Question

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

• A. Relative change in the radii of two consequtive orbitals does not depend on $Z$
• B. Relative change in the radii of two consequtive orbitals varies as $\frac{1}{n}$
• C. Relative change in the energy of two consequtive orbitals varies as $1/n^3$
• D. Relative change in the angular momenta of two consequtive orbitals varies as $\frac{1}{n}$

Answer 1

Answer: (A), (B), (D)

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

Radius of the $n^{th}$ state $r_n=0.529\frac{n^2}{Z_e}$ $⟹r_n\propto \frac{n^2}{Z}$

$\therefore$ $\frac{r_{n+1}-r_n}{r_n}\propto \frac{\frac{(n+1{)}^2}{Z}-\frac{n^2}{Z}}{\frac{n^2}{Z}}$

OR $\frac{\Delta r_n}{r_n}\propto \frac{(n+1{)}^2-n^2}{n^2}=\frac{2n+1}{n^2}=\frac{2}{n}$ ($\because 2n+1=2n$ for $n>>1$ )

$⟹\frac{\Delta r_n}{r_n}\propto \frac{1}{n}$ and independent of $Z$.
Thus options A and B are correct.


Energy of $n^{th}$ state $E_n=-13.6\frac{Z^2}{n^2}$ $⟹E_n\propto \frac{Z^2}{n^2}$


$\therefore$ $\frac{E_{n+1}-E_n}{E_n}\propto \frac{\frac{Z^2}{(n+1{)}^2}-\frac{Z^2}{n^2}}{\frac{Z^2}{n^2}}$

OR $\frac{\Delta E_n}{E_n}\propto \frac{(n+1{)}^2-n^2}{n^2}=\frac{2n+1}{n^2}=\frac{2}{n}$ ($\because 2n+1=2n$ and $n+1=n$ for $n>>1$ )

$⟹\frac{\Delta E_n}{E_n}\propto \frac{1}{n}$ and independent of $Z$.
Thus option C is wrong.


Angular momentum $L_n=n\hslash$
$\therefore$ $\frac{\Delta L_n}{L_n}=\frac{(n+1)\hslash -n\hslash }{n\hslash }=\frac{1}{n}$ $⟹$ $\frac{\Delta L_n}{L_n}\propto \frac{1}{n}$
Thus option D is also correct.