Subtopic: Lagrangian & Hamiltonian Dynamics & Special Relativity
Question Statement:
An electron with mass m and charge q is in the spin up state (1
0) at time t = 0. A
constant magnetic field is applied along the y-axis, B = B0 ĵ, where B0 is a
constant. The Hamiltonian of the system is H = - ℏ𝜔𝜎y, where 𝜔 =
qB0
2m > 0 and
𝜎y = (0 -i
i 0 ). The minimum time after which the electron will be in the spin down
state along the x-axis, i.e.,
1
√2 ( 1
-1) , is
Options:
(A) 𝜋
8𝜔
(B) 𝜋
4𝜔
(C) 𝜋
2𝜔
(D) 𝜋
𝜔
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.45)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Classical Mechanics** within *Section 2: Classical Mechanics* (2 Marks, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Classical 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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