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.
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,
![{\displaystyle [J_{x},J_{y}]=iJ_{z},\quad [J_{z},J_{x}]=iJ_{y},\quad [J_{y},J_{z}]=iJ_{x},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/174ba69063f90a76d451af6dea3b8e8519411e7c)
where i is the purely imaginary number and Planck's constant ħ has been set equal to one. The Casimir operator
![{\displaystyle J^{2}=J_{x}^{2}+J_{y}^{2}+J_{z}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1335a7eaecf80587621548a6d7f8fe69593acc)
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
![{\displaystyle J^{2}|jm\rangle =j(j+1)|jm\rangle ,\quad J_{z}|jm\rangle =m|jm\rangle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52dcc616da854cd6c6a37bd746d0893ad61ae731)
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.
A 3-dimensional rotation operator can be written as
![{\displaystyle {\mathcal {R}}(\alpha ,\beta ,\gamma )=e^{-i\alpha J_{z}}e^{-i\beta J_{y}}e^{-i\gamma J_{z}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fde179cd39ef848a6a7c7c3e02c4c91c5697fd17)
where α, β, γ are Euler angles (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 2j + 1 in this spherical basis with elements
![{\displaystyle D_{m'm}^{j}(\alpha ,\beta ,\gamma )\equiv \langle jm'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|jm\rangle =e^{-im'\alpha }d_{m'm}^{j}(\beta )e^{-im\gamma },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53fd7befce1972763f7f53f5bcf4dd158c324b55)
where
![{\displaystyle d_{m'm}^{j}(\beta )=\langle jm'|e^{-i\beta J_{y}}|jm\rangle =D_{m'm}^{j}(0,\beta ,0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c67153e772a45d5befff41554c133ab956c9d9)
is an element of the orthogonal Wigner's (small) d-matrix.
That is, in this basis,
![{\displaystyle D_{m'm}^{j}(\alpha ,0,0)=e^{-im'\alpha }\delta _{m'm}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c8c8f809213d88521d7535d1d5949af8ee7635b)
is diagonal, like the γ matrix factor, but unlike the above β factor.
Wigner (small) d-matrix
Wigner gave the following expression:[1]
![{\displaystyle d_{m'm}^{j}(\beta )=[(j+m')!(j-m')!(j+m)!(j-m)!]^{\frac {1}{2}}\sum _{s=s_{\mathrm {min} }}^{s_{\mathrm {max} }}\left[{\frac {(-1)^{m'-m+s}\left(\cos {\frac {\beta }{2}}\right)^{2j+m-m'-2s}\left(\sin {\frac {\beta }{2}}\right)^{m'-m+2s}}{(j+m-s)!s!(m'-m+s)!(j-m'-s)!}}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a61f599b558e2540e6d57e30d7f7c4dd2c1fa9b)
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
![{\displaystyle k=\min(j+m,j-m,j+m',j-m').}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1622a7c09ee4ead55ae1afcc437db957b15cab47)
If
![{\displaystyle k={\begin{cases}j+m:&a=m'-m;\quad \lambda =m'-m\\j-m:&a=m-m';\quad \lambda =0\\j+m':&a=m-m';\quad \lambda =0\\j-m':&a=m'-m;\quad \lambda =m'-m\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aac7da3c59e1509e97e25b1a06e5ca18405356b2)
Then, with
the relation is
![{\displaystyle d_{m'm}^{j}(\beta )=(-1)^{\lambda }{\binom {2j-k}{k+a}}^{\frac {1}{2}}{\binom {k+b}{b}}^{-{\frac {1}{2}}}\left(\sin {\frac {\beta }{2}}\right)^{a}\left(\cos {\frac {\beta }{2}}\right)^{b}P_{k}^{(a,b)}(\cos \beta ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57c54f1d263decec22ebd4489618d3e8fe53d49a)
where
It is also useful to consider the relations
, where
and
, which lead to:
![{\displaystyle d_{m'm}^{j}(\beta )=(-1)^{\frac {m-m'-|m-m'|}{2}}\left[{\frac {(j+M)!(j-M)!}{(j+N)!(j-N)!}}\right]^{\frac {1}{2}}\left(\sin {\frac {\beta }{2}}\right)^{|m-m'|}\left(\cos {\frac {\beta }{2}}\right)^{|m+m'|}P_{j-M}^{(|m-m'|,|m+m'|)}(\cos \beta ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff3f03eaf352e2d7b353dc5ab4a30ca23e25a3dc)
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
![{\displaystyle {\begin{aligned}{\hat {\mathcal {J}}}_{1}&=i\left(\cos \alpha \cot \beta {\frac {\partial }{\partial \alpha }}+\sin \alpha {\partial \over \partial \beta }-{\cos \alpha \over \sin \beta }{\partial \over \partial \gamma }\right)\\{\hat {\mathcal {J}}}_{2}&=i\left(\sin \alpha \cot \beta {\partial \over \partial \alpha }-\cos \alpha {\partial \over \partial \beta }-{\sin \alpha \over \sin \beta }{\partial \over \partial \gamma }\right)\\{\hat {\mathcal {J}}}_{3}&=-i{\partial \over \partial \alpha }\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72ad149c7fdecd836ac1b42468be3faefc4711c4)
which have quantum mechanical meaning: they are space-fixed rigid rotor angular momentum operators.
Further,
![{\displaystyle {\begin{aligned}{\hat {\mathcal {P}}}_{1}&=i\left({\cos \gamma \over \sin \beta }{\partial \over \partial \alpha }-\sin \gamma {\partial \over \partial \beta }-\cot \beta \cos \gamma {\partial \over \partial \gamma }\right)\\{\hat {\mathcal {P}}}_{2}&=i\left(-{\sin \gamma \over \sin \beta }{\partial \over \partial \alpha }-\cos \gamma {\partial \over \partial \beta }+\cot \beta \sin \gamma {\partial \over \partial \gamma }\right)\\{\hat {\mathcal {P}}}_{3}&=-i{\partial \over \partial \gamma },\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05ab04fafd28c2d9b22e31da817f817e83230fbe)
which have quantum mechanical meaning: they are body-fixed rigid rotor angular momentum operators.
The operators satisfy the commutation relations
![{\displaystyle \left[{\mathcal {J}}_{1},{\mathcal {J}}_{2}\right]=i{\mathcal {J}}_{3},\qquad {\hbox{and}}\qquad \left[{\mathcal {P}}_{1},{\mathcal {P}}_{2}\right]=-i{\mathcal {P}}_{3},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2326a21d58e05ac4f1307e913b9885b33458a16f)
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,
![{\displaystyle \left[{\mathcal {P}}_{i},{\mathcal {J}}_{j}\right]=0,\quad i,j=1,2,3,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6154fa1b814eb691be07d38efdd69930fb5d03cd)
and the total operators squared are equal,
![{\displaystyle {\mathcal {J}}^{2}\equiv {\mathcal {J}}_{1}^{2}+{\mathcal {J}}_{2}^{2}+{\mathcal {J}}_{3}^{2}={\mathcal {P}}^{2}\equiv {\mathcal {P}}_{1}^{2}+{\mathcal {P}}_{2}^{2}+{\mathcal {P}}_{3}^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7eb355fcfc06aef1084f4da01564cd0833384c41)
Their explicit form is,
![{\displaystyle {\mathcal {J}}^{2}={\mathcal {P}}^{2}=-{\frac {1}{\sin ^{2}\beta }}\left({\frac {\partial ^{2}}{\partial \alpha ^{2}}}+{\frac {\partial ^{2}}{\partial \gamma ^{2}}}-2\cos \beta {\frac {\partial ^{2}}{\partial \alpha \partial \gamma }}\right)-{\frac {\partial ^{2}}{\partial \beta ^{2}}}-\cot \beta {\frac {\partial }{\partial \beta }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b02331dfe0d639cd2e04239501c9683a6568cce)
The operators
act on the first (row) index of the D-matrix,
![{\displaystyle {\begin{aligned}{\mathcal {J}}_{3}D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}&=m'D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}\\({\mathcal {J}}_{1}\pm i{\mathcal {J}}_{2})D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}&={\sqrt {j(j+1)-m'(m'\pm 1)}}D_{m'\pm 1,m}^{j}(\alpha ,\beta ,\gamma )^{*}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52e39358e4e1308192c7f4e1d80cdb242bd30718)
The operators
act on the second (column) index of the D-matrix,
![{\displaystyle {\mathcal {P}}_{3}D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}=mD_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef753b1b40bf17dbab43fb5f60f2cdad862f2f71)
and, because of the anomalous commutation relation the raising/lowering operators are defined with reversed signs,
![{\displaystyle ({\mathcal {P}}_{1}\mp i{\mathcal {P}}_{2})D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}={\sqrt {j(j+1)-m(m\pm 1)}}D_{m',m\pm 1}^{j}(\alpha ,\beta ,\gamma )^{*}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/080ab1efecd56175bb304e83379aa3de6dc05e7f)
Finally,
![{\displaystyle {\mathcal {J}}^{2}D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}={\mathcal {P}}^{2}D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}=j(j+1)D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1715b74a7c8425a7fa722266d01bbd69f5f992a2)
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 operator T,
![{\displaystyle \langle jm'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|jm\rangle =\langle jm'|T^{\dagger }{\mathcal {R}}(\alpha ,\beta ,\gamma )T|jm\rangle =(-1)^{m'-m}\langle j,-m'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|j,-m\rangle ^{*},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb55cf261ceb966e4a7a43021eddee95a986468d)
or
![{\displaystyle D_{m'm}^{j}(\alpha ,\beta ,\gamma )=(-1)^{m'-m}D_{-m',-m}^{j}(\alpha ,\beta ,\gamma )^{*}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb22f2e20a44a0745ee736c152290ef45ac00164)
Here, we used that
is anti-unitary (hence the complex conjugation after moving
from ket to bra),
and
.
A further symmetry implies
![{\displaystyle (-1)^{m'-m}D_{mm'}^{j}(\alpha ,\beta ,\gamma )=D_{m'm}^{j}(\gamma ,\beta ,\alpha )~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c00f9907360ce47abd23eacf4031032e8a0d677b)
Orthogonality relations
The Wigner D-matrix elements
form a set of orthogonal functions of the Euler angles
and
:
![{\displaystyle \int _{0}^{2\pi }d\alpha \int _{0}^{\pi }d\beta \sin \beta \int _{0}^{2\pi }d\gamma \,\,D_{m'k'}^{j'}(\alpha ,\beta ,\gamma )^{\ast }D_{mk}^{j}(\alpha ,\beta ,\gamma )={\frac {8\pi ^{2}}{2j+1}}\delta _{m'm}\delta _{k'k}\delta _{j'j}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8cfc0c1de5b7bf3ac0edebadc1c24f0ed326501)
This is a special case of the Schur orthogonality relations.
Crucially, by the Peter–Weyl theorem, 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:[3]
![{\displaystyle \sum _{k}D_{m'k}^{j}(\alpha ,\beta ,\gamma )^{*}D_{mk}^{j}(\alpha ,\beta ,\gamma )=\delta _{m,m'},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a1f3c3d970e24ed5fcc70142aefc4fe55318cd3)
![{\displaystyle \sum _{k}D_{km'}^{j}(\alpha ,\beta ,\gamma )^{*}D_{km}^{j}(\alpha ,\beta ,\gamma )=\delta _{m,m'}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c43c9562e28332a79101edd00ebe7563fccbee73)
The group characters for SU(2) only depend on the rotation angle β, being class functions, so, then, independent of the axes of rotation,
![{\displaystyle \chi ^{j}(\beta )\equiv \sum _{m}D_{mm}^{j}(\beta )=\sum _{m}d_{mm}^{j}(\beta )={\frac {\sin \left({\frac {(2j+1)\beta }{2}}\right)}{\sin \left({\frac {\beta }{2}}\right)}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cae68d532007ca3f33b35f8ac4baae1a02ef584)
and consequently satisfy simpler orthogonality relations, through the Haar measure of the group,[4]
![{\displaystyle {\frac {1}{\pi }}\int _{0}^{2\pi }d\beta \sin ^{2}\left({\frac {\beta }{2}}\right)\chi ^{j}(\beta )\chi ^{j'}(\beta )=\delta _{j'j}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5955ccc10cbf12fe78dd907cfeb91ace948f40ca)
The completeness relation (worked out in the same reference, (3.95)) is
![{\displaystyle \sum _{j}\chi ^{j}(\beta )\chi ^{j}(\beta ')=\delta (\beta -\beta '),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/689d2afaf1bf6a26890753f772540fb87e9b0261)
whence, for
![{\displaystyle \sum _{j}\chi ^{j}(\beta )(2j+1)=\delta (\beta ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/155bc0a492f154b88b03aa7e8d388553cab0f47f)
Kronecker product of Wigner D-matrices, Clebsch-Gordan series
The set of Kronecker product matrices
![{\displaystyle \mathbf {D} ^{j}(\alpha ,\beta ,\gamma )\otimes \mathbf {D} ^{j'}(\alpha ,\beta ,\gamma )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c103138f36e0a7bfb41056fc20a4406cd4e86326)
forms a reducible matrix representation of the groups SO(3) and SU(2). Reduction into irreducible components is by the following equation:[3]
![{\displaystyle D_{mk}^{j}(\alpha ,\beta ,\gamma )D_{m'k'}^{j'}(\alpha ,\beta ,\gamma )=\sum _{J=|j-j'|}^{j+j'}\langle jmj'm'|J\left(m+m'\right)\rangle \langle jkj'k'|J\left(k+k'\right)\rangle D_{\left(m+m'\right)\left(k+k'\right)}^{J}(\alpha ,\beta ,\gamma )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d12ca93ad82b52559d74bb3b6af0a297e749d327)
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 and associated Legendre polynomials, normalized to unity and with Condon and Shortley phase convention:
![{\displaystyle D_{m0}^{\ell }(\alpha ,\beta ,\gamma )={\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell }^{m*}(\beta ,\alpha )={\sqrt {\frac {(\ell -m)!}{(\ell +m)!}}}\,P_{\ell }^{m}(\cos {\beta })\,e^{-im\alpha }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/757c680f6a1d19382224507bcde7bb8da6a23c8c)
This implies the following relationship for the d-matrix:
![{\displaystyle d_{m0}^{\ell }(\beta )={\sqrt {\frac {(\ell -m)!}{(\ell +m)!}}}\,P_{\ell }^{m}(\cos {\beta }).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f55456d4b0b3401417da07346d07978972f0ea6)
A rotation of spherical harmonics
then is effectively a composition of two rotations,
![{\displaystyle \sum _{m'=-\ell }^{\ell }Y_{\ell }^{m'}(\theta ,\phi )~D_{m'~m}^{\ell }(\alpha ,\beta ,\gamma ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a0e934c2427b53b3991195f782ba909d4a6aff0)
When both indices are set to zero, the Wigner D-matrix elements are given by ordinary Legendre polynomials:
![{\displaystyle D_{0,0}^{\ell }(\alpha ,\beta ,\gamma )=d_{0,0}^{\ell }(\beta )=P_{\ell }(\cos \beta ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b059d327da0f73054d99afde13198d77f37d061e)
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
![{\displaystyle \left(Y_{\ell }^{m}\right)^{*}=(-1)^{m}Y_{\ell }^{-m}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8493edab34cafc712c6bce708a3cc86ffade3d00)
There exists a more general relationship to the spin-weighted spherical harmonics:
[5]
Connection with transition probability under rotations
The absolute square of an element of the D-matrix,
![{\displaystyle F_{mm'}(\beta )=|D_{mm'}^{j}(\alpha ,\beta ,\gamma )|^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9adb41569d2aead69f6cbb20a9ddbfd06635e80)
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]
![{\displaystyle \sum _{m'=-j}^{j}F_{mm'}(\beta )f_{\ell }^{j}(m')=P_{\ell }(\cos \beta )f_{\ell }^{j}(m)\qquad (\ell =0,1,\ldots ,2j).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fecafe241af78da67a66846b31375cce4ce7266a)
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
![{\displaystyle D_{mm'}^{\ell }(\alpha ,\beta ,\gamma )\approx e^{-im\alpha -im'\gamma }J_{m-m'}(\ell \beta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5788325121218b06c03a88f14866637a8971fb89)
where
is the Bessel function 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
![{\displaystyle {\begin{aligned}d_{{\frac {1}{2}},{\frac {1}{2}}}^{\frac {1}{2}}&=\cos {\frac {\theta }{2}}\\[6pt]d_{{\frac {1}{2}},-{\frac {1}{2}}}^{\frac {1}{2}}&=-\sin {\frac {\theta }{2}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2e5474f4cbdd9e476644650d447f05a2bf6bcf2)
for j = 1
![{\displaystyle {\begin{aligned}d_{1,1}^{1}&={\frac {1}{2}}(1+\cos \theta )\\[6pt]d_{1,0}^{1}&=-{\frac {1}{\sqrt {2}}}\sin \theta \\[6pt]d_{1,-1}^{1}&={\frac {1}{2}}(1-\cos \theta )\\[6pt]d_{0,0}^{1}&=\cos \theta \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/016c3cd83ccd7f285334e8f03a4413183fc12ac4)
for j = 3/2
![{\displaystyle {\begin{aligned}d_{{\frac {3}{2}},{\frac {3}{2}}}^{\frac {3}{2}}&={\frac {1}{2}}(1+\cos \theta )\cos {\frac {\theta }{2}}\\[6pt]d_{{\frac {3}{2}},{\frac {1}{2}}}^{\frac {3}{2}}&=-{\frac {\sqrt {3}}{2}}(1+\cos \theta )\sin {\frac {\theta }{2}}\\[6pt]d_{{\frac {3}{2}},-{\frac {1}{2}}}^{\frac {3}{2}}&={\frac {\sqrt {3}}{2}}(1-\cos \theta )\cos {\frac {\theta }{2}}\\[6pt]d_{{\frac {3}{2}},-{\frac {3}{2}}}^{\frac {3}{2}}&=-{\frac {1}{2}}(1-\cos \theta )\sin {\frac {\theta }{2}}\\[6pt]d_{{\frac {1}{2}},{\frac {1}{2}}}^{\frac {3}{2}}&={\frac {1}{2}}(3\cos \theta -1)\cos {\frac {\theta }{2}}\\[6pt]d_{{\frac {1}{2}},-{\frac {1}{2}}}^{\frac {3}{2}}&=-{\frac {1}{2}}(3\cos \theta +1)\sin {\frac {\theta }{2}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3abde770b0ef91fba9d3d318f7cc82387361e913)
for j = 2[8]
![{\displaystyle {\begin{aligned}d_{2,2}^{2}&={\frac {1}{4}}\left(1+\cos \theta \right)^{2}\\[6pt]d_{2,1}^{2}&=-{\frac {1}{2}}\sin \theta \left(1+\cos \theta \right)\\[6pt]d_{2,0}^{2}&={\sqrt {\frac {3}{8}}}\sin ^{2}\theta \\[6pt]d_{2,-1}^{2}&=-{\frac {1}{2}}\sin \theta \left(1-\cos \theta \right)\\[6pt]d_{2,-2}^{2}&={\frac {1}{4}}\left(1-\cos \theta \right)^{2}\\[6pt]d_{1,1}^{2}&={\frac {1}{2}}\left(2\cos ^{2}\theta +\cos \theta -1\right)\\[6pt]d_{1,0}^{2}&=-{\sqrt {\frac {3}{8}}}\sin 2\theta \\[6pt]d_{1,-1}^{2}&={\frac {1}{2}}\left(-2\cos ^{2}\theta +\cos \theta +1\right)\\[6pt]d_{0,0}^{2}&={\frac {1}{2}}\left(3\cos ^{2}\theta -1\right)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4371f5663c4a9625357ce17d64397dd63f981117)
Wigner d-matrix elements with swapped lower indices are found with the relation:
![{\displaystyle d_{m',m}^{j}=(-1)^{m-m'}d_{m,m'}^{j}=d_{-m,-m'}^{j}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3583cd6541490a4efa12eb0081c27c52d6f0898a)
Symmetries and special cases
![{\displaystyle {\begin{aligned}d_{m',m}^{j}(\pi )&=(-1)^{j-m}\delta _{m',-m}\\[6pt]d_{m',m}^{j}(\pi -\beta )&=(-1)^{j+m'}d_{m',-m}^{j}(\beta )\\[6pt]d_{m',m}^{j}(\pi +\beta )&=(-1)^{j-m}d_{m',-m}^{j}(\beta )\\[6pt]d_{m',m}^{j}(2\pi +\beta )&=(-1)^{2j}d_{m',m}^{j}(\beta )\\[6pt]d_{m',m}^{j}(-\beta )&=d_{m,m'}^{j}(\beta )=(-1)^{m'-m}d_{m',m}^{j}(\beta )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1485b033a390e8fae8a19860ef2a90509dc5ddba)
See also
References
- ^ Wigner, E. P. (1951) [1931]. Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren. Braunschweig: Vieweg Verlag. OCLC 602430512. Translated into English by Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. Translated by Griffin, J.J. Elsevier. 2013 [1959]. ISBN 978-1-4832-7576-5.
- ^ Biedenharn, L. C.; Louck, J. D. (1981). Angular Momentum in Quantum Physics. Reading: Addison-Wesley. ISBN 0-201-13507-8.
- ^ a b Rose, Morris Edgar (1995) [1957]. Elementary theory of angular momentum. Dover. ISBN 0-486-68480-6. OCLC 31374243.
- ^ Schwinger, J. (January 26, 1952). On Angular Momentum (Technical report). Harvard University, Nuclear Development Associates. doi:10.2172/4389568. NYO-3071, TRN: US200506%%295.
- ^ Shiraishi, M. (2013). "Appendix A: Spin-Weighted Spherical Harmonic Function" (PDF). Probing the Early Universe with the CMB Scalar, Vector and Tensor Bispectrum (PhD). Nagoya University. pp. 153–4. ISBN 978-4-431-54180-6.
- ^ Meckler, A. (1958). "Majorana formula". Physical Review. 111 (6): 1447. doi:10.1103/PhysRev.111.1447.
- ^ 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. S2CID 121648820.
- ^ 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.
External links
- Amsler, C.; et al. (Particle Data Group) (2008). "PDG Table of Clebsch-Gordan Coefficients, Spherical Harmonics, and d-Functions" (PDF). Physics Letters B667.