A particle of mass m is moving in a circular orbit under the...

A particle of mass

m

is moving in a circular orbit under the influence of the central force

F(r)=-k r

, corresponding to the potential energy

V(r)=k r^2 / 2

, where

k

is a positive force constant and

r

is the radial distance from the origin. According to the Bohr's quantization rule, the angular momentum of the particle is given by

L=n \hbar

, where

\hbar=h /(2 \pi), h

is the Planck's constant, and

n

a positive integer. If

v

and

E

are the speed and total energy of the particle, respectively, then which of the following expression(s) is(are) correct?

Options

A. $r^2=n \hbar \sqrt{\frac{1}{m k}}$

The central force will provide necessary centripetal force

\Rightarrow \mathrm{kr}=\frac{\mathrm{mv}^2}{\mathrm{r}}

or,

\mathrm{kr}^2=\mathrm{mv}^2

...(1) By quantisation rule

\mathrm{n} \hbar=\mathrm{mvr}

or,

\frac{\mathrm{n} \hbar}{\mathrm{r}}=\mathrm{mv}

...(2)

\begin{aligned} & \frac{(1)}{(2)^2} \Rightarrow \frac{\mathrm{kr}^2}{\frac{\mathrm{n}^2 \hbar^2}{\mathrm{r}^2}}=\frac{\mathrm{mv}^2}{\mathrm{~m}^2 \mathrm{v}^2} \\ & \Rightarrow \frac{\mathrm{k}}{\mathrm{n}^2 \hbar^2} \mathrm{r}^4=\frac{1}{\mathrm{~m}} \\ & \Rightarrow \mathrm{r}=\left(\frac{\mathrm{n}^2 \hbar^2}{\mathrm{~km}}\right)^{\frac{1}{4}} \Rightarrow \mathrm{r}^2=\frac{\mathrm{n} \hbar}{\sqrt{\mathrm{mk}}}\end{aligned}

\begin{aligned} & \text{(2) Using (1), } \mathrm{K} \cdot \frac{\mathrm{n} \hbar}{\sqrt{\mathrm{mk}}}=\mathrm{mv}^2 \\ & \Rightarrow \mathrm{v}^2=\mathrm{n} \hbar \sqrt{\frac{\mathrm{k}}{\mathrm{m}^3}}\end{aligned}

(3)

\frac{\mathrm{L}}{\mathrm{mr}^2}=\frac{\mathrm{mvr}}{\mathrm{mr}^2}=\frac{\mathrm{v}}{\mathrm{r}}=\sqrt{\frac{\mathrm{k}}{\mathrm{m}}}

from (1)

\begin{aligned} & \text{(4) } \mathrm{E}=\frac{1}{2} \mathrm{mv}^2+\frac{1}{2} \mathrm{kr}^2=\frac{\mathrm{n} \hbar}{2} \sqrt{\frac{\mathrm{k}}{\mathrm{m}}}+\frac{1}{2} \mathrm{k} \frac{\mathrm{n} \hbar}{\sqrt{\mathrm{mk}}} \\ & \mathrm{E}=\mathrm{n} \hbar \sqrt{\frac{\mathrm{k}}{\mathrm{m}}}\end{aligned}