### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Governing Concept & Domain Context:** * This question evaluates **Fundamentals of Fourier and Laplace transforms** under *Fourier and Laplace transforms* in GATE PH. 2. **Stage 2: Mathematical Derivation:** * Zero-point energy: $E_0 = \left(0 + \frac{1}{2}\right)\hbar\omega = \frac{1}{2}\hbar\omega$. Option A is correct. 3. **Stage 3: Distractor Elimination Forensics:** * The evaluated correct answer is derived using standard physical equations. * Distractor values correspond to typical unit conversion omissions or inverted formulas. 4. **Stage 4: Common Student Trap:** * *Trap:* Assuming the ground state energy is zero as in classical mechanics. *Reference: George B. Arfken, Hans J. Weber, Mathematical Methods for Physicists (Academic Press).*