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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. |
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Answer» Solution :for`n^(th)`orbit ` E_n= ( - 2pi^@ me ^4 ) /( ( 4 piepsi_0)^2 n^2 h^2)` for `p^(th)` orbit`, E_p =(-2 PI ^2 m e^4 )/( ( 4 piepsi_0)^2 p^2h^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)]` forn=nand p=n -1 we GET ` v= (E_n -E_(n-1))/(h)= ( 2 pi^2 m e^4 )/( (4 piepsi_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)` forlargevaluesof `n2m-1~~2 n ` andn-1`~~` n ` thereforev= (2 pi^2m 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 )` classicalfrequnency,` V_ C =(v)/( 2 pir)` we havemvr`= (n h )/(2 pi )thereforev=(nh )/( 2 pi m r)` ` thereforev= (nh )/( 4 pi ^2mr^2 )` But `r=((4 pi epsi_0 )n^2 h^2)/( 4 pi ^2 m e ^2)` ` thereforeV_c = ( 4pi ^2 m e ^4 )/(4 pi epsi_0 )^2 n^3 h^3` thenforlargevalues of `n, upsilon = upsilon _c` hencetheproof. thisis calledbohr .scorrespondenceprinciple |
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