Subtopic: Wave Mechanics, Operators & Perturbation Theory
Question Statement:
An electron in the Coulomb field of a proton is in the following state of coherent
superposition of orthonormal states 𝜓nlm
Ψ = 1
3 𝜓100 + 1
√3
𝜓210 - √5
3 𝜓320
Let E1, E2, and E3 represent the first three energy levels of the system. A sequence
of measurements is done on the same system at different times. Energy is measured
first at time t1 and the outcome is E2. Then total angular momentum is measured at
time t2 > t1 and finally energy is measured again at t3 > t2. The probability of
finding the system in a state with energy E2 after the final measurement is P/9.
The value of P is _____ (in integer).
Official Answer Key:
2.5
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2024 Q.64)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Quantum Mechanics** within *Section 4: Quantum Mechanics* (2 Marks, 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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