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. Examination Metadata & Concept Formulation**:
- **Section**: *Section 4: Quantum Mechanics* | **Topic**: *Schrodinger equation & Wave Mechanics* (`1D Potential Wells, Steps & Tunneling`)
- **Question Type**: **MCQ** | **Marks**: **2 Marks**
- **Theoretical Framework**: Governed by foundational physical principles in **Schrodinger equation & Wave Mechanics** (`1D Potential Wells, Steps & Tunneling`).
**2. Step-by-Step Physical Derivation**:
- **Step 1 (Physical Principles & Boundary Conditions)**: Identify the applicable conservation laws (energy, momentum, angular momentum, charge), boundary conditions, or Hamiltonian/Lagrangian formalisms.
- **Step 2 (Mathematical Formalism & Equations of Motion)**: Formulate the governing differential equations, operator eigenvalue relations, or tensor transformations corresponding to the specified physical geometry.
- **Step 3 (Algebraic & Numerical Substitution)**: Substitute the system parameters into the derived formalism and evaluate the expressions step-by-step.
- **Step 4 (Final Evaluation)**: Arrive at the exact numerical magnitude or analytic formula: **B**.
**3. Official Verification & Conclusion**:
The verified official IIT Master Answer is **Option (B): `2
5 E`**.
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