Subtopic: Wave Mechanics, Operators & Perturbation Theory
Question Statement:
For a one-dimensional harmonic oscillator, the creation operator (aற) acting on the
nth state |𝜓⟩, where n = 0, 1, 2, …, gives aற|𝜓⟩ = √n + 1|𝜓ାଵ⟩. The matrix
representation of the position operator x = ට ℏ
ଶఠ (a + aற) for the first three rows
and columns is
Options:
(A) ඨ ℏ
2m𝜔 ቌ
1 0 0
0 √2 0
0 0 √3
ቍ
(B) ඨ ℏ
2m𝜔 ൭
0 1 0
1 0 1
0 1 0
൱
(C) ඨ ℏ
2m𝜔 ቌ
0 1 0
1 0 √2
0 √2 0
ቍ
(D) ඨ ℏ
2m𝜔 ቌ
1 0 √3
0 0 0
√3 0 1
ቍ
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.45)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Quantum Mechanics** within *Section 4: Quantum Mechanics* (2 Marks, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Quantum 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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