GATE 2023 PH – Question 52
Two nonidentical spin-half particles are in $|\uparrow\downarrow\rangle$. For $H=4\lambda\mathbf S_1\cdot\mathbf S_2/\hbar^2$, what is the energy expectation?
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Correct answer: (A) $-\lambda$
Explanation
With $H=\dfrac{4\lambda}{\hbar^2}\mathbf S_1\cdot\mathbf S_2$ and
$$\mathbf S_1\cdot\mathbf S_2=S_{1x}S_{2x}+S_{1y}S_{2y}+S_{1z}S_{2z}.$$
**Expectation value in $|\uparrow\downarrow\rangle$.**
- $\langle S_{1z}S_{2z}\rangle=\left(\tfrac\hbar2\right)\left(-\tfrac\hbar2\right)=-\dfrac{\hbar^2}{4}$.
- $S_{1x}S_{2x}+S_{1y}S_{2y}=\tfrac12\left(S_{1+}S_{2-}+S_{1-}S_{2+}\right)$ changes $|\uparrow\downarrow\rangle$ into $|\downarrow\uparrow\rangle$, which is orthogonal to the original state, so its expectation value is 0.
$$\langle H\rangle=\frac{4\lambda}{\hbar^2}\left(-\frac{\hbar^2}{4}\right)=\mathbf{-\lambda}\quad(\text{option A}).$$
(The state is not an eigenstate of $H$, but the expectation value is well defined.)