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)

$F _{ e }=\frac{ mv ^{2}}{ r }$
$ \Rightarrow \frac{ mv ^{2}}{ r }=\frac{ k ( Ze )( e )}{ r ^{2}}$
$\frac{1}{2} mv ^{2}=\frac{ KZe ^{2}}{2 r }$.....(i) (Kinetic energy)
Potential energy $=\frac{ Kq _{1} q _{2}}{ r }$
$=\frac{ K ( Ze )(- e )}{ r }$....(ii)
Total energy $= KE + PE $
$=-\frac{ KZe ^{2}}{2 r }=-\frac{ E _{0}}{ n }$
$\therefore r \propto n$
As kinetic energy $=\frac{ KZe ^{2}}{2 r }$
$ \Rightarrow KE \propto \frac{1}{ n }$
or $v ^{2} \propto \frac{1}{ n } $
$\Rightarrow v ^{2} \propto \frac{1}{\sqrt{ n }}$
$L = mvr$
$L \propto vr$
$L \propto \frac{1}{\sqrt{ n }}( n ) \propto \sqrt{ n }$
$\Rightarrow L \propto \sqrt{ n }$