When did Bohr hydrogen atom approaches classical condition



Here our problem is to prove that the Bohr hydrogen atom approaches classical conditions when n becomes very large and small quantum jumps are involved.

To prove this let us compute the frequency of a photon that is emitted in the transition between the adjacent state n_{k}=n and n_{i}=n-1 when n\gg 1.

we define Rydberg’s constant as

R=\frac{2\pi ^{2}me^{4}}{h^{3}c}

So,

{E_k} = \frac{{ch}}{{n_k^2}}R

and

{E_i} = \frac{{ch}}{{n_i^2}}R

Therefore the frequency of the emitted photon is

\nu  = \frac{{n_k^2 - n_i^2}}{{n_k^2n_i^2}}cR = \frac{{\left( {{n_k} + {n_i}} \right)\left( {{n_k} - {n_i}} \right)}}{{n_k^2n_i^2}}cR

{{n_k} - {n_i} = 1} , so for n\gg 1 we have

{n_k} + {n_i} \cong 2n and n_k^2n_i^2 \cong {n^4}

Therefore ,

\nu  = \frac{{2cR}}{{{n^3}}}

According to classical theory of electromagnetism , a rotating charge with a frequency f will emit a radiation of frequency f. On the other hand , using the Bohr hydrogen model , the orbital frequency of the electron around the nucleus is

{f_n} = \frac{{{\nu _n}}}{{2\pi {r_n}}} = \frac{{4{\pi ^2}m{e^4}}}{{{n^3}{h^3}}}

or

{f_n} = \frac{{2cR}}{{{n^3}}}, which is identical to \nu





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