Question

What do you mean by the dual nature of electrons?

Answer 1

Dual nature of an electron can be understood by that like light, electrons too have a dual nature or property. When an electron moves it can be considered moving as a particle and also as a wave both at the same time.

Complete step by step answer:
French physicist named Louis de-Broglie in 1924 said electrons have dual nature, as both particle and wave nature can be seen in an electron.
It stated that when small particles like electrons can pass wave nature when they are in motion, it was a new theory.
Now, de-Broglie introduced the wavelength
$\lambda = h/mv …………………………………………….. (1)$;
Where, $m$ is mass of particle, $v$is velocity and $h$ is planck's constant.
Derivation of de-Broglie’s equation:
According to planck's equation
$E = hf = hc/\lambda - (2)$
Where, $E$is energy, $f$ is frequency and $c$ is speed of light.
Now, according to Einstein’s mass energy relationship
$hc/\lambda = m{c^2}$
$hc = m{c^2}\lambda$
Change the above equation with respect to $\lambda$
$\lambda = hc/m{c^2}$
Cancel $c$ from numerator and denominator
$\lambda = h/mc - (4)$
This is the same as de-Broglie’s equation.
Also, Davison and Germer, by performing diffraction effects on an electron beam, verified the dual nature of the electron. Consider an electron accelerated at potential $V$, than its kinetic energy is
$1/2(m{v^2}) = eV$
$m{V^2} = 2eV$
$mV = 2e$
Now, by multiplying $mV$ on both the sides
${(mV)^2} = 2eVm$
taking$\sqrt {}$on both the sides
$mV = \sqrt {2eVm} - (5)$
Substituting equation $(5)$ and $(1)$ we get
$\lambda = h/\sqrt {2eVm} - (6)$
Now, Bohr’s theory can also be related to de-Broglie’s equation, therefore wavelength of an electron is calculated by Bohr’s orbit and can be related to its circumference that is $2\pi r$, and by multiplying with a whole number $n$.
$2\pi r = n\lambda$
Now, from de-Broglie’s equation, by applying
$2\pi r = nh/mV$
Then, $h/mV = 2\pi r/n$
Now, by interchanging quantity we get
$mVr = nh/2\pi - (7)$
Which is considered as Bohr’s equation.
Now, de-Broglie is not significant in daily life because it is considered for micro-particle and not on macro-particle.

Note: Dual nature of electrons can be related to dual nature of light. Light behaves as both particles and waves. It acts as an electromagnetic wave when in transmission in space, and interacting with electrons acts as if a particle i.e. photon.