Subtopic: Wave Mechanics, Operators & Perturbation Theory
Question Statement:
A gas of non-interacting 4He atoms (of mass m) is in a three-dimensional trap whose
energy levels can be approximated by those of a harmonic oscillator potential
V(x, y, z) =
1
2 m𝜔2(x2 + y2 + z2). The chemical potential of the gas at T=0 K is
Options:
(A) 0
(B) 1
2 ℏ𝜔
(C) 3
2 ℏ𝜔
(D) 3ℏ𝜔
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2026 Q.20)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Quantum Mechanics** within *Section 4: Quantum Mechanics* (1 Mark, 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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