Obtain an expression for the frequency of radiation emitted when a hydrogen atom de-excites from level n to level (n-1). For large n, show that this frequency equals the classical frequency of revolution of the electron in the orbit.

Explanation

for `n^(th)` orbit ` E_n = ( - 2pi ^@ me ^4 ) /( ( 4 pi epsi_0)^2 n^2 h^2)`
for `p^(th)` orbit `, E_p = (-2 pi ^2 m e^4 )/( ( 4 pi epsi_0)^2 p^2 h^2) `
frequency `, v = (E_a-E_p) /( h ) = ( 2pi ^2m e^4)/((4 pi epsi_0 )^2 h^2) [ (1)/(p^2) - (1)/( n^2)]`
for n=n and p=n -1 we get
` v= (E_n -E_(n-1))/(h) = ( 2 pi ^2 m e^4 )/( (4 pi epsi_0 )^2 h^3 ) [ (1)/( (n-1)^2)-(1)/(n^2)]`
i.e, ` v= (2 pi ^2 m e^4 )/( (4 pi epsi _0 ) ^2 h^3) . ((2n-1))/(n^2 (n-1)^2)`
for large values of `n2m-1~~ 2 n ` and n-1 `~~` n
` therefore v= (2 pi ^2 m e ^4 )/( (4 pi epsi_0)h^3).( 2n )/( h^4) = ( 4 pi ^2 me ^4 )/((4 pi epsi_0)^2 n^3 h^3 )`
classical frequnency ,` V_ c =(v )/( 2 pi r)`
we have mvr `= (n h )/(2 pi ) therefore v= (nh )/( 2 pi m r)`
` therefore v= (nh )/( 4 pi ^2 mr^2 )`
But `r= ((4 pi epsi_0 )n^2 h^2)/( 4 pi ^2 m e ^2)`
` therefore V_c = ( 4 pi ^2 m e ^4 )/(4 pi epsi_0 )^2 n^3 h^3`
then for large values of `n, upsilon = upsilon _c` hence the proof .
this is called bohr .s correspondence principle