(a) State Bohr’s postulate to define stable- orbits in hydrogen atom. How does de Broglie’s hypothesis explain the stability of these orbits ?

(b) A hydrogen atom initially in the ground state absorbs a photon which excites it to then n = 4 level. Estimate the frequency of the photon.

(a) Bohr’s postulate for stable orbits in hydrogen atom : An electron can revolve only in those
circular orbits in which its angular momentum is an integral multiple of h/2π, where h is Planck’s constant.
If n is the principal quantum number of orbit, then an electron can revolve only in certain orbits or definite radii. These are called stable orbits.

De Broglie explanation of stability of orbits:
According to de Broglie, orbiting electron around the nucleus is associated with a stationary
wave. Electron wave is a circular standing wave. Since destructive interference will occur if a
standing wave does not close upon itself, only those de Broglie waves exist for which the circumference of. circular orbit contains a whole number of wavelengths i.e., for orbit circumference of nth orbit as 2nπrn

$\begin{array}{l}2{\mathrm{\pi r}}_{\mathrm{n}}=\mathrm{n\lambda }\\ 2{\mathrm{\pi r}}_{\mathrm{n}}=\mathrm{n}\left(\frac{\mathrm{h}}{\mathrm{mv}}\right)\\ {\mathrm{mvr}}_{\mathrm{n}}=\mathrm{n}\left(\frac{\mathrm{h}}{2\mathrm{\pi }}\right)\end{array}$
$\mathrm{which} \mathrm{is} \mathrm{quantum} \mathrm{condition} \mathrm{proposed} \mathrm{by} \mathrm{Bohr}.$

$\left(\mathrm{b}\right) \mathrm{In} \mathrm{ground} \mathrm{state}, \mathrm{n}=1$
In excited state, n = 4
$\frac{1}{\mathrm{\lambda }}=\mathrm{R}\left[\frac{1}{(1{)}^2}-\frac{1}{(4{)}^2}\right]$ Where R is 
                                   Rydberg constant

$\begin{array}{l}\frac{1}{\mathrm{\lambda }}=1.097\times {10}^7\times \frac{15}{16}=10284375\\ =1.028\times {10}^7{\mathrm{m}}^{-1}\end{array}$
Frequency, $\begin{array}{l}\mathrm{V}=\frac{\mathrm{c}}{\mathrm{\lambda }}=3\times {10}^8\times 1.028\times {10}^7\\ =3.09\times {10}^{15}\mathrm{Hz}\end{array}$