Word Count : 502

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To solve for the angular momentum states \( |j, m_j \rangle \) and determine the eigenvalues of \( \hat{J}^2 \) and \( \hat{J}_z \), let’s break it down step by step: ### Given: - \( j = \frac{5}{2} \), so \( m_j \) can range from \( -j \) to \( +j \) in integer steps. - The eigenvalue equation for \( \hat{J}^2 \) is: \[ \hat{J}^2 |j, m_j \rangle = j(j+1)\hbar^2 |j, m_j \rangle \] - The eigenvalue equation for \( \hat{J}_z \) is: \[ \hat{J}_z |j, m_j \rangle = m_j \hbar |j, m_j \rangle \] where \( m_j \) is the magnetic quantum number. ### Step 1: Determine \( |j, m_j \rangle \) states For \( j = \frac{5}{2} \), the possible values __________ _________ ______ ____ ____ ________ _________ _____.
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