Which of the following option(s) is/are independent of both `n` and `Z` for `H-`like species?
`U_(n) =` Potential energy of electron in `n^(th)` orbit
`KE_(n) =` Kinetic energy of electron in `n^(th)` orbit
`l_(n) =` Angular momentam of electoron in `n^(th)` orbit
`v_(n) =` Velcity of electron in `n^(th)` orbit
`f_(n) =` Frequency of electron in `n^(th)` orbit
`T_(n) =` Time period of revolution of electron in `n^(th)` orbit

Options

A. `(r_(n))/(U_(n))`
B. `r_(n) xx KE_(n) xx l_(n)`
C. `(U_(n) xx T_(n))/(l_(n))`
D. `(l_(n) xx f_(n))/(v_(n)^(2))`

Explanation

To solve the question of which of the following options is/are independent of both `n` and `Z` for `H-`like species, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding the Variables**:
- `U_n`: Potential energy of electron in the `n`th orbit.
- `KE_n`: Kinetic energy of electron in the `n`th orbit.
- `L_n`: Angular momentum of electron in the `n`th orbit.
- `v_n`: Velocity of electron in the `n`th orbit.
- `f_n`: Frequency of electron in the `n`th orbit.
- `T_n`: Time period of revolution of electron in the `n`th orbit.

2. **Analyzing Each Option**:
- **Option A: \( U_n \)**
The potential energy is given by:
\[ U_n = -\frac{2E_0 Z^2}{n^2} \]
This expression clearly depends on both `n` and `Z`, so it is not independent.

- **Option B: \( KE_n \)**
The kinetic energy is given by:
\[ KE_n = \frac{E_0 Z^2}{n^2} \]
This also depends on both `n` and `Z`, so it is not independent.

- **Option C: \( L_n \)**
The angular momentum is given by:
\[ L_n = \frac{nh}{2\pi} \]
This expression only depends on `n` and is independent of `Z`, but since it depends on `n`, it is not independent of both.

- **Option D: \( v_n \)**
The velocity is given by:
\[ v_n = \frac{Z e^2}{h} \cdot \frac{1}{n} \]
This expression depends on both `n` and `Z`, so it is not independent.

- **Option E: \( f_n \)**
The frequency is given by:
\[ f_n = \frac{E_0 Z^2}{h} \]
This expression depends on `Z`, so it is not independent.

- **Option F: \( T_n \)**
The time period is given by:
\[ T_n = \frac{1}{f_n} = \frac{h}{E_0 Z^2} \]
This expression also depends on `Z`, so it is not independent.

3. **Finding Combinations**:
- **Option G: \( U_n \cdot T_n / L_n \)**
We can analyze this combination:
\[ \frac{U_n \cdot T_n}{L_n} = \frac{-\frac{2E_0 Z^2}{n^2} \cdot \frac{h}{E_0 Z^2}}{\frac{nh}{2\pi}} = \frac{-4\pi}{n^3} \]
This expression depends on `n` but not on `Z`.

- **Option H: \( L_n \cdot f_n / v_n^2 \)**
Analyzing this combination:
\[ \frac{L_n \cdot f_n}{v_n^2} = \frac{\frac{nh}{2\pi} \cdot \frac{E_0 Z^2}{h}}{\left(\frac{Z e^2}{h} \cdot \frac{1}{n}\right)^2} \]
This simplifies to:
\[ \frac{nh E_0 Z^2}{2\pi} \cdot \frac{n^2 h^2}{Z^2 e^4} = \frac{n^3 E_0 h}{2\pi e^4} \]
This expression depends on `n` but not on `Z`.

4. **Conclusion**:
After analyzing all options, we find that:
- The combinations \( U_n \cdot T_n / L_n \) and \( L_n \cdot f_n / v_n^2 \) are independent of `Z` but dependent on `n`.

### Final Answer:
The options that are independent of both `n` and `Z` are:
- **None of the options provided are completely independent of both `n` and `Z`.**