Quantum States and Energy Levels of Single Electron Helium Ion

To determine the correct statements regarding the electronic state $\Psi$ of the $\text{He}^+$ ion, we analyze the given parameters step-by-step:

1. Principal Quantum Number ($n$): The energy of a hydrogen-like species in the $n^{\text{th}}$ quantum state is given by: $E_n = -13.6 \times \frac{Z^2}{n^2} \text{ eV}$

   For a helium ion ($\text{He}^+$), the atomic number is $Z = 2$. Given that the energy of the state $\Psi$ is $E_n = -3.4 \text{ eV}$: $-3.4 = -13.6 \times \frac{2^2}{n^2}$ $-3.4 = -\frac{54.4}{n^2}$ $n^2 = \frac{54.4}{3.4} = 16 \implies n = 4$

2. Identification of the Electronic State:

   • Principal quantum number, $n = 4$
   • Azimuthal quantum number, $l = 2$ (which corresponds to a $d$ orbital)
   • Magnetic quantum number, $m_l = 0$

   Therefore, the state $\Psi$ represents a $4d$ state. $\implies$ Statement (A) is correct.

3. Calculation of Nodes:

   • Angular nodes: The number of angular nodes is equal to the azimuthal quantum number $l$: $\text{Number of angular nodes} = l = 2$ $\implies$ Statement (B) is correct.

   • Radial nodes: The number of radial nodes is given by the formula $n - l - 1$: $\text{Number of radial nodes} = 4 - 2 - 1 = 1$ $\implies$ Statement (C) is incorrect (it has $1$ radial node, not $3$).

4. Effective Nuclear Charge: Since $\text{He}^+$ is a single-electron system, there are no other electrons to cause screening or shielding effects. Hence, the effective nuclear charge experienced by the electron is exactly equal to the actual nuclear charge, $Ze = 2e$. $\implies$ Statement (D) is incorrect.

Correct Options: A and B