Question

An electron and a proton are separated by a distance $r$ so that the potential energy between them is $U=k\log r$, where $k$ is a constant. The expression for various energy levels of the above said hypothetical atom is:

• A. $\frac{k}{2}[1+\log \frac{n^2{h}^2}{4{\pi }^{2}mk}]$
• B. $2k[2+\log \frac{n^2{h}^2}{4{\pi }^{2}mk}]$
• C. $k[2+\log \frac{n^2{h}^2}{4{\pi }^{2}mk}]$
• D. $\frac{k}{2}[1+\log \frac{n^2{h}^2}{2{\pi }^{2}mk}]$

Answer 1

The correct option is A $\frac{k}{2}[1+\log \frac{n^2{h}^2}{4{\pi }^{2}mk}]$

$U=klogr$

$⟹F=-\frac{dU}{dr}=-\frac{k}{r}$

$⟹\frac{k}{r}=\frac{m{v}^2}{r}$

$⟹v=\sqrt{\frac{k}{m}}$

Since angular momentum is quantized:

$mvr=\frac{nh}{2\pi }$

$⟹r=\frac{nh}{2\pi m}\sqrt{\frac{m}{k}}$

Thus, the energy of the system $=\frac{1}{2}m{v}^2+U$

$=\frac{1}{2}m(\sqrt{\frac{k}{m}}{)}^2+klog(\frac{nh\sqrt{m}}{2\pi m\sqrt{k}})$

$=\frac{k}{2}+klog(\frac{n^2{h}^2}{4{\pi }^{2}mk}{)}^{1/2}$

$=\frac{k}{2}[1+log(\frac{n^2{h}^2}{4{\pi }^{2}mk}\left)\right]$