Calculate the matrix elements for J2 for a system of two spin half particles.
For a system of two spin-(\frac{1}{2}) particles, the total angular momentum operator is [ \vec{J} = \vec{S}_1 + \vec{S}_2 ] and the square of the total angular momentum is [ J^2 = (\vec{S}_1 + \vec{S}_2)^2 = S_1^2 + S_2^2 + 2 \vec{S}_1 \cdot \vec{S}_2. ] Each spin-(\frac{1}{2}) particle has [ S_1^2 = S_2^2 = s(s+1)\hbar^2 = \frac{3}{4}\hbar^2. ] The basis for two spin-(\frac{1}{2}) particles can be written as the product states: ______ _______ ______ ____ _____ ____ _____ _____ _____ ___ _________ _________.
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