Subtopic: Wave Mechanics, Operators & Perturbation Theory
Question Statement:
The wavefunction of a particle in an infinite one-dimensional potential well at time
t is
Ψ(x, t) = √2
3 e-iE1t ℏ⁄ 𝜓1(x) + 1
√6
ei𝜋 6⁄ e-iE2t ℏ⁄ 𝜓2(x) + 1
√6
ei𝜋 4⁄ e-iE3t ℏ⁄ 𝜓3(x)
where 𝜓1, 𝜓2 and 𝜓3 are the normalized ground state, the normalized first excited
state and the normalized second excited state, respectively. E1, E2 and E3 are the
eigen-energies corresponding to 𝜓1, 𝜓2 and 𝜓3, respectively. The expectation value
of energy of the particle in state Ψ(x, t) is
Options:
(A) 17
6 E1
(B) 2
3 E1
(C) 3
2 E1
(D) 14 E1
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2024 Q.13)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Quantum Mechanics** within *Section 4: Quantum Mechanics* (1 Mark, 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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