Question

Instead of angular momentum quantization a sudent posits that energy is quantized as $E=-E_0/n\left(E_0>0\right)$ and $n$ is a positive integer. Which of the following options is correct?

• A. The radius of the electron orbit is $r\propto \sqrt{n}$.
• B. The speed of the electron is $v\propto \sqrt{n}$.
• C. The angular speed of the electron is $\omega \propto 1/n$
• D. The angular momentum of the electron is $\propto \sqrt{n}$

Answer 1

Answer: (D)

$F_e=\frac{m{v}^2}{r}$
$\Rightarrow \frac{m{v}^2}{r}=\frac{k(Ze)(e)}{r^2}$
$\frac{1}{2}m{v}^2=\frac{KZ{e}^2}{2r}$.....(i) (Kinetic energy)
Potential energy $=\frac{K{q}_1{q}_2}{r}$
$=\frac{K(Ze)(-e)}{r}$....(ii)
Total energy $=KE+PE$
$=-\frac{KZ{e}^2}{2r}=-\frac{E_0}{n}$
$\therefore r\propto n$
As kinetic energy $=\frac{KZ{e}^2}{2r}$
$\Rightarrow KE\propto \frac{1}{n}$
or $v^2\propto \frac{1}{n}$
$\Rightarrow v^2\propto \frac{1}{\sqrt{n}}$
$L=mvr$
$L\propto vr$
$L\propto \frac{1}{\sqrt{n}}(n)\propto \sqrt{n}$
$\Rightarrow L\propto \sqrt{n}$