de-Broglie equation

QUESTION: How the idea of dual nature of matter was deduced from the dual nature of light?
Answer: The idea of dual nature of matter particles like electron, proton, neutron was deduced by de-Broglie. He derived an equation, known as de-Broglie equation, which shows the relationship between wave length and momentum of electron. The de-Broglie equation is derived by comparing the Planck’s equation of energy and Einstein equation of energy. For example,
According to Planck’s equation,
E = hν ……….. (1)
Where E = energy of photon, h= Planck’s constant, ν = frequency of photon According to Einstein’s equation,
E =mc^2 ……….. (2)
Where m= mass of the particle converted into energy E, c = velocity of light By comparing equation (1) and equation(2);
hν=mc^2
or

hν/c = mc
or

h/(c⁄ν) = mc
or

h/mc = c⁄ν ……….. (3)

Since, c⁄ν = λ
So equation (3) can be written as
h/mc = λ ……….. (3)

where λ is wave length associated with particle of mass m moving with velocity of light (c). For electron of mass m and velocity v and equation (3) can be written as

λ = h/mv

This is known as de–Broglie equation. Here wave length of electron is inversely proportional to its momentum.

Question: According to de-Broglie’s idea only microscopic particles have waves. Comment upon it.
Answer: According to de-Broglie’s equation, wave length of electron is inversely proportional to its momentum.

i.e. λ = h/mv Now consider a macroscopic body i.e. a stone of mass of 0.001 Kilogram moving with a velocity of 10 m/s. Its wave length will be:

λ = h/mv = (6.626 x10^-34 Js )/(0.001 kg x 10m/s)

= 6.66 x 10^-30 m

This wave length is so small that it cannot be measured by any possible method. On the other hand, in case of microscopic particle i.e. electron moving with velocity 2.18 x 106 ms-1. Its wave length will be:

λ = (6.626 x10^-34 Js )/(9.11 x10^(-31 ) kg x 2.18 x10^(6 ) m/s)

= 0.33 nm This is comparable to wave length of x-ray which is measurable. Due to this reason, only microscopic particles are thought to behave as waves.

Question: How does the duel nature of matter was verified?

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