This is an old revision of this page, as edited by I am One of Many(talk | contribs) at 09:12, 16 November 2016(Reverted 4 edits by 137.204.192.37 (talk): Provide an explanation and source for all mathematical changes on the talk page. (TW)). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.
Revision as of 09:12, 16 November 2016 by I am One of Many(talk | contribs)(Reverted 4 edits by 137.204.192.37 (talk): Provide an explanation and source for all mathematical changes on the talk page. (TW))
The Wigner D-matrix is a matrix in an irreducible representation of the groups SU(2) and SO(3). The complex conjugate of the D-matrix is an eigenfunction of the Hamiltonian of spherical and symmetric rigid rotors. The matrix was introduced in 1927 by Eugene Wigner.
Definition of the Wigner D-matrix
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. In all cases the three operators satisfy the following commutation relations,
commutes with all generators of the Lie algebra. Hence it may be diagonalized together with .
That is, it can be shown that there is a complete set of kets 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.
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
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,
and
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 algebra's generated by and .
An important property of the Wigner D-matrix follows from the commutation of
with the time reversal operator,
or
Here we used that is anti-unitary (hence the complex conjugation after moving
from ket to bra), and .
Orthogonality relations
The Wigner D-matrix elements form a complete 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:
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
^Wigner, E. P. (1931). Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren. Braunschweig: Vieweg Verlag. Translated into English by Griffin, J. J. (1959). Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. New York: Academic Press.
^Biedenharn, L. C.; Louck, J. D. (1981). Angular Momentum in Quantum Physics. Reading: Addison-Wesley. ISBN0-201-13507-8.
^Rose, M. E. Elementary Theory of Angular Momentum. New York, JOHN WILEY & SONS, 1957.
^Edén, M. (2003). "Computer simulations in solid-state NMR. I. Spin dynamics theory". Concepts Magn. Reson. 17A (1): 117–154. doi:10.1002/cmr.a.10061.