Modern Physics 1 Question 38
35. Highly excited states for hydrogen-like atoms (also called Rydberg states) with nuclear charge Ze are defined by their principle quantum number $n$, where $n \gg 1$. Which of the following statement(s) is (are) true?
(2016 Adv.)
(a) Relative change in the radii of two consecutive orbitals does not depend on $Z$
(b) Relative change in the radii of two consecutive orbitals varies as $1 / n$
(c) Relative change in the energy of two consecutive orbitals varies as $1 / n^{3}$
(d) Relative change in the angular momenta of two consecutive orbitals varies as $1 / n$
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Solution:
- As radius $r \propto \frac{n^{2}}{z}$
$$ \Rightarrow \quad \frac{\Delta r}{r}=\frac{\frac{n+1}{z}^{2}-\frac{n}{z}^{2}}{\frac{n}{z}^{2}}=\frac{2 n+1}{n^{2}} \approx \frac{2}{n} \propto \frac{1}{n} $$
Energy, $\quad E \propto \frac{z^{2}}{n^{2}}$
$$ \begin{aligned} \Rightarrow \quad \frac{\Delta E}{E} & =\frac{\frac{z^{2}}{n^{2}}-\frac{z^{2}}{(n-1)^{2}}}{\frac{z^{2}}{(n+1)^{2}}} \\ & =\frac{(n+1)^{2}-n^{2}}{n^{2} \cdot(n+1)^{2}} \cdot(n+1)^{2} \\ \Rightarrow \quad \frac{\Delta E}{E} & =\frac{2 n+1}{n^{2}} \simeq \frac{2 n}{n^{2}} \propto \frac{1}{n} \end{aligned} $$
Angular momentum, $L=\frac{n h}{2 \pi}$
$$ \Rightarrow \quad \frac{\Delta L}{L}=\frac{\frac{(n+1)}{2 \pi}-\frac{n h}{2 \pi}}{\frac{n h}{2 \pi}}=\frac{1}{n} \propto \frac{1}{n} $$
