Assume a hypothetical hydrogen atom in which the potential energy between electron and proton at separation $r$ is given by $U=k\left({\log }_{e}r-\frac{1}{2}\right)$, where $k$ is a constant. For such a hypothetical hydrogen atom, calculate the radius of nth Bohr's orbit and energy levels.

Force of interaction between electron and proton is given by

       $F=-\frac{dU}{dr}=-\frac{k}{r}$

Force is negative. It means there is attraction between the particles and they are bound to each other. This force provides the necessary centripetal force for the electron. So, we have

      $\frac{m{v}^2}{r}=\frac{k}{r}$

According to Bohr's assumption, we have

       $mvr=n\left(\frac{h}{2\pi }\right)$

Solving equations (1) and (2), we get

        $r=\frac{nh}{2\pi \sqrt{mk}}\text{ and }v=\sqrt{\frac{k}{m}}$

Since, $E=U+\frac{1}{2}m{v}^2$

$\Rightarrow E=k{\log }_{e}r-\frac{k}{2}+\frac{k}{2}=k{\log }_{e}r$

So, $r_n=\frac{nh}{2\pi \sqrt{mk}}$ and $E_n=k{\log }_e\left(\frac{nh}{2\pi \sqrt{mk}}\right)$