GATE PH 2022 • Q26
MSQ • 1 Mark
Section 1: Mathematical Physics
Mathematical Physics
Subtopic: Complex Analysis, Transforms & Differential Equations
Question Statement:
If g(k) is the Fourier transform of f(x), then which of the following are true?
Options:
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(A) g(-k) = +g∗(k) implies f(x) is real
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(B) g(-k) = -g∗(k) implies f(x) is purely imaginary
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(C) g(-k) = +g∗(k) implies f(x) is purely imaginary
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(D) g(-k) = -g∗(k) implies f(x) is real
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.26)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Mathematical Physics** within *Section 1: Mathematical Physics* (1 Mark, MSQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Mathematical Physics:
- 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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