GATE PH 2026 • Q41
MCQ • 2 Marks
Section 4: Quantum Mechanics
Quantum Mechanics
Subtopic: Wave Mechanics, Operators & Perturbation Theory
Question Statement:
The energy E and degeneracy d of the second excited state of a three-dimensional,
isotropic quantum harmonic oscillator with angular frequency 𝜔 are
Options:
-
(A) E = 7
2 ℏ𝜔, d = 6
-
(B) E = 7
2 ℏ𝜔, d = 3
-
(C) E = 5
2 ℏ𝜔, d = 3
-
(D) E = 5
2 ℏ𝜔, d = 6
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2026 Q.41)**
**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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