According to classical physics, an electron in periodic motion with emit electromagnetic radiation with the same frequency as that of its revolution. Compute this value for hydrogen atom in nth quantum theory permit emission of such photons due to transition between adjoining orbits ? Discuss the result obtained.

Explanation

The orbital frequency of `n^(th)` orbit is given by
`v_(n) = ("electron speed)/(orbital circumference")`
`= (e^(2))/(2 nh epsilon_(0)) xx (1)/(2 pi) xx (pi me^(2))/(n^(2) epsilon_(0) h^(2)) = (me^(4))/(4 epsilon_(0)^(2)n^(3) h^(3))`
`[As K = (1)/(4 pi epsilon_(0)]`
From the lows of electromagnetic theory, the frequency of the radition emitted by this electron will also be `v_(n)`
According to Bohr's theory , the frequency of the radition for the adjoining orbits is given by
`v_(0) = (E_(n) - E_(n-1))/(h) = (me^(4))/(8 epsilon_(0)^(2) h^(2)) [(1)/((n -1)^(2)) - (1)/(n^(2))]`
`(me^(4))/(8 epsilon_(0)^(2) h^(2)) ((2n - 1))/((n - 1)^(2) xx n^(2))`
A comprison of this expression with the classic expression above show that the difference in their prediction will be large for small n, i.e. , for `n = 2, 1//n^(3) = 1//8` and `(2 n - 1) //2 n^(2) (n- 1)^(2) = 3//8`, so that the frequency given by quantum theory is `3` times that calculate from classic theory. However , for very large `n` , the quantum expression reduced to that obtained from classical theory because for `n to oo`
`((2n - 1)/(2n^(2) (n - 1)^(2) xx (2 n)/(2 n^(2) n^(2)))) = (1)/(n^(3))`
Consequentiy , the predictions of Bohr's theory agrees with that of the classical theory in the limit of very large quantum numbers . This correspondence is called as bohr's correspondenceprincipal.