Subtopic: Lagrangian & Hamiltonian Dynamics & Special Relativity
Question Statement:
A system of three non -identical spin
1
2 particles has the Hamiltonian
H =
A
ℏ2 ( S 1 + S 2) ⋅ S 3, where S 1, S 2 and S 3 are the spin operators of particles
labelled 1, 2 and 3 respectively and A is a constant with appropriate dimensions.
The set of possible energy eigenvalues of the system is
Options:
(A) 0, A
2 , -A
(B) 0, A
2 , - A
2
(C) 0, 3A
2 , - A
2
(D) 0, - 3A
2 , A
2
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.46)**
**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)**.
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.