Question

Assume an imaginary world. Where angular momentum is quantized to even multiple $h$. Find the longest possible wavelength emitted by Hydrogen in the visible spectrum. (in nano metres)

Answer 1

$mvr=2nh$

$mv=\frac{2nh}{r}$

$m{v}^2=\frac{m^2{v}^2}{n}=\frac{(2nh{)}^2}{m{r}^2}$

$\frac{Z{e}^2}{4\pi {\epsilon }_0{r}^2}=\frac{m{v}^2}{r}$

$r=\frac{(2nh{)}^{2}4\pi {\epsilon }_0}{mZ{e}^2}$

$BE=KE+PE$ =$\frac{-Z{e}^2}{8\pi {\epsilon }_{0}r}=\frac{-Z^2{e}^{4}m}{8\pi {\epsilon }_0\left(2nh{)}^2\right)4\pi {\epsilon }_0}$ $=\frac{-Z^2{e}^{4}m}{32{\epsilon }_0{n}^2{h}^2}$

$BE=\frac{-3.4}{n^2}eV$ for Hydrogen.

To find longest wavelength: $hv$ = 3.4 $[1-\frac{1}{4}]$$=3.4\times \frac{3}{4}=2.55$