GATE PH 2022 • Q14
MCQ • 1 Mark
Section 4: Quantum Mechanics
Angular momentum & Hydrogen Atom
Subtopic: Addition of Angular Momenta & Clebsch-Gordan Coefficients
Question Statement:
An excited state of Ca atom is [Mg]3p54s23d1. The spectroscopic terms corresponding to
the total orbital angular momentum are
Options:
-
(A) S, P, and D
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(B) P, D, and F
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(C) P and D
-
(D) S and P
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.14)**
**1. Atomic Physics & Spectroscopic Term Formulation**:
- Given electronic configuration of Ca: $[\text{Mg}] 3p^5 4s^2 3d^1$.
- Closed subshells ($[\text{Mg}]$ and $4s^2$) contribute $L = 0, S = 0$.
- Optically active electrons/holes:
- Subshell $3p^5$ has one hole with orbital angular momentum $l_1 = 1$.
- Subshell $3d^1$ has one electron with orbital angular momentum $l_2 = 2$.
**2. Step-by-Step Addition of Angular Momenta**:
- By the Clebsch-Gordan series for orbital angular momenta:
$L = |l_1 - l_2|, |l_1 - l_2| + 1, \dots, l_1 + l_2$
$L = |1 - 2|, |1 - 2| + 1, \dots, 1 + 2 = 1, 2, 3$
- Spectroscopic notation mapping:
- $L = 1 \implies \text{P}$
- $L = 2 \implies \text{D}$
- $L = 3 \implies \text{F}$
- Therefore, the spectroscopic terms corresponding to the total orbital angular momentum are **P, D, and F**.
**3. Official Verification**:
The verified official IIT Master Answer is **Option (B)** (`P, D, and F`).
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