Q.(a) State Bohr's postulate of quantisation of angular momentum of the orbiting electron in hydrogen atom.

Bohr postulated that angular momentum is quantised in integer multiples of h/2π; combining this with the Coulomb-force-provides-centripetal-force condition gives the orbit radius formula.

1. Bohr's quantisation postulate: Bohr postulated that an electron can revolve only in certain discrete, stable, non-radiating orbits around the nucleus for which its orbital angular momentum is an integral multiple of $h/2\pi$: $mvr = \dfrac{nh}{2\pi}, \quad n = 1, 2, 3,\ldots$ where $m$ is the electron's mass, $v$ its speed, $r$ the orbit radius, $h$ Planck's constant, and $n$ the principal quantum number. While in such an orbit, the electron does not radiate energy, despite being accelerated (this was Bohr's ad hoc departure from classical electrodynamics).
2. Derivation of the orbit radius: For an electron of charge $-e$ revolving around a nucleus of charge $+e$ (hydrogen atom) in a circular orbit of radius $r$, the electrostatic (Coulomb) force provides the centripetal force: $\dfrac{1}{4\pi\varepsilon_0}\dfrac{e^2}{r^2} = \dfrac{mv^2}{r} \implies v^2 = \dfrac{e^2}{4\pi\varepsilon_0 mr} \quad \text{...(i)}$ From Bohr's quantisation condition: …