Subtopic: Wave Mechanics, Operators & Perturbation Theory
Question Statement:
A particle has wavefunction
𝜓(x, y, z) = N z e-𝛼(x2+y2+z2) ,
where N is a normalization constant and 𝛼 is a positi ve constant. In this state,
which one of the following options represent s the eigenvalues of L2 and Lz
respectively?
Some values of Yℓ
m are :
Y0
0 = √
1
4𝜋 , Y1
0 = √
3
4𝜋 cos 𝜃 , Y1
±1 = ∓√
3
8𝜋 sin 𝜃 e±i𝜙
Options:
(A) 0 and 0
(B) ℏ2 and -ℏ
(C) 2ℏ2 and 0
(D) ℏ2 and ℏ
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2023 Q.49)**
**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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