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?

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