The Wigner D-matrix is a
unitary matrix
In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if
U^* U = UU^* = UU^ = I,
where is the identity matrix.
In physics, especially in quantum mechanics, the conjugate transpose is ...
in an
irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W ...
of the groups
SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
and
SO(3)
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition.
By definition, a rotation about the origin is a tr ...
. It was introduced in 1927 by
Eugene Wigner
Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his con ...
, 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 rotor
In rotordynamics, the rigid rotor is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rigi ...
s. The letter stands for ''Darstellung'', which means "representation" in German.
Definition of the Wigner D-matrix
Let be generators of the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
of SU(2) and SO(3). In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, these three operators are the components of a vector operator known as ''angular momentum''. Examples are the
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
of an electron in an atom,
electronic spin, and the angular momentum of a
rigid rotor
In rotordynamics, the rigid rotor is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rigi ...
.
In all cases, the three operators satisfy the following
commutation relations
In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example,
hat x,\hat p_ ...
,
:
where ''i'' is the purely
imaginary number
An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . Fo ...
and Planck's constant has been set equal to one. The
Casimir operator
In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the Center (ring theory), center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared ang ...
:
commutes with all generators of the Lie algebra. Hence, it may be diagonalized together with .
This defines the
spherical basis
In pure and applied mathematics, particularly quantum mechanics and computer graphics and their applications, a spherical basis is the basis used to express spherical tensors. The spherical basis closely relates to the description of angular mo ...
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, .
A 3-dimensional
rotation operator can be written as
:
where ''α'', ''β'', ''γ'' are
Euler angles
The Euler angles are three angles introduced by Leonhard Euler to describe the Orientation (geometry), orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189†...
(characterized by the keywords: z-y-z convention, right-handed frame, right-hand screw rule, active interpretation).
The Wigner D-matrix is a unitary square matrix of dimension 2''j'' + 1 in this spherical basis with elements
:
where
:
is an element of the orthogonal Wigner's (small) d-matrix.
That is, in this basis,
:
is diagonal, like the ''γ'' matrix factor, but unlike the above ''β'' factor.
Wigner (small) d-matrix
Wigner gave the following expression:
:
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 Euler angles are three angles introduced by Leonhard Euler to describe the Orientation (geometry), orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189†...
, 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
In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P_n^(x)
are a class of Classical orthogonal polynomials, classical orthogonal polynomials. They are orthogonal with respect to the weight
(1-x)^\alpha(1+x)^\beta ...
with nonnegative
and
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
In rotordynamics, the rigid rotor is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rigi ...
angular momentum operators.
Further,
:
which have quantum mechanical meaning: they are body-fixed
rigid rotor
In rotordynamics, the rigid rotor is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rigi ...
angular momentum operators.
The operators satisfy the
commutation relations
In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example,
hat x,\hat p_ ...
:
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
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W ...
of the isomorphic
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
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
.
A further symmetry implies
:
Orthogonality relations
The Wigner D-matrix elements
form a set of orthogonal functions of the Euler angles
and
:
:
This is a special case of the
Schur orthogonality relations In mathematics, the Schur orthogonality relations, which were proven by Issai Schur through Schur's lemma, express a central fact about representations of finite groups.
They admit a generalization to the case of compact groups in general, and in ...
.
Crucially, by the
Peter–Weyl theorem
In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, ...
, they further form a ''complete'' set.
The fact that
are matrix elements of a unitary transformation from one spherical basis
to another
is represented by the relations:
:
:
The
group character
In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about ...
s for SU(2) only depend on the rotation angle ''β'', being
class functions, so, then, independent of the axes of rotation,
:
and consequently satisfy simpler orthogonality relations, through the
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
This measure was introduced by Alfréd Haar in 1933, though ...
of the group,
:
The completeness relation (worked out in the same reference, (3.95)) is
:
whence, for
:
Kronecker product of Wigner D-matrices, Clebsch-Gordan series
The set of
Kronecker product
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to ...
matrices
:
forms a reducible matrix representation of the groups SO(3) and SU(2). Reduction into irreducible components is by the following equation:
[
:
The symbol is a Clebsch–Gordan coefficient.
]
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
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics form a ...
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 physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...
:
:
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''-''z''
convention is used frequently in molecular physics.
From the time-reversal property of the Wigner D-matrix follows immediately
:
There exists a more general relationship to the spin-weighted spherical harmonics
In special functions, a topic in mathematics, spin-weighted spherical harmonics are generalizations of the standard spherical harmonics and—like the usual spherical harmonics—are functions on the sphere. Unlike ordinary spherical harmonics, t ...
:
:
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:
:
Here, the eigenvector, , is a scaled and shifted discrete Chebyshev polynomial, and the corresponding eigenvalue, , is the Legendre polynomial.
Relation to Bessel functions
In the limit when we have
:
where is the Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
and is finite.
List of d-matrix elements
Using sign convention of Wigner, et al. the d-matrix elements
for ''j'' = 1/2, 1, 3/2, and 2 are given below.
for ''j'' = 1/2
:
for ''j'' = 1
:
for ''j'' = 3/2
:
for ''j'' = 2
:
Wigner d-matrix elements with swapped lower indices are found with the relation:
:
Symmetries and special cases
:
See also
* Clebsch–Gordan coefficients
In physics, the Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics. They appear as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. In ...
* Tensor operator
In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the ...
* Symmetries in quantum mechanics
References
External links
* {{cite web , first=C. , last=Amsler , author2=''et al.'' (Particle Data Group) , title=PDG Table of Clebsch-Gordan Coefficients, Spherical Harmonics, and d-Functions , date=2008 , work=Physics Letters B667 , url=http://pdg.lbl.gov/2008/reviews/clebrpp.pdf
Representation theory of Lie groups
Matrices
Special hypergeometric functions
Rotational symmetry