Subtopic: Wave Mechanics, Operators & Perturbation Theory
Question Statement:
The wavefunction for a particle is given by the form e-(i𝛼x+𝛽), where 𝛼 and 𝛽 are
real constants. In which one of the following potentials V(x), the particle is moving?
Options:
(A) V(x) ∝ 𝛼2x2
(B) V(x) ∝ e-𝛼x
(C) V(x) = 0
(D) V(x) ∝ sin(𝛼x)
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2024 Q.29)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Quantum Mechanics** within *Section 4: Quantum Mechanics* (1 Mark, 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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