Properties of Rydberg States in Hydrogen Like Atoms

To determine which of the given options are correct for a hydrogen-like atom in a highly excited Rydberg state ($n \gg 1$) with nuclear charge $Ze$, let us analyze each physical quantity step-by-step using the Bohr model.

---

1. Relative Variation in Orbital Radius ($r_n$)

The radius of the $n$-th Bohr orbit is given by: $r_n = \frac{n^2 h^2 \varepsilon_0}{\pi m Z e^2} \propto \frac{n^2}{Z}$

The relative variation in the radii of two consecutive orbitals ($n$ and $n+1$) is: $\frac{\Delta r_n}{r_n} = \frac{r_{n+1} - r_n}{r_n} = \frac{\frac{(n+1)^2}{Z} - \frac{n^2}{Z}}{\frac{n^2}{Z}} = \frac{(n+1)^2 - n^2}{n^2} = \frac{2n + 1}{n^2}$

• Dependence on $Z$: The term $Z$ cancels out completely in the expression for $\frac{\Delta r_n}{r_n}$. Therefore, the relative variation in radii is independent of $Z$. $\text{This makes } \mathbf{(A)} \text{ a correct statement.}$

• Dependence on $n$: For $n \gg 1$, we can simplify: $\frac{\Delta r_n}{r_n} = \frac{2n + 1}{n^2} \approx \frac{2n}{n^2} = \frac{2}{n} \propto \frac{1}{n}$ Alternatively, using differential calculus for large $n$: $\frac{\mathrm{d}r}{r} = \mathrm{d}(\ln r) = \mathrm{d}\left(\ln C + 2\ln n - \ln Z\right) = 2 \frac{\mathrm{d}n}{n}$ For consecutive orbitals, $\mathrm{d}n = 1$, so: $\frac{\Delta r}{r} \approx \frac{2}{n} \propto \frac{1}{n}$ $\text{This makes } \mathbf{(B)} \text{ a correct statement.}$

---

2. Relative Variation in Energy ($E_n$)

The total energy of an electron in the $n$-th orbit is given by: $E_n = -\frac{m Z^2 e^4}{8 \varepsilon_0^2 h^2 n^2} \propto -\frac{1}{n^2}$

Taking the magnitude of the relative variation in energy for two consecutive orbitals using differentials (valid for $n \gg 1$): $\left| \frac{\mathrm{d}E}{E} \right| = \left| \mathrm{d}(\ln |E|) \right| = \left| \mathrm{d}\left(\text{constant} - 2\ln n\right) \right| = \left| -2 \frac{\mathrm{d}n}{n} \right| = \frac{2}{n}$

For consecutive orbitals ($\mathrm{d}n = 1$): $\left| \frac{\Delta E_n}{E_n} \right| \approx \frac{2}{n} \propto \frac{1}{n}$

(Note: While the absolute energy difference $\Delta E_n \propto \frac{1}{n^3}$, the relative variation $\frac{\Delta E_n}{E_n}$ is proportional to $\frac{1}{n}$, not $\frac{1}{n^3}$.)

$\text{This makes } \mathbf{(C)} \text{ an incorrect statement.}$

---

3. Relative Variation in Angular Momentum ($L_n$)

According to Bohr's quantization condition, the orbital angular momentum of an electron in the $n$-th state is: $L_n = \frac{n h}{2\pi} \propto n$

The relative variation in angular momentum between two consecutive orbitals is: $\frac{\Delta L_n}{L_n} = \frac{L_{n+1} - L_n}{L_n} = \frac{(n+1) - n}{n} = \frac{1}{n}$

Thus, the relative variation in angular momentum is proportional to $\frac{1}{n}$.

$\text{This makes } \mathbf{(D)} \text{ a correct statement.}$

---

Conclusion

The correct options are A, B, and D.