Highly excited states for hydrogen like atoms (also called Rydberg states) with nuclear charge Ze are defined by their principal quantum number n, where n>>1. Which of the following statement (s) is (are) true ?

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Relative change in the energy of two consecutive orbitals varies as $\frac{1}{n^{3}}$

Relative change in the radii of two consecutive orbitals varies as $\frac{1}{n}$

Relative change in the radii of two consecutive orbitals does not depend on Z

Relative change in the angular momenta of the consecutive orbitals varies as $\frac{1}{n}$

answer is B, C, D.

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Detailed Solution

Radius $r α \frac{n^{2}}{z}$ $\frac{Δ r}{r} = \frac{\frac{{(n + 1)}^{2}}{z} - \frac{n^{2}}{z}}{\frac{n^{2}}{z}} = \frac{{(n + 1)}^{2} - n^{2}}{n^{2}} = \frac{2 n + 1}{n^{2}} \approx \frac{2 n}{n^{2}} (∵ n > > 1)$ $\frac{Δ r}{r} α \frac{1}{n}$

Energy , $E α \frac{z^{2}}{n^{2}}$

$\begin{array}{l} \frac{Δ E}{E} = \frac{\frac{z^{2}}{n^{2}} - \frac{z^{2}}{{(n + 1)}^{2}}}{\frac{z^{2}}{n^{2}}} = \frac{{(n + 1)}^{2} - n^{2}}{{(n + 1)}^{2}} \\ = \frac{2 n + 1}{{(n + 1)}^{2}} \approx \frac{2 n}{n^{2}} \\ \frac{Δ E}{E} α \frac{1}{n} \end{array}$

Angular momentum, $L = \frac{n h}{2 π}$
$\begin{array}{l} \frac{Δ L}{L} = \frac{\frac{(n + 1) h}{2 π} - \frac{n h}{2 π}}{\frac{n h}{2 π}} = \frac{n + 1 - n}{n} = \frac{1}{n} \\ \frac{Δ L}{L} α \frac{1}{n} \end{array}$