Explain the quantization of angular momentum to behave an electron as wave in a atom.

Explanation

To introduce the atomic model Bohr in his second hypothesis said that the angular momentum of the electron orbiting around the nucleus is quantised.
Louis de Broglie explain this in 1923.
According to de Broglie.s hypothesis material particles have a wave nature. Davisson and Germer experimentally verified the wave nature.
Bohr proposed that the electron in a circular orbit should be as matter wave.
The waves travelling on a string, particle waves too can lead to standing waves under resonant condition.
When a string is plucked, a vast number of waves (wavelengths) are excited only those waves survive which have nodes at the ends and form the standing waves in the string. It means that in a string, standing waves are formed when the total distance travelled by a wave down the string and back is one wavelength, two wavelengths or any integral number of wavelength.
Waves with other wavelength interfere with themselves upon reflection and their amplitudes quickly drop to zero. Under this position the electron cannot remain in such orbit.
For an electron moving in `n^(th)` circular orbit of radius `r_(n)` the total distance is the circumference of the orbit.
`:.` Circumference of orbit = n wavelength
`:. 2pi r_(n)= n lambda...(1) "where" n=1,2,3,...`

Figure shows the standing wave located at the circular orbit for the four wavelength.
Here n = 4
The relation of de Broglie wavelength and momentum
`lambda=(h)/(p)=(h)/(mv_(n))....2 [ :. p=mv_(n)]`
From equation (1) and (2),
`2pi r_(n)=n((h)/(mv_(n)))` which is the second hypothesis of Bohr atomic model. Momentum is always multiple integer of `(h)/(2pi)`. Hence, de Broglie explained the 211 quantization of the angular momentum of the orbiting electrons. The quantised electron orbits and energy states are due to the wave nature of the electron and only resonant standing waves can persist.