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 generated by dimensional angular momentum operators, defineThe irreducible spin representation matrix isThe factors of are required because . One may equivalently use dimensionless generators .
For spin one-half, the middle matrix isA full rotation acts as . Only integer-spin representations descend to the three-dimensional special orthogonal group; half-integer representations distinguish the two lifts of the same spatial rotation.