Word Count : 486

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Let's solve this step by step manually, using standard angular momentum algebra. --- ### i) Angular momentum states for ( j = 1/2 ) and matrix elements of (\hat{J}_x, \hat{J}_y, \hat{J}_z) For ( j = 1/2 ), the possible ( m_j ) values are: [ m_j = +\frac{1}{2}, -\frac{1}{2} ] So the states are: [ \left| \frac{1}{2}, \frac{1}{2} \right\rangle, \quad \left| \frac{1}{2}, -\frac{1}{2} \right\rangle ] Step 1: (\hat{J}_z) matrix By definition: [ \hat{J}_z \left| j, m_j \right\rangle = \hbar m_j \left| j, m_j \right\rangle ] So in the basis ( { \left| \frac{1}{2}, \frac{1}{2} \right\rangle, \left| \frac{1}{2}, __________ ______ ________ ____ ______ ________ __________ ___ ______.
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