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Question 46
GATE PH 2025 • Q46
MCQ • 2 Marks
Section 1: Mathematical Physics
Linear Algebra & Vector Calculus
Subtopic: Matrices, Eigenvalues & Eigenvectors
Question Statement:
A system of three non -identical spin
1
2 particles has the Hamiltonian
H =
A
\hbar2 ( S 1 + S 2) ⋅ S 3, where S 1, S 2 and S 3 are the spin operators of particles
labelled 1, 2 and 3 respectively and A is a constant with appropriate dimensions.
The set of possible energy eigenvalues of the system is
Options:
(A) 0, A
2 , -A
(B) 0, A
2 , - A
2
(C) 0, 3A
2 , - A
2
(D) 0, - 3A
2 , A
2
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.46)**
**1. Quantum Mechanical Spin Coupling**:
- Three non-identical spin-1/2 particles with Hamiltonian:
$H = \frac{A}{\hbar^2}(\vec{S}_1 + \vec{S}_2) \cdot \vec{S}_3$
**2. Angular Momentum Addition**:
- Define intermediate coupled spin: $\vec{S}_{12} = \vec{S}_1 + \vec{S}_2$.
- Total spin of the three-particle system: $\vec{S} = \vec{S}_{12} + \vec{S}_3$.
- Squaring both sides:
$\vec{S}^2 = \vec{S}_{12}^2 + \vec{S}_3^2 + 2\vec{S}_{12} \cdot \vec{S}_3 \implies \vec{S}_{12} \cdot \vec{S}_3 = \frac{1}{2}\left(\vec{S}^2 - \vec{S}_{12}^2 - \vec{S}_3^2\right)$
- Hamiltonian eigenvalues:
$E = \frac{A}{2}\Big[S(S + 1) - S_{12}(S_{12} + 1) - S_3(S_3 + 1)\Big]$
Since particle 3 has spin $1/2$, $S_3(S_3 + 1) = \frac{1}{2}\left(\frac{3}{2}\right) = \frac{3}{4}$.
- Coupling particles 1 and 2 ($1/2 \otimes 1/2 = 0 \oplus 1$):
- **Case 1: $S_{12} = 0$ (Singlet)**:
$S = 0 \otimes 1/2 = 1/2 \implies S(S + 1) = 3/4$
$E = \frac{A}{2}\left[\frac{3}{4} - 0 - \frac{3}{4}\right] = 0$
- **Case 2: $S_{12} = 1$ (Triplet)**:
Total spin $S \in \{1 - 1/2, 1 + 1/2\} = \{1/2, 3/2\}$.
- For $S = 3/2$: $S(S + 1) = \frac{3}{2}\left(\frac{5}{2}\right) = \frac{15}{4}$.
$E = \frac{A}{2}\left[\frac{15}{4} - 2 - \frac{3}{4}\right] = \frac{A}{2}\left[\frac{12}{4} - 2\right] = \frac{A}{2}(3 - 2) = \frac{A}{2}$
- For $S = 1/2$: $S(S + 1) = \frac{3}{4}$.
$E = \frac{A}{2}\left[\frac{3}{4} - 2 - \frac{3}{4}\right] = \frac{A}{2}(-2) = -A$
- The distinct energy eigenvalues are **$0, \frac{A}{2}, -A$**.
**3. Official Verification**:
The verified official IIT answer is **Option (A)** (`0, A/2 , -A`).
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