The ground state energy of hydrogen atom is `-13.6 eV`. Consider an electronic state `Psi` of `He^(+)` whose energy, azimuthal quantum number and magnetic quantum number are `-3.4 eV, 2` and 0, respectively. Which of the following statement(s) is(are) true for the state `Psi`?

Options

A. It has 3 radial nodes
B. It has 2 angular nodes
C. The nuclear charge experienced by the electron in this state is less than 2e, where e is the magnitude of the electronic charge
D. It is a 4d state

Answer

To solve the problem, we need to analyze the given electronic state of the He⁺ ion and determine which statements about it are true based on the provided energy, azimuthal quantum number (l), and magnetic quantum number (m).

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Energy (E) = -3.4 eV
- Azimuthal quantum number (l) = 2
- Magnetic quantum number (m) = 0

2. **Use the Energy Formula for Hydrogen-like Atoms:**
The energy of an electron in a hydrogen-like atom is given by the formula:
\[ E = -\frac{13.6 \, \text{eV} \cdot Z^2}{n^2} \]
where Z is the atomic number and n is the principal quantum number.

3. **Determine the Principal Quantum Number (n):**
For He⁺, Z = 2. We can rearrange the formula to find n:
\[ -3.4 = -\frac{13.6 \cdot 2^2}{n^2} \]
Simplifying this gives:
\[ 3.4 = \frac{54.4}{n^2} \]
\[ n^2 = \frac{54.4}{3.4} = 16 \]
\[ n = 4 \]

4. **Determine the Orbital Type:**
The azimuthal quantum number l = 2 corresponds to the d subshell. Since n = 4 and l = 2, this means the electron is in the 4d state.

5. **Calculate the Number of Angular Nodes:**
The number of angular nodes is equal to the azimuthal quantum number (l):
\[ \text{Number of angular nodes} = l = 2 \]

6. **Calculate the Number of Radial Nodes:**
The number of radial nodes can be calculated using the formula:
\[ \text{Number of radial nodes} = n - l - 1 = 4 - 2 - 1 = 1 \]

7. **Assess the Nuclear Charge Experienced by the Electron:**
In the case of He⁺, there is only one electron, and it experiences the full nuclear charge of +2e without any shielding effects from other electrons. Therefore, the nuclear charge experienced is equal to the atomic number Z = 2.

### Conclusion:
Based on the calculations:
- The azimuthal quantum number indicates that the state belongs to the d subshell.
- The number of angular nodes is 2.
- The number of radial nodes is 1.
- The nuclear charge experienced by the electron is equal to 2e.

### True Statements:
- The state belongs to the 4d subshell (True).
- The number of angular nodes is 2 (True).
- The nuclear charge experienced is equal to 2e (True).

### Final Answer:
The true statements are:
- The state belongs to the 4d subshell.
- The number of angular nodes is 2.