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

Explanation

To determine which statements are true for the electronic state \( \Psi \) of \( \text{He}^+ \) with the given energy, azimuthal quantum number, and magnetic quantum number, we can follow these steps:

### Step 1: Understand the Energy of the Electron in a Hydrogen-like Atom
The energy of an electron in a hydrogen-like atom is given by the formula:
\[ E_n = -\frac{13.6 Z^2}{n^2} \text{ eV} \]
where \( Z \) is the atomic number and \( n \) is the principal quantum number.

### Step 2: Identify the Atomic Number and Energy
For \( \text{He}^+ \), the atomic number \( Z = 2 \). The given energy of the state \( \Psi \) is \( -3.4 \text{ eV} \).

### Step 3: Set Up the Equation
Substituting the values into the energy formula:
\[ -3.4 = -\frac{13.6 \times 2^2}{n^2} \]

### Step 4: Solve for \( n^2 \)
Rearranging the equation gives:
\[ 3.4 = \frac{54.4}{n^2} \]
Multiplying both sides by \( n^2 \) and rearranging:
\[ 3.4 n^2 = 54.4 \]
\[ n^2 = \frac{54.4}{3.4} = 16 \]
Thus, \( n = 4 \).

### Step 5: Determine the Azimuthal Quantum Number
The azimuthal quantum number \( l \) is given as 2. For \( l = 2 \), the corresponding subshell is \( d \).

### Step 6: Identify the State
Since \( n = 4 \) and \( l = 2 \), the state is referred to as \( 4d \).

### Step 7: Calculate the Number of Radial and Angular Nodes
- **Radial Nodes**: The formula for radial nodes is given by \( n - l - 1 \):
\[ \text{Radial Nodes} = 4 - 2 - 1 = 1 \]
- **Angular Nodes**: The number of angular nodes is equal to \( l \):
\[ \text{Angular Nodes} = l = 2 \]

### Step 8: Analyze the Nuclear Charge
For \( \text{He}^+ \), there is only one electron, so the effective nuclear charge experienced by the electron is equal to the actual nuclear charge, which is \( 2e \). There is no shielding effect since there are no other electrons.

### Conclusion
Based on the analysis:
1. The state \( \Psi \) is indeed a \( 4d \) state.
2. The nuclear charge experienced by the electron is equal to \( 2e \), not less than \( 2e \).
3. The number of radial nodes is 1 and the number of angular nodes is 2.

### True Statements
- The state \( \Psi \) is a \( 4d \) state.
- The nuclear charge is equal to \( 2e \).
- The number of radial nodes is 1.
- The number of angular nodes is 2.