SATHEE: Modern Physics 1 Question 38

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 $n ≫ 1$ . 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 $1 / n$

(d) Relative change in the angular momenta of two consecutive orbitals varies as $1 / n$

Show Answer

Solution:

As radius $r \propto \frac{n^{2}}{z}$

$\Rightarrow \frac{Δ r}{r} = \frac{{\frac{n + 1}{z}}^{2} - {\frac{n}{z}}^{2}}{{\frac{n}{z}}^{2}} = \frac{2 n + 1}{n^{2}} \approx \frac{2}{n} \propto \frac{1}{n}$

Energy, $E \propto \frac{z^{2}}{n^{2}}$

$\begin{aligned} \Rightarrow \frac{Δ E}{E} & = \frac{\frac{z^{2}}{n^{2}} - \frac{z^{2}}{(n - 1)^{2}}}{\frac{z^{2}}{(n + 1)^{2}}} \\ = \frac{(n + 1)^{2} - n^{2}}{n^{2} \cdot (n + 1)^{2}} \cdot (n + 1)^{2} \\ \Rightarrow \frac{Δ E}{E} & = \frac{2 n + 1}{n^{2}} ≃ \frac{2 n}{n^{2}} \propto \frac{1}{n} \end{aligned}$

Angular momentum, $L = \frac{n h}{2 π}$

$\Rightarrow \frac{Δ L}{L} = \frac{\frac{(n + 1)}{2 π} - \frac{n h}{2 π}}{\frac{n h}{2 π}} = \frac{1}{n} \propto \frac{1}{n}$

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 $n ≫ 1$ . Which of the following statement(s) is (are) true?

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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≫1. Which of the following statement(s) is (are) true?

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

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 $n ≫ 1$ . Which of the following statement(s) is (are) true?