Question

What are de-Broglie waves? Establish the de-Broglie wavelength equation.

Answer 1

Some events of light are only described on the basis of Quantum theory as photoelectric effect, Compton Effect and Raman Effect etc. not by the wave theory. And these events are proof of the particle nature of light. Due to this light is considered as dual nature behaviour.

Complete step by step answer:
There were a lot of problems created among scientists by the dual nature of light. In $1924$ Louis de-Broglie (a French Scientist) presented the logic that, “A moving particle is always associated with a wave. Hence Photon also consists of the properties of waves. This concept of de-Broglie is also known as the dual nature of particles.
He also states that any moving particle i.e. electrons, photons are always associated with a wave. And these waves are called matter waves or de-Broglie waves. And the wavelengths of these waves are known as de-Broglie wavelengths.
Expression for de-Broglie wavelength
According to the Quantum theory of light, the energy of a Photon of frequency $\upsilon$ is –
$E = h\upsilon$ ……….(i)
Where $h$ is Planck’s constant.
But according to the special theory of relativity given by Einstein, the energy of Photon of mass $m$is –
$E = m{c^2}$ ……...(ii)
Where $c$ is the speed of light.
Comparing eq. (i) and (ii)
$\Rightarrow h\upsilon = m{c^2}$
But we know that $\upsilon = c/\lambda$, Where $\lambda$ is the wavelength. So, substituting this value in above equation, we get
$\Rightarrow \dfrac{{hc}}{\lambda } = m{c^2}$
$\Rightarrow \dfrac{h}{\lambda } = mc$
$\Rightarrow \dfrac{h}{{mc}} = \lambda$
But we know that $mc = p$
$\Rightarrow \lambda = \dfrac{h}{p}$
This is the associated wavelength with Photon. And $p$ is the momentum of Photon.

Additional information:
we can also find the de-Broglie wavelength associated with electrons by their kinetic energy i.e.
$\lambda = \dfrac{h}{{\sqrt {2mK} }}$
Here $K$ is the kinetic energy of the electrons.

Note:
The experimental verification of de-Broglie wavelength is presented by Davisson and Germer in $1927$. And the wavelength of an electron at $54$ volt is $1.67{A^ \circ }$. The de-Broglie wavelength does not depend on the charge of the particle.