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### Uncertainty relation results in rejection of semi classic Bohr model for hydrogen atom

Question
Show that the uncertainty relation forces us to reject the semi classical Bohr model for the hydrogen atom
Solution
In Bohr atom model we deal with the electron as a classical particle. The allowed orbits are defined by the quantization rules:
The radius r of a circular orbit and the momentum $p = mv$ of the rotating electron must satisfy $pr = n\hbar (n = 1,2,3,....)$. To consider an electron’s model in classical terms , the uncertainties in its position and momentum must be negligible when compared to  $r$ and $p$. In other words $\Delta x \ll r$ and $\Delta p \ll p$
This implies
$\frac{{\Delta x}}{r}\frac{{\Delta p}}{p} \ll 1$                     (1)
On the other hand , the uncertainty relation imposes
$\frac{{\Delta x}}{r}\frac{{\Delta p}}{p} \ge \frac{\hbar }{{rp}} \Rightarrow \frac{{\Delta x\Delta p}}{{rp}} \ge \frac{1}{n}$                           (2)
So, equation 1 is incompatible with 2 , unless $n \gg 1$