Two identical particles with a fixed total energy E = 2\hbar\omega are in thermal equilibrium in a
one-dimensional harmonic oscillator potential
1
2 m\omega2x2. Let the entropy of the particles
be denoted by SF if they are fermions with spin
1
2 (sz = ±
\hbar
2 ) and SB if they are
bosons with spin 0. Then, which of the following options is correct?
(kB is the Boltzmann constant)
Options:
(A) SF = kB ln 2, SB = kB ln 2
(B) SF = 2kB ln 2, SB = 0
(C) SF = 4kB ln 2, SB = 0
(D) SF = 2kB ln 2, SB = kB ln 2
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2026 Q.42)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *Section 4: Quantum Mechanics* | **Topic**: *Harmonic oscillator* (`Algebraic Method with Ladder Operators`)
- **Question Type**: **MCQ** | **Marks**: **2 Marks**
- **Theoretical Framework**: Governed by foundational physical principles in **Harmonic oscillator** (`Algebraic Method with Ladder Operators`).
**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): `SF = 2kB ln 2, SB = 0`**.
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.