Subtopic: Wave Mechanics, Operators & Perturbation Theory
Question Statement:
A particle of mass m is moving inside a hollow spherical shell of radius a so that
the potential is
V(r) = ቄ 0 for r < a
∞ for r ≥ a
The ground state energy and wavefunction of the particle are E and R(r),
respectively. Then which of the following options are correct?
Options:
(A) E = ℏଶ 𝜋ଶ
2maଶ
(B) - ℏଶ
2m
1
rଶ
d
dr ൬rଶ dR
dr ൰ = ER (r < a)
(C) - ℏଶ
2m
1
rଶ
dଶR
drଶ = ER (r < a)
(D) R(r) = 1
r sin ቀ𝜋r
a ቁ (r < a)
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.48)**
**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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