Subtopic: Lagrangian & Hamiltonian Dynamics & Special Relativity
Question Statement:
If the perturbation V = 𝜆x3 is added to the Hamiltonian of a one-dimensional harmonic
oscillator, the matrix element ⟨m|V|0⟩ is/are non-zero for which of the following states?
Here, the eigenstates of the harmonic oscillator are denoted by |n⟩.
Options:
(A) |m = 3⟩
(B) |m = 1⟩
(C) |m = 2⟩
(D) |m = 5⟩
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2026 Q.26)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Classical Mechanics** within *Section 2: Classical Mechanics* (1 Mark, 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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