 m/s

Example 4: In order to be detectable wavelength (1 m) what would be the velocity of the particle with a particle having a mass of 1020kg?

Solution:

Mass of the object 10 kg,

λ =1 m

we know the wavelength of a particle is given by,

λ = h/mv

then

v = h/λm

v = 6.64×10-34/1020

v = 6.64×10-14 m/s.

Example 5: What would be the mass of the particle having a wavelength of 1.3 ×10-33m and moving with a velocity of 10 m/s?

Solution:

Mass of the object = ?

λ = 1.3 ×10-33m

We know the wavelength of a particle is given by,

m = h/λv

then

m = (6.64×10-34)/(1.3 ×10-33)10.

m = 1/20.

Example 6: What is the mass of the particle if its rest energy is 6×102 j?

Solution:

Given the rest  energy,   6×102 j

Since we know from mass-energy relation,

E = mc2

m = E/c2

m = (6×102) /(3×108)2

m = 0. 666×10-14kg

## FAQs Wave Particle Duality

Question 1: What is Wave-Particle Duality?

Wave-Particle Duality explains how light behaves both as particles and waves is one of the main concept of quantum mechanics.

Question 2: How can we proves the Particle nature of light?

Particle nature of light is proved by Photoelectric effect which is given by Albert Einstein.

Question 3: How can we proves the Wave nature of light?

Wave nature of light is proved by Reflection, Refraction and by other properties of light.

Question 4: What is the mass of the photon particle having a frequency of  1022 Hz ?

Given the frequency, 1022Hz

Since we know that photon’s energy is

E = hv ……………..(1)

and also the mass-energy relation

E = mc2 …………….(2)

from (1)and (2)

mc2 = hv

m = hv/c2

putting all the values

m = (6.64×10-34) (1022)/(3×108)2

m = 0.73×10-4kg

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