A particle of mass

Question:

A particle of mass $\mathrm{m}$ moves in a circular orbit in a central potential field $\mathrm{U}(\mathrm{r})=\mathrm{U}_{0} \mathrm{r}^{4}$. If Bohr's quantization conditions are applied, radii of possible orbitals $r_{n}$ vary with $n^{1 / \alpha}$, where $\alpha$ is ___________

Solution:

$\mathrm{F}=\frac{-\mathrm{dU}}{\mathrm{dr}}=-4 \mathrm{U}_{0} \mathrm{r}^{3}=\frac{\mathrm{mv}^{2}}{\mathrm{r}}$

$m v^{2}=4 U_{0} r^{4}$

$V \propto r^{2}$

$m v r=\frac{n h}{2 \pi}$

$\mathrm{r}^{3} \propto \mathrm{n}$

$r \propto n 1 / 3$

$=3$

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