Highly excited states for hydrogen like atom (alos called Ryburg states) with nucleus Charge Ze are defined by their principal qunatum number n, where `n lt lt 1`. Which of the following statement(s) is (are) true?

Options

A. Releative 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`
C. Relative change in the energy of two consecutive orbitals varies as `1//n^(2)`
D. Relative change in the angular momenta of two consecutive orbitals varies as `1//n`

Explanation

To solve the question regarding the highly excited states (Rydberg states) for hydrogen-like atoms with nucleus charge \( Ze \), we need to analyze the given statements one by one.

### Step-by-Step Solution:

1. **Understanding the Rydberg Formula for Radius**:
The radius of the nth orbit for a hydrogen-like atom is given by:
\[ R_n = \frac{0.529 \, n^2}{Z} \]
where \( n \) is the principal quantum number and \( Z \) is the atomic number.

2. **Analyzing the First Statement**:
The first statement claims that the relative change in the radii of two consecutive orbitals does not depend on \( Z \).
- For two consecutive orbits \( n \) and \( n+1 \):
\[ R_{n+1} = \frac{0.529 \, (n+1)^2}{Z} \]
The change in radius \( \Delta R \) is:
\[ \Delta R = R_{n+1} - R_n = \frac{0.529 \, ((n+1)^2 - n^2)}{Z} \]
The expression simplifies to:
\[ \Delta R = \frac{0.529 \, (2n + 1)}{Z} \]
- The relative change in radius is:
\[ \frac{\Delta R}{R_n} = \frac{(2n + 1)}{n^2} \]
- Since \( Z \) cancels out, the first statement is **true**.

3. **Analyzing the Second Statement**:
The second statement claims that the relative change in radii of two consecutive orbitals varies as \( \frac{1}{n} \).
- From the previous calculation, we found:
\[ \frac{\Delta R}{R_n} = \frac{(2n + 1)}{n^2} \]
- For \( n \) very small, we can approximate:
\[ \frac{(2n + 1)}{n^2} \approx \frac{2}{n} \]
- Thus, the second statement is also **true**.

4. **Analyzing the Third Statement**:
The third statement discusses the relative change in energy of two consecutive orbitals.
- The energy of the nth orbital is given by:
\[ E_n = -\frac{13.6 Z^2}{n^2} \]
- The change in energy \( \Delta E \) is:
\[ \Delta E = E_{n+1} - E_n = -\frac{13.6 Z^2}{(n+1)^2} + \frac{13.6 Z^2}{n^2} \]
- Simplifying gives:
\[ \Delta E = 13.6 Z^2 \left( \frac{1}{n^2} - \frac{1}{(n+1)^2} \right) = 13.6 Z^2 \cdot \frac{2n + 1}{n^2(n+1)^2} \]
- The relative change in energy is:
\[ \frac{\Delta E}{E_n} = \frac{2n + 1}{(n + 1)^2} \]
- As \( n \) becomes very small, this does not simplify to \( \frac{1}{n^2} \) but rather to \( \frac{2}{n} \). Thus, the third statement is **false**.

5. **Analyzing the Fourth Statement**:
The fourth statement claims that the relative change in angular momentum of two consecutive orbitals varies as \( \frac{1}{n} \).
- The angular momentum for the nth orbital is given by:
\[ L_n = n \hbar \]
- The change in angular momentum \( \Delta L \) is:
\[ \Delta L = L_{n+1} - L_n = (n + 1)\hbar - n\hbar = \hbar \]
- The relative change is:
\[ \frac{\Delta L}{L_n} = \frac{\hbar}{n\hbar} = \frac{1}{n} \]
- Thus, the fourth statement is **true**.

### Conclusion:
The true statements are A, B, and D.