GATE 2025 PH – Question 46
Three non-identical spin-half particles have Hamiltonian $H=\frac A{\hbar^2}(\mathbf S_1+\mathbf S_2)\cdot\mathbf S_3$, where the spin operators label particles 1,2,3 and $A$ has appropriate dimensions. The possible energy eigenvalues are
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Correct answer: (A) $0,A/2,-A$
Explanation
Let $\mathbf S_{12}=\mathbf S_1+\mathbf S_2$ and $\mathbf J=\mathbf S_{12}+\mathbf S_3$. Then
$$\mathbf S_{12}\cdot\mathbf S_3=\frac12\left[J^2-S_{12}^2-S_3^2\right],$$
so, in units with $\hbar=1$,
$$E=\frac A2\left[J(J+1)-s_{12}(s_{12}+1)-\tfrac34\right].$$
Two spin-½ particles couple to $s_{12}=0$ or $1$.
- **$s_{12}=0$:** $J=\tfrac12$, so $E=\dfrac A2\left[\tfrac34-0-\tfrac34\right]=\mathbf0$.
- **$s_{12}=1$, $J=\tfrac32$:** $E=\dfrac A2\left[\tfrac{15}4-2-\tfrac34\right]=\dfrac A2(1)=\mathbf{\dfrac A2}$.
- **$s_{12}=1$, $J=\tfrac12$:** $E=\dfrac A2\left[\tfrac34-2-\tfrac34\right]=\dfrac A2(-2)=\mathbf{-A}$.
The eigenvalues are $0$, $A/2$ and $-A$ (option A).