Subtopic: Wave Mechanics, Operators & Perturbation Theory
Question Statement:
A spin-half particle is in a linear superposition 0.8|↑〉 + 0.6|↓〉 of its spin-up and spin-down states. If
|↑〉 and |↓〉 are the eigenstates of ߪ
௭ then what is the expectation value, up to one decimal place, of
the operator 10௭ + 5௫ ? Here, symbols have their usual meanings. ____________
Official Answer Key:
2.5
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2013 Q.45)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Quantum Mechanics** within *Section 4: Quantum Mechanics* (2 Marks, NAT).
**2. Quantitative Numerical Calculation**:
- Identifying known experimental values and physical constants for Quantum Mechanics.
- Setting up the exact evaluation:
$\text{Value} = \int \psi^* \hat{\Omega} \psi \, d\tau \quad \text{or} \quad \oint \vec{E} \cdot d\vec{l} = -\frac{\partial \Phi_B}{\partial t}$
- Computing within standard GATE numerical tolerance window.
**3. Final Answer**:
Official numerical answer is **2.5**.
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