Subtopic: Wave Mechanics, Operators & Perturbation Theory
Question Statement:
A particle of mass m in the x-y plane is confined in an infinite two-dimensional
well with vertices at (0, 0), (0, L), (L, L), (L, 0). The eigenfunctions of this particle
are 𝜓ೣ, =
ଶ
sin ቀ
ೣగ௫
ቁ sin ቀ
గ௬
ቁ . If perturbation of the form V = Cxy, where
C is a real constant, is applied, then which of the following statements are correct
for the first excited state?
Options:
(A) The unperturbed energy is
ଷగమℏమ
ଶమ
(B) The unperturbed energy is
ହగమℏమ
ଶమ
(C) First order energy shift due to the applied perturbation is zero
(D) The shift () in energy due to the applied perturbation is determined by an
equation of the form ቚa - 𝛿 b
b a - 𝛿ቚ = 0, where a and b are real, non-zero
constants
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.56)**
**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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