GATE PH 2022 • Q23
MSQ • 1 Mark
Section 4: Quantum Mechanics
Angular momentum & Hydrogen Atom
Subtopic: Spin Angular Momentum & Pauli Spin Matrices
Question Statement:
Pauli spin matrices satisfy
(A) σα σβ - σβ σα = i εαβγ σγ
(B) σα σβ - σβ σα = 2i εαβγ σγ
(C) σα σβ + σβ σα = εαβγ σγ
(D) σα σβ + σβ σα = 2 δαβ
Options:
-
(A) σα σβ - σβ σα = i εαβγ σγ
-
(B) σα σβ - σβ σα = 2i εαβγ σγ
-
(C) σα σβ + σβ σα = εαβγ σγ
-
(D) σα σβ + σβ σα = 2 δαβ
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.23)**
**1. Pauli Spin Matrix Algebra**:
- The standard Pauli matrices $\sigma_x, \sigma_y, \sigma_z$ satisfy the algebraic identities:
$\sigma_\alpha \sigma_\beta = \delta_{\alpha\beta} \mathbb{I} + i \epsilon_{\alpha\beta\gamma} \sigma_\gamma$
**2. Detailed Step-by-Step Derivation**:
- **Commutator**:
$\sigma_\alpha \sigma_\beta - \sigma_\beta \sigma_\alpha = [\sigma_\alpha, \sigma_\beta] = 2i \epsilon_{\alpha\beta\gamma} \sigma_\gamma$
- Statement (A) has coefficient $i$ (Missing factor of 2). Incorrect.
- **Statement (B) is TRUE**: $\sigma_\alpha \sigma_\beta - \sigma_\beta \sigma_\alpha = 2i\epsilon_{\alpha\beta\gamma}\sigma_\gamma$.
- **Anticommutator**:
$\sigma_\alpha \sigma_\beta + \sigma_\beta \sigma_\alpha = \{\sigma_\alpha, \sigma_\beta\} = 2 \delta_{\alpha\beta} \mathbb{I}$
- Statement (C) is incorrect.
- **Statement (D) is TRUE**: $\sigma_\alpha \sigma_\beta + \sigma_\beta \sigma_\alpha = 2\delta_{\alpha\beta}$.
**3. Official Verification**:
The verified official IIT Master Answer is **Options (B, D)**.
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