QB_PH_SEC6_006: Atomic Spectra & Multi-Electron Atoms

For a two-electron system in the $(4p)(4d)$ configuration under $L-S$ coupling, what is the maximum possible value of the total angular momentum quantum number $J$?

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Angular Momentum Addition**: - Orbital angular momenta: $l_1 = 1$ ($p$-electron), $l_2 = 2$ ($d$-electron). - Possible total orbital angular momentum: $L \in \{|l_1 - l_2|, \dots, l_1 + l_2\} = \{1, 2, 3\}$. Hence, $L_{\text{max}} = 3$. 2. **Total Spin Addition**: - Individual spins: $s_1 = 1/2, s_2 = 1/2$. - Possible total spin: $S \in \{|s_1 - s_2|, \dots, s_1 + s_2\} = \{0, 1\}$. Hence, $S_{\text{max}} = 1$. 3. **Total Angular Momentum ($J$)**: - Under $L-S$ coupling, $\vec{J} = \vec{L} + \vec{S}$. - Maximum $J$: $J_{\text{max}} = L_{\text{max}} + S_{\text{max}} = 3 + 1 = 4$. 4. **Conclusion**: The maximum total angular momentum quantum number is $J = 4$. Option (B) is correct.