Subtopic: Wave Mechanics, Operators & Perturbation Theory
Question Statement:
Two projectile protons P1 and P2 both with spin up (along the +z direction) are
scattered from another fixed target proton T with spin up at rest in the xy plane, as
shown in the figure. They scatter one at a time. The nuclear interaction potential
between both the projectiles and the target proton is šLā ā S , where Lā is the orbital
angular momentum of the system with respect to the target, S is the spin angular
momentum of the system and š is a negative constant in appropriate units. Which
one of the following is correct?
Options:
(A) P1 will be scattered in the +y direction (upward) and P2 will be scattered in the -y
direction (downward)
(B) P1 will be scattered in the +y direction (upward) and P2 will be scattered in the
+y direction (upward)
(C) P1 will be scattered in the -y direction (downward) and P2 will be scattered in the
+y direction (upward)
(D) P1 will be scattered in the -y direction (downward) and P2 will be scattered in
the -y direction (downward)
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.39)**
**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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