The notation $^4D_{5/2}$ for an atomic state provides key quantum information:

• The superscript 4 represents the spin multiplicity ($2S+1 = 4$), which means the total spin quantum number is $S = 3/2$.
• The letter D indicates the total orbital angular momentum quantum number ($L = 2$).
• The subscript 5/2 indicates the total angular momentum quantum number ($J = 5/2$).

Analyzing Statement 1: Quantum Numbers

Statement 1 proposes $L=2, S=1/2$, and $J=5/2$. While $L=2$ and $J=5/2$ are correct for $^4D_{5/2}$, the spin quantum number $S=1/2$ would yield a spin multiplicity of $2(1/2)+1 = 2$, corresponding to a 2D state. The $^4D_{5/2}$ state requires $S=3/2$. Thus, statement 1 is incorrect.

Analyzing Statement 2: Electronic Configuration Origin

Statement 2 suggests that the $^4D_{5/2}$ state can originate from an $s^1p^2$ electronic configuration. This configuration involves electrons in s and p orbitals. The combination of angular momenta from these electrons can lead to various spectroscopic terms. Based on established principles of atomic spectroscopy, the $s^1p^2$ configuration is capable of producing states that correspond to the $^4D_{5/2}$ term.

Therefore, statement 2 is correct.

Analyzing Statement 3: Magnetic Field Splitting

When an atom is placed in a magnetic field, its energy levels split due to the Zeeman effect. The number of resulting sublevels depends on the total angular momentum $J$. Specifically, there are $2J+1$ possible values for the magnetic quantum number $M_J$, ranging from $-J$ to $+J$. For the $^4D_{5/2}$ state, $J=5/2$.

The number of split levels is calculated as:

The state splits into 6 levels, not 5. Hence, statement 3 is incorrect.

Analyzing Statement 4: Spectral Transition Possibility

Statement 4 considers a spectral transition from the $^4D_{5/2}$ state to the $^4P_{3/2}$ state. Such transitions must adhere to specific selection rules for electric dipole radiation:

• Change in total angular momentum: $\Delta J = 0, \pm 1$ (but $J=0 \to J=0$ is forbidden).
• Change in total spin: $\Delta S = 0$.
• Change in total orbital angular momentum: $\Delta L = 0, \pm 1$.

For the transition $^4D_{5/2} \to ^4P_{3/2}$:

• Initial state: $L=2, S=3/2, J=5/2$.
• Final state: $L=1, S=3/2, J=3/2$.

Checking the rules:

• $\Delta J = J_{final} - J_{initial} = 3/2 - 5/2 = -1$. This is allowed.
• $\Delta S = S_{final} - S_{initial} = 3/2 - 3/2 = 0$. This is allowed.
• $\Delta L = L_{final} - L_{initial} = 1 - 2 = -1$. This is allowed.

Since all selection rules are satisfied, this spectral transition is permitted. Therefore, statement 4 is correct.

Conclusion

The correct statements regarding the $^4D_{5/2}$ state are statements B and D.