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  • Suppose that a bound state is formed by a particle moving in a potential given by <img alt="\mathrm{U}=\frac{1}{2} m \omega_0^2 r^2," src="https://entrancecorner.oncodecogs.com/gif.latex?

Suppose that a bound state is formed by a particle moving in a potential given by

\mathrm{U}=\frac{1}{2} m \omega_0^2 r^2, where \mathrm{r} is the distance of the particle from the origin, and \mathrm{m} is the mass of the particle. Using Bohr's rule of quantization of angular momentum, then the allowed radius of the nth orbit is
 

Option: 1

\sqrt{\frac{n\hbar}{m \omega_o}}

 


Option: 2

2 \sqrt{\frac{n\hbar}{m \omega_o}}


Option: 3

3 \sqrt{\frac{n\hbar}{m \omega_o}}


Option: 4

4 \sqrt{\frac{n\hbar}{m \omega_o}}


Answers (1)

best_answer

The force acting on the particle is in the radial direction, given by,

\mathrm{F}=-\frac{d u}{d r}=-\frac{d}{d r}\left(\frac{1}{2} m \omega_0^2 r^2\right)=-m \omega_0^2 r
We can, therefore, write,
m \omega_o^2 r^2=m \frac{v^2}{r} \quad \quad \quad \dots(i)
or
\begin{aligned} & \mathrm{mv}^2=m \omega_0^2 r^2 \quad \Rightarrow \quad \mathrm{v}=\omega_0 \mathrm{r} \\ & \mathrm{mvr}=\frac{n h}{k \pi}=n \hbar \quad \quad \ldots \text { (2) } \end{aligned}
Putting in (2)

m \omega_0^2 r^2=\mathrm{n} \hbar
or       \quad \mathrm{r}_{\mathrm{n}}=\sqrt{\frac{n \hbar}{m \omega_0}}

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mansi

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