Subtopic: Wave Mechanics, Operators & Perturbation Theory
Question Statement:
A spin
1
2 particle is in a spin up state along the x-axis (with unit vector x̂) and
is denoted as |
1
2 ,
1
2〉x. What is the probability of finding the parti cle to be in a
spin up state along the direction x̂′, which lies in the xy-plane and makes an
angle 𝜃 with respect to the positive x-axis, if such a measurement is made?
Options:
(A) 1
2 cos2 𝜃
4
(B) cos2 𝜃
4
(C) 1
2 cos2 𝜃
2
(D) cos2 𝜃
2
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2023 Q.53)**
**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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