A two -level quantum system has energy eigenvalues E1 and E2. A perturbing
potential H′ = 𝜆Δ𝜎x is introduced, where Δ is a constant having dimensions of
energy, 𝜆 is a small dimensionless parameter, and 𝜎x = (0 1
1 0). The magnitudes of
the first and the second order corrections to E1 due to H′, respectively, are
Options:
(A) 0 and
𝜆2Δ2
|E1-E2|
(B) |𝜆Δ|
2 and
𝜆2Δ2
|E1-E2|
(C) |𝜆Δ| and
𝜆2Δ2
|E1-E2|
(D) 0 and
1
2
𝜆2Δ2
|E1-E2|
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.44)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Nuclear & Particle Physics** within *Section 9: Nuclear and Particle Physics* (2 Marks, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Nuclear & Particle Physics:
- 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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