Show that the first few frequencies of light that are emitted when electrons fall to the nth level from levels higher than n are approximate harmonics (i.e. in the ratio 1 : 2: 3…) when n >>1.
When an electron falls from (n+n′)th to nth energy level, the frequency of radiations in spectrum H - atom like atoms is given as ν=CRZ2[(n+n′)1−n21] Here n >> n' n′=1,2,3,.... R = Rydberg's constant ν=CRZ2⎣⎢⎢⎢⎢⎡n2[1+n2n′]21−n21⎦⎥⎥⎥⎥⎤=CRZ2[n21[1+nn′]−2−n21] Neglecting the higher terms as n >> n' =CRZ2[n21[1−n2n′]−n21]=CRZ2[n21−n32n′−n21] =n3−CRZ22n′=[n32CRZ2]n′ So the first few frequencies of light that is emitted when electrons fall from (n+n′) to nth energy level are in the ratio of n′=1:2:3,.... when n>>1.
