Subtopic: Wave Mechanics, Operators & Perturbation Theory
Question Statement:
Consider a particle in a two dimensional infinite square well potential of side L,
with 0 ≤ x ≤ L and 0 ≤ y ≤ L. The wavefunction of the particle is zero only
along the line y =
L
2 , apart from the boundaries of the well. If the energy of the
particle in this state is E, what is the energy of the ground state?
Options:
(A) 1
4 E
(B) 2
5 E
(C) 3
8 E
(D) 1
2 E
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2023 Q.51)**
**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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