Subtopic: Lagrangian & Hamiltonian Dynamics & Special Relativity
Question Statement:
Consider two non-identical spin
1
2 particles labelled 1 and 2 in the spin product
state |
1
2 ,
1
2〉 |
1
2 , -
1
2〉. The Hamiltonian of the system is
H =
4λ
ℏ2 S 1 ⋅ S 2 ,
where S 1 and S 2 are the spin operators of particles 1 and 2, respectively, and 𝜆
is a constant with appropriate dimensions. What is the expectation value of H in
the above state?
Options:
(A) -𝜆
(B) -2𝜆
(C) 𝜆
(D) 2𝜆
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2023 Q.52)**
**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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