GATE 2023 PH – Question 53

Correct answer: (D) $\cos^2(\theta/2)$

Explanation

Let the spin be up along $x$, $|+x\rangle$. We want the probability to find it up along an axis in the $xy$-plane at angle $\theta$ to $x$.

**Spin-½ state along a direction $\hat n$:** for two directions separated by an angle $\theta$, the overlap of the "up" states is
$$|\langle+\hat n|+\hat x\rangle|=\cos\frac\theta2 .$$

(Reason: the spin-½ spinor for the direction at polar angle $\vartheta$ is $\left(\cos\tfrac\vartheta2,\ \sin\tfrac\vartheta2\,e^{i\phi}\right)$, so half-angles appear.)

**Probability:**
$$P=\cos^2\frac\theta2\quad(\text{option D}).$$

Check: $\theta=0$ gives 1, $\theta=90^\circ$ gives $\tfrac12$, and $\theta=180^\circ$ gives 0.