Question 237: Atoms & Nuclei

Question

A particle of mass m moves in a circular orbit in a central potential field U(r) = $${1 \over 2}$$ kr
2
. If Bohr 's
quantization conditions are applied, radii of possible orbitls and energy levels vary with quantum number n as :

Accepted Answer

Force due to this field, F = $$ - {{\partial U} \over {\partial r}}$$
F = $$ - {\partial  \over {\partial r}}\left( {{1 \over 2}k{r^2}} \right)$$ = -kr
For circular orbit, $${{m{v^2}} \over r}$$ = -kr
$$ \Rightarrow $$ v $$ \propto $$ r ..... (1)
Fron BohrÃ¢ÂÂs quantization condition
mvr = $${{nh} \over {2\pi }}$$ .....(2)
From (1) and (2),
$${r_n}$$ $$ \propto $$ $${n^{{1 \over 2}}}$$
Given, U(r) = $${1 \over 2}$$kr
2
$$ \Rightarrow $$ E
n
= $$ - {1 \over 2}U\left( r \right)$$ = $$ - {1 \over 4}k{r^2}$$
$$ \Rightarrow $$ E
n
$$ \propto $$ n

Answer

r
n
$$ \propto $$ $$\sqrt n $$, E
n
$$ \propto $$ n

Answer

r
n
$$ \propto $$ $$\sqrt n $$, E
n
$$ \propto $$ $${1 \over n}$$

Answer

r
n
$$ \propto $$ n, E
n
$$ \propto $$ n

Answer

r
n
$$ \propto $$ n
2
, E
n
$$ \propto $$ $${1 \over {{n^2}}}$$

Atoms & Nuclei

A particle of mass m moves in a circular orbit in a central potential field U(r) = ${1 \over 2}$ kr 2 . If Bohr 's quantization conditions are applied, radii of possible orbitls and energy levels vary with quantum number n as :

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