A spin $\dfrac{1}{2}$ particle is in a spin up state along the $x$-axis (with unit vector $\hat{x}$) and is denoted as $\left|\dfrac{1}{2}, \dfrac{1}{2}\right\rangle_x$. What is the probability of finding the particle to be in a spin up state along the direction $\hat{x'}$, which lies in the $xy$-plane and makes an angle $\theta$ with respect to the positive $x$-axis, if such a measurement is made?
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For spin-1/2 particles, the probability amplitude of finding the particle in a spin state along a new direction is given by the square of the cosine of half the angle between the two directions.
For a spin $\dfrac{1}{2}$ particle, the probability amplitude for finding the particle in the state $\left| \dfrac{1}{2}, \dfrac{1}{2} \right\rangle$ along an axis making an angle $\theta$ with respect to the $x$-axis is given by:
\[
\left| \langle \dfrac{1}{2}, \dfrac{1}{2} \mid \hat{x'} \rangle \right|^2 = \cos^2{\left(\dfrac{\theta}{2}\right)}.
\]
Thus, the probability of finding the particle in the spin up state along the direction $\hat{x'}$ is $\cos^2{\left(\dfrac{\theta}{2}\right)}$, which corresponds to option (D).
