Introduction

The problem asks to analyze the properties of highly excited states (Rydberg states) of a hydrogen-like atom with nuclear charge $Ze$ and principal quantum number $n \gg 1$. We need to determine which of the given statements about the relative change in radius, energy, and angular momentum for consecutive orbitals are true.

First, let's recall the basic formulas from the Bohr model for a hydrogen-like atom:

• Radius of the n-th orbit: $r_n = a_0 \frac{n^2}{Z}$, where $a_0$ is the Bohr radius.
• Energy of the n-th state: $E_n = -E_0 \frac{Z^2}{n^2}$, where $E_0 \approx 13.6$ eV.
• Angular momentum of the n-th orbit: $L_n = n \frac{h}{2\pi} = n\hbar$.

The relative change in a quantity $X$ between two consecutive orbitals ($n$ and $n+1$) is defined as $\frac{\Delta X_n}{X_n} = \frac{X_{n+1} - X_n}{X_n}$. Since $n \gg 1$, we can use approximations like $n+1 \approx n$ and $2n+1 \approx 2n$.

Step-by-step Analysis of Options

1. Analysis of Option A and B: Relative change in radii

The radius of the n-th orbit is $r_n = a_0 \frac{n^2}{Z}$. The radius of the (n+1)-th orbit is $r_{n+1} = a_0 \frac{(n+1)^2}{Z}$.

The change in radius is: $\Delta r_n = r_{n+1} - r_n = a_0 \frac{(n+1)^2}{Z} - a_0 \frac{n^2}{Z} = \frac{a_0}{Z}((n+1)^2 - n^2) = \frac{a_0}{Z}(n^2 + 2n + 1 - n^2) = \frac{a_0}{Z}(2n+1)$

The relative change in radius is: $\frac{\Delta r_n}{r_n} = \frac{\frac{a_0}{Z}(2n+1)}{a_0 \frac{n^2}{Z}} = \frac{2n+1}{n^2}$

• Statement A: The expression for the relative change, $\frac{2n+1}{n^2}$, does not contain the nuclear charge $Z$. Therefore, the relative change in the radii of two consecutive orbitals does not depend on Z. Statement A is true.

• Statement B: For highly excited states, $n \gg 1$. We can approximate the relative change: $\frac{\Delta r_n}{r_n} = \frac{2n+1}{n^2} \approx \frac{2n}{n^2} = \frac{2}{n}$ The relative change in radii varies as $\frac{1}{n}$. Statement B is true.

2. Analysis of Option C: Relative change in energy

The energy of the n-th state is $E_n = -E_0 \frac{Z^2}{n^2}$. The energy of the (n+1)-th state is $E_{n+1} = -E_0 \frac{Z^2}{(n+1)^2}$.

The change in energy is: $\Delta E_n = E_{n+1} - E_n = -E_0 Z^2 \left( \frac{1}{(n+1)^2} - \frac{1}{n^2} \right) = -E_0 Z^2 \left( \frac{n^2 - (n+1)^2}{n^2(n+1)^2} \right) = -E_0 Z^2 \left( \frac{-2n-1}{n^2(n+1)^2} \right) = E_0 Z^2 \frac{2n+1}{n^2(n+1)^2}$

The relative change in energy is typically considered for the magnitude of the energy, $|E_n| = E_0 \frac{Z^2}{n^2}$. $\frac{\Delta E_n}{|E_n|} = \frac{E_0 Z^2 \frac{2n+1}{n^2(n+1)^2}}{E_0 \frac{Z^2}{n^2}} = \frac{2n+1}{(n+1)^2}$ For $n \gg 1$: $\frac{\Delta E_n}{|E_n|} \approx \frac{2n}{n^2} = \frac{2}{n}$ The relative change in energy varies as $\frac{1}{n}$.

• Statement C: It claims the relative change varies as $\frac{1}{n^3}$. This is incorrect. The absolute change in energy $\Delta E_n$ varies as $\frac{1}{n^3}$ for large $n$ (since $\Delta E_n \approx E_0 Z^2 \frac{2n}{n^4} = \frac{2E_0 Z^2}{n^3}$), but the question asks for the relative change. Statement C is false.

3. Analysis of Option D: Relative change in angular momentum

The angular momentum of the n-th orbit is $L_n = n\hbar$. The angular momentum of the (n+1)-th orbit is $L_{n+1} = (n+1)\hbar$.

The change in angular momentum is: $\Delta L_n = L_{n+1} - L_n = (n+1)\hbar - n\hbar = \hbar$

The relative change in angular momentum is: $\frac{\Delta L_n}{L_n} = \frac{\hbar}{n\hbar} = \frac{1}{n}$

• Statement D: The relative change in angular momentum is exactly $\frac{1}{n}$. Thus, it varies as $\frac{1}{n}$. Statement D is true.

Conclusion

Based on the analysis:

• Statement A is true.
• Statement B is true.
• Statement C is false.
• Statement D is true.

The correct options are A, B, and D.