Dmatrices

Dmatrices, commonly called Wigner D-matrices, are the matrix elements of the irreducible representations of the rotation group SO(3) (and its double cover SU(2)). They describe how angular momentum states transform under three-dimensional rotations and are fundamental in quantum mechanics, spectroscopy, and molecular physics. For a given total angular momentum j (a nonnegative integer or half-integer), the D-matrix has dimension (2j+1) with row and column indices m and m' running from -j to +j.

The Wigner D-matrix for a rotation specified by the Euler angles (α, β, γ) is written as D^j_{m m'}(α, β, γ) = e^{-i m α} d^j_{m m'}(β) e^{-i m' γ}, where d^j_{m m'}(β) are the small-d functions depending only on the polar angle β. Explicit forms of d^j_{m m'}(β) exist for each j, and simple closed forms are available for small j (for example j = 0, 1/2, 1). The complete D^j(α, β, γ) acts as the matrix representing the rotation in the j-th spin or orbital angular momentum space.

Key properties include unitarity and orthogonality, with the group multiplication rule D^j(R1) D^j(R2) = D^j(R1 R2) and the orthogonality relation ∫ D^j_{m m'}(R) [D^{j'}_{n n'}(R)]^* dR = (8π^2/(2j+1)) δ_{j j'} δ_{m n} δ_{m' n'}. The D-matrices form a complete basis for rotating tensor operators and are used to couple angular momenta, transform spherical tensors, and analyze rotational spectra and transition amplitudes in various physical systems.