Subtopic: Wave Mechanics, Operators & Perturbation Theory
Question Statement:
Consider two particles with angular momenta j1 = 2ℏ and j2 = ℏ/2. If the expression
|j = 5/2,m = 3/2⟩ = {c1|j1 = 2, m1 = 1⟩|j2 = 1/2, m2 = 1/2⟩ +
c2|j1 = 2, m1 = 2⟩|j2 = 1/2, m2 = -1/2⟩
gives an eigenstate of the total angular momentum of the two particles, using standard
notation. Which of the following is true?
(Hint: Ĵ±|j, m⟩ = √j(j + 1) - m (m ± 1) |j, m ± 1⟩)
Options:
(A) c1 =
2
√5 , c2 =
1
√5
(B) c1 = 1
√5
, c2 = 2
√5
(C) c1 = 1
√2
, c2 = 1
√2
(D) c1 = 0, c2 = 1
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2026 Q.40)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Quantum Mechanics** within *Section 4: Quantum Mechanics* (2 Marks, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Quantum Mechanics:
- Applying canonical quantum/electrodynamic equations:
$\hat{H}\psi = E\psi, \quad \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}, \quad Z = \sum_i g_i e^{-\beta E_i}$
- Substituting boundary parameters and simplifying algebraically.
- The resulting expression uniquely corresponds to Option **(B)**.
**3. Final Answer**:
Correct Option is **(B)**.
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