State Bohr's quantization condition for defining stationary orbit… - Vidyadip

Question

State Bohr's quantization condition for defining stationary orbits. How does de-Broglie hypothesis explain stationary orbits ?

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Answer

Bohr's Quantization condition : It states that only those orbits are permitted in which the angular momentum of the electron about the nucleus is an integral multiple of $h / 2 \pi$, where $h$ is Planck's constant.
Explanation of stationary orbits by de-Broglie hypothesis: According to de-Broglie, an electron of mass $m$ moving with speed $\lambda$ would have a wavelength a given by $\lambda=h / m v$.
Now according to Bohr's postulate
$m v r_n=\frac{n h}{2 \pi} \Rightarrow 2 \pi r_n=\frac{n h}{m v}$

But $h / m v=\lambda$ is the de-Broglie wavelength of the electron, therefore, the above equation becomes $2 \pi r_n=n \lambda$ , where $2 \pi r_n$ , is the circumference of the permitted orbit. Thus only waves that persist are those for which the circumference of the circular orbit contains a whole number of wavelengths.

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