QB_PH_SEC6_003: Atomic Spectra & Multi-Electron Atoms

Calculate the Landé $g$-factor for the atomic state $^2P_{3/2}$. (Round off your answer to two decimal places)

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Quantum Numbers**: From the term symbol $^2P_{3/2}$: - Multiplicity $2S + 1 = 2 \implies S = 1/2$ - State letter $P \implies L = 1$ - Total angular momentum $J = 3/2$ 2. **Landé Formula Application**: $g = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)}$ 3. **Evaluation**: $J(J+1) = \frac{3}{2} \times \frac{5}{2} = \frac{15}{4}$ $S(S+1) = \frac{1}{2} \times \frac{3}{2} = \frac{3}{4}$ $L(L+1) = 1 \times 2 = 2$ $\text{Numerator} = \frac{15}{4} + \frac{3}{4} - 2 = \frac{18}{4} - 2 = 4.5 - 2 = 2.5 = \frac{5}{2}$ $\text{Denominator} = 2 \times \frac{15}{4} = \frac{15}{2}$ $g = 1 + \frac{5/2}{15/2} = 1 + \frac{1}{3} = \frac{4}{3} \approx 1.33$ 4. **Official Range**: $[1.32, 1.34]$.