The Wigner D-matrix is a mathematical construct used primarily in quantum mechanics and in the field of representation theory of the rotation group SO(3). It plays a significant role in angular momentum theory, particularly in the description of quantum states associated with rotations. ### Definition The Wigner D-matrix is defined for a specific angular momentum quantum state characterized by two quantum numbers: the total angular momentum \( j \) and the magnetic quantum number \( m \).
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For active rotations with , the irreducible spin representation matrix is , where . Its spin-one-half middle matrix is . A rotation acts by ; only integer-spin representations descend to the special orthogonal group in three dimensions.