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





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