Irreducible representation of the rotation group SO
The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3). It was introduced in 1927 by Eugene Wigner, and plays a fundamental role in the quantum mechanical theory of angular momentum. The complex conjugate of the D-matrix is an eigenfunction of the Hamiltonian of spherical and symmetric rigid rotors. The letter D stands for Darstellung, which means "representation" in German.
Let Jx, Jy, Jz be generators of the Lie algebra of SU(2) and SO(3). In quantum mechanics, these three operators are the components of a vector operator known as angular momentum. Examples are the angular momentum of an electron in an atom, electronic spin, and the angular momentum of a rigid rotor.
commutes with all generators of the Lie algebra. Hence, it may be diagonalized together with Jz.
This defines the spherical basis used here. That is, there is a complete set of kets (i.e. orthonormal basis of joint eigenvectors labelled by quantum numbers that define the eigenvalues) with
where j = 0, 1/2, 1, 3/2, 2, ... for SU(2), and j = 0, 1, 2, ... for SO(3). In both cases, m = −j, −j + 1, ..., j.
The sum over s is over such values that the factorials are nonnegative, i.e. , .
Note: The d-matrix elements defined here are real. In the often-used z-x-z convention of Euler angles, the factor in this formula is replaced by causing half of the functions to be purely imaginary. The realness of the d-matrix elements is one of the reasons that the z-y-z convention, used in this article, is usually preferred in quantum mechanical applications.
The d-matrix elements are related to Jacobi polynomials with nonnegative and [2] Let
If
Then, with the relation is
where
Properties of the Wigner D-matrix
The complex conjugate of the D-matrix satisfies a number of differential properties that can be formulated concisely by introducing the following operators with
which have quantum mechanical meaning: they are space-fixed rigid rotor angular momentum operators.
Further,
which have quantum mechanical meaning: they are body-fixed rigid rotor angular momentum operators.
and the corresponding relations with the indices permuted cyclically. The satisfy anomalous commutation relations (have a minus sign on the right hand side).
The two sets mutually commute,
and the total operators squared are equal,
Their explicit form is,
The operators act on the first (row) index of the D-matrix,
The operators act on the second (column) index of the D-matrix,
and, because of the anomalous commutation relation the raising/lowering operators are defined with reversed signs,
Finally,
In other words, the rows and columns of the (complex conjugate) Wigner D-matrix span irreducible representations of the isomorphic Lie algebras generated by and .
An important property of the Wigner D-matrix follows from the commutation of with the time reversal operatorT,
or
Here, we used that is anti-unitary (hence the complex conjugation after moving from ket to bra), and .
A further symmetry implies
Orthogonality relations
The Wigner D-matrix elements form a set of orthogonal functions of the Euler angles and :
Relation to spherical harmonics and Legendre polynomials
For integer values of , the D-matrix elements with second index equal to zero are proportional to spherical harmonics and associated Legendre polynomials, normalized to unity and with Condon and Shortley phase convention:
This implies the following relationship for the d-matrix:
A rotation of spherical harmonics then is effectively a composition of two rotations,
When both indices are set to zero, the Wigner D-matrix elements are given by ordinary Legendre polynomials:
In the present convention of Euler angles, is a longitudinal angle and is a colatitudinal angle (spherical polar angles in the physical definition of such angles). This is one of the reasons that the z-y-zconvention is used frequently in molecular physics. From the time-reversal property of the Wigner D-matrix follows immediately
Connection with transition probability under rotations
The absolute square of an element of the D-matrix,
gives the probability that a system with spin prepared in a state with spin projection along some direction will be measured to have a spin projection along a second direction at an angle to the first direction. The set of quantities itself forms a real symmetric matrix, that depends only on the Euler angle , as indicated.
Remarkably, the eigenvalue problem for the matrix can be solved completely:[6][7]
Here, the eigenvector, , is a scaled and shifted discrete Chebyshev polynomial, and the corresponding eigenvalue, , is the Legendre polynomial.
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↑ Mermin, N.D.; Schwarz, G.M. (1982). "Joint distributions and local realism in the higher-spin Einstein-Podolsky-Rosen experiment". Foundations of Physics. 12 (2): 101. doi:10.1007/BF00736844. S2CID121648820.
↑ Edén, M. (2003). "Computer simulations in solid-state NMR. I. Spin dynamics theory". Concepts in Magnetic Resonance Part A. 17A (1): 117–154. doi:10.1002/cmr.a.10061.
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