Derivation of Standard Treatment of Spin Addition From Probability Amplitudes

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In a recent paper, we introduced a new way of treating systems of compounded angular momentum. We obtained the probability amplitudes for measurements on the systems and used these to derive the matrix treatment of compounded spin. However, the matrix forms are in 3- and 4- dimensional space and are therefore entirely different from the standard forms. This raises the question of the connection between these forms and the standard forms. In this paper, we answer this question. We not only derive the standard matrix treatment of spin addition - we discover a more generalized form of the theory. We apply the new generalized theory to the singlet and triplet states arising from the addition of the spins of two systems of spin 1/2 each. We obtain new generalized forms of the vectors and operators for these cases, and show that they reduce to the standard forms in the appropriate limit.