Consider a spin \( S = \hbar/2 \) particle in the state \( | \phi \rangle = \frac{1}{\sqrt{3}} | 2 \rangle + i | 2 \rangle \). The probability that a measurement finds the state with \( S_x = + \hbar/2 \) is
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For spin-1/2 particles, probabilities of measurements are calculated by projecting the state onto the relevant eigenstates of the operator.
Step 1: The given state \( | \phi \rangle \) is a superposition of spin states. To find the probability of measuring \( S_x = + \hbar/2 \), we project the state onto the eigenstate of \( S_x \).
Step 2: The probability is given by the square of the absolute value of the overlap between \( | \phi \rangle \) and the \( S_x = + \hbar/2 \) eigenstate. Using the properties of spin-1/2 particles and the normalization, we calculate:
\[
P(S_x = + \hbar/2) = \left| \langle + | \phi \rangle \right|^2 = \frac{17}{18}
\]
Thus, the correct answer is (D).
