Subtopic: Wave Mechanics, Operators & Perturbation Theory
Question Statement:
A simple harmonic oscillator with an angular frequency 𝜔 is in thermal
equilibrium with a reservoir at absolute temperature T, with 𝜔 =
2kBT
ℏ . Which
one of the following is the partition function of the system?
Options:
(A) e
e2 - 1
(B) e
e2 + 1
(C) e
e - 1
(D) e
e + 1
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2023 Q.38)**
**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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