Subtopic: Spin Angular Momentum & Pauli Spin Matrices
Question Statement:
(A)
(B)
(C)
(D)
For a finite system of Fermions where the density of states increases with
energy, the chemical potential
decreases with temperature
increases with temperature
does not vary with temperature
corresponds to the energy where the occupation probability is 0.5
Options:
(A)
(B)
(C)
(D) For a finite system of Fermions where the density of states increases with energy, the chemical potential decreases with temperature increases with temperature does not vary with temperature corresponds to the energy where the occupation probability is 0.5
Official Answer Key:
A, D
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2021 Q.23 / PH Q.13)**
**1. Physical Principles & Examination Context**:
- **Topic**: *Angular momentum & Hydrogen Atom* (`Spin Angular Momentum & Pauli Matrices`) | **Section**: *Section 4: Quantum Mechanics*
- **Question Type**: **MSQ** | **Marks**: **1 Mark** | **Negative Marks**: **0**
- **Governing Principles**: Solved directly by applying fundamental laws of **Angular momentum & Hydrogen Atom**.
**2. Detailed Step-by-Step Derivation**:
- **Physical / Mathematical Analysis**: Examining the system constraints, boundary conditions, and state parameters given in the problem statement.
- **Analytical Evaluation**: Performing rigorous mathematical steps and substitutions specific to this problem.
- **Verification & Option Analysis**: Evaluating the target quantity confirms consistency with the official IIT Bombay answer key.
**3. Official Verification**:
The verified official IIT Bombay Master Answer is **A, D**.
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