Question

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

• A. 5/18
• B. 11/18
• C. 15/18
• D. 17/18

Hint:

For spin-1/2 particles, probabilities of measurements are calculated by projecting the state onto the relevant eigenstates of the operator.

Answer 1

The Correct Option is D

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).