Subtopic: Wave Mechanics, Operators & Perturbation Theory
Question Statement:
An electron in the ground state of the hydrogen atom has the wave function
Ѱ(rr⃗) = 1
�𝜋𝜋aa0
3
ee-�rr aa0� �
where aa0 is constant. The expectation value of the operator 2 2ˆ r z Q- =, where θcosr z= is
(Hint: ∫ ee-𝛼𝛼 rr rrnn ddrr =
𝛤𝛤(nn)
𝛼𝛼 nn +1 =
(nn-1)!
𝛼𝛼 nn +1
∞
0 )
(A) -aa0
2/2 (B) -aa0
2 (C) -3aa0
2/2 (D) -2aa0
2
Official Answer Key:
2.5
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2014 Q.27)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Quantum Mechanics** within *Section 4: Quantum Mechanics* (1 Mark, NAT).
**2. Quantitative Numerical Calculation**:
- Identifying known experimental values and physical constants for Quantum Mechanics.
- Setting up the exact evaluation:
$\text{Value} = \int \psi^* \hat{\Omega} \psi \, d\tau \quad \text{or} \quad \oint \vec{E} \cdot d\vec{l} = -\frac{\partial \Phi_B}{\partial t}$
- Computing within standard GATE numerical tolerance window.
**3. Final Answer**:
Official numerical answer is **2.5**.
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