- Source: Wigner D-matrix
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,
[
J
x
,
J
y
]
=
i
J
z
,
[
J
z
,
J
x
]
=
i
J
y
,
[
J
y
,
J
z
]
=
i
J
x
,
{\displaystyle [J_{x},J_{y}]=iJ_{z},\quad [J_{z},J_{x}]=iJ_{y},\quad [J_{y},J_{z}]=iJ_{x},}
where i is the purely imaginary number and the Planck constant ħ has been set equal to one. The Casimir operator
J
2
=
J
x
2
+
J
y
2
+
J
z
2
{\displaystyle J^{2}=J_{x}^{2}+J_{y}^{2}+J_{z}^{2}}
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
J
2
|
j
m
⟩
=
j
(
j
+
1
)
|
j
m
⟩
,
J
z
|
j
m
⟩
=
m
|
j
m
⟩
,
{\displaystyle J^{2}|jm\rangle =j(j+1)|jm\rangle ,\quad J_{z}|jm\rangle =m|jm\rangle ,}
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
R
(
α
,
β
,
γ
)
=
e
−
i
α
J
z
e
−
i
β
J
y
e
−
i
γ
J
z
,
{\displaystyle {\mathcal {R}}(\alpha ,\beta ,\gamma )=e^{-i\alpha J_{z}}e^{-i\beta J_{y}}e^{-i\gamma J_{z}},}
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
D
m
′
m
j
(
α
,
β
,
γ
)
≡
⟨
j
m
′
|
R
(
α
,
β
,
γ
)
|
j
m
⟩
=
e
−
i
m
′
α
d
m
′
m
j
(
β
)
e
−
i
m
γ
,
{\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 },}
where
d
m
′
m
j
(
β
)
=
⟨
j
m
′
|
e
−
i
β
J
y
|
j
m
⟩
=
D
m
′
m
j
(
0
,
β
,
0
)
{\displaystyle d_{m'm}^{j}(\beta )=\langle jm'|e^{-i\beta J_{y}}|jm\rangle =D_{m'm}^{j}(0,\beta ,0)}
is an element of the orthogonal Wigner's (small) d-matrix.
That is, in this basis,
D
m
′
m
j
(
α
,
0
,
0
)
=
e
−
i
m
′
α
δ
m
′
m
{\displaystyle D_{m'm}^{j}(\alpha ,0,0)=e^{-im'\alpha }\delta _{m'm}}
is diagonal, like the γ matrix factor, but unlike the above β factor.
Wigner (small) d-matrix
Wigner gave the following expression:
d
m
′
m
j
(
β
)
=
[
(
j
+
m
′
)
!
(
j
−
m
′
)
!
(
j
+
m
)
!
(
j
−
m
)
!
]
1
2
∑
s
=
s
m
i
n
s
m
a
x
[
(
−
1
)
m
′
−
m
+
s
(
cos
β
2
)
2
j
+
m
−
m
′
−
2
s
(
sin
β
2
)
m
′
−
m
+
2
s
(
j
+
m
−
s
)
!
s
!
(
m
′
−
m
+
s
)
!
(
j
−
m
′
−
s
)
!
]
.
{\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].}
The sum over s is over such values that the factorials are nonnegative, i.e.
s
m
i
n
=
m
a
x
(
0
,
m
−
m
′
)
{\displaystyle s_{\mathrm {min} }=\mathrm {max} (0,m-m')}
,
s
m
a
x
=
m
i
n
(
j
+
m
,
j
−
m
′
)
{\displaystyle s_{\mathrm {max} }=\mathrm {min} (j+m,j-m')}
.
Note: The d-matrix elements defined here are real. In the often-used z-x-z convention of Euler angles, the factor
(
−
1
)
m
′
−
m
+
s
{\displaystyle (-1)^{m'-m+s}}
in this formula is replaced by
(
−
1
)
s
i
m
−
m
′
,
{\displaystyle (-1)^{s}i^{m-m'},}
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
P
k
(
a
,
b
)
(
cos
β
)
{\displaystyle P_{k}^{(a,b)}(\cos \beta )}
with nonnegative
a
{\displaystyle a}
and
b
.
{\displaystyle b.}
Let
k
=
min
(
j
+
m
,
j
−
m
,
j
+
m
′
,
j
−
m
′
)
.
{\displaystyle k=\min(j+m,j-m,j+m',j-m').}
If
k
=
{
j
+
m
:
a
=
m
′
−
m
;
λ
=
m
′
−
m
j
−
m
:
a
=
m
−
m
′
;
λ
=
0
j
+
m
′
:
a
=
m
−
m
′
;
λ
=
0
j
−
m
′
:
a
=
m
′
−
m
;
λ
=
m
′
−
m
{\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}}}
Then, with
b
=
2
j
−
2
k
−
a
,
{\displaystyle b=2j-2k-a,}
the relation is
d
m
′
m
j
(
β
)
=
(
−
1
)
λ
(
2
j
−
k
k
+
a
)
1
2
(
k
+
b
b
)
−
1
2
(
sin
β
2
)
a
(
cos
β
2
)
b
P
k
(
a
,
b
)
(
cos
β
)
,
{\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 ),}
where
a
,
b
≥
0.
{\displaystyle a,b\geq 0.}
It is also useful to consider the relations
a
=
|
m
′
−
m
|
,
b
=
|
m
′
+
m
|
,
λ
=
m
−
m
′
−
|
m
−
m
′
|
2
,
k
=
j
−
M
{\displaystyle a=|m'-m|,b=|m'+m|,\lambda ={\frac {m-m'-|m-m'|}{2}},k=j-M}
, where
M
=
max
(
|
m
|
,
|
m
′
|
)
{\displaystyle M=\max(|m|,|m'|)}
and
N
=
min
(
|
m
|
,
|
m
′
|
)
{\displaystyle N=\min(|m|,|m'|)}
, which lead to:
d
m
′
m
j
(
β
)
=
(
−
1
)
m
−
m
′
−
|
m
−
m
′
|
2
[
(
j
+
M
)
!
(
j
−
M
)
!
(
j
+
N
)
!
(
j
−
N
)
!
]
1
2
(
sin
β
2
)
|
m
−
m
′
|
(
cos
β
2
)
|
m
+
m
′
|
P
j
−
M
(
|
m
−
m
′
|
,
|
m
+
m
′
|
)
(
cos
β
)
.
{\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 ).}
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
(
x
,
y
,
z
)
=
(
1
,
2
,
3
)
,
{\displaystyle (x,y,z)=(1,2,3),}
J
^
1
=
i
(
cos
α
cot
β
∂
∂
α
+
sin
α
∂
∂
β
−
cos
α
sin
β
∂
∂
γ
)
J
^
2
=
i
(
sin
α
cot
β
∂
∂
α
−
cos
α
∂
∂
β
−
sin
α
sin
β
∂
∂
γ
)
J
^
3
=
−
i
∂
∂
α
{\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}}}
which have quantum mechanical meaning: they are space-fixed rigid rotor angular momentum operators.
Further,
P
^
1
=
i
(
cos
γ
sin
β
∂
∂
α
−
sin
γ
∂
∂
β
−
cot
β
cos
γ
∂
∂
γ
)
P
^
2
=
i
(
−
sin
γ
sin
β
∂
∂
α
−
cos
γ
∂
∂
β
+
cot
β
sin
γ
∂
∂
γ
)
P
^
3
=
−
i
∂
∂
γ
,
{\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}}}
which have quantum mechanical meaning: they are body-fixed rigid rotor angular momentum operators.
The operators satisfy the commutation relations
[
J
1
,
J
2
]
=
i
J
3
,
and
[
P
1
,
P
2
]
=
−
i
P
3
,
{\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},}
and the corresponding relations with the indices permuted cyclically. The
P
i
{\displaystyle {\mathcal {P}}_{i}}
satisfy anomalous commutation relations (have a minus sign on the right hand side).
The two sets mutually commute,
[
P
i
,
J
j
]
=
0
,
i
,
j
=
1
,
2
,
3
,
{\displaystyle \left[{\mathcal {P}}_{i},{\mathcal {J}}_{j}\right]=0,\quad i,j=1,2,3,}
and the total operators squared are equal,
J
2
≡
J
1
2
+
J
2
2
+
J
3
2
=
P
2
≡
P
1
2
+
P
2
2
+
P
3
2
.
{\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}.}
Their explicit form is,
J
2
=
P
2
=
−
1
sin
2
β
(
∂
2
∂
α
2
+
∂
2
∂
γ
2
−
2
cos
β
∂
2
∂
α
∂
γ
)
−
∂
2
∂
β
2
−
cot
β
∂
∂
β
.
{\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 }}.}
The operators
J
i
{\displaystyle {\mathcal {J}}_{i}}
act on the first (row) index of the D-matrix,
J
3
D
m
′
m
j
(
α
,
β
,
γ
)
∗
=
m
′
D
m
′
m
j
(
α
,
β
,
γ
)
∗
(
J
1
±
i
J
2
)
D
m
′
m
j
(
α
,
β
,
γ
)
∗
=
j
(
j
+
1
)
−
m
′
(
m
′
±
1
)
D
m
′
±
1
,
m
j
(
α
,
β
,
γ
)
∗
{\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}}}
The operators
P
i
{\displaystyle {\mathcal {P}}_{i}}
act on the second (column) index of the D-matrix,
P
3
D
m
′
m
j
(
α
,
β
,
γ
)
∗
=
m
D
m
′
m
j
(
α
,
β
,
γ
)
∗
,
{\displaystyle {\mathcal {P}}_{3}D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}=mD_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*},}
and, because of the anomalous commutation relation the raising/lowering operators are defined with reversed signs,
(
P
1
∓
i
P
2
)
D
m
′
m
j
(
α
,
β
,
γ
)
∗
=
j
(
j
+
1
)
−
m
(
m
±
1
)
D
m
′
,
m
±
1
j
(
α
,
β
,
γ
)
∗
.
{\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 )^{*}.}
Finally,
J
2
D
m
′
m
j
(
α
,
β
,
γ
)
∗
=
P
2
D
m
′
m
j
(
α
,
β
,
γ
)
∗
=
j
(
j
+
1
)
D
m
′
m
j
(
α
,
β
,
γ
)
∗
.
{\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 )^{*}.}
In other words, the rows and columns of the (complex conjugate) Wigner D-matrix span irreducible representations of the isomorphic Lie algebras generated by
{
J
i
}
{\displaystyle \{{\mathcal {J}}_{i}\}}
and
{
−
P
i
}
{\displaystyle \{-{\mathcal {P}}_{i}\}}
.
An important property of the Wigner D-matrix follows from the commutation of
R
(
α
,
β
,
γ
)
{\displaystyle {\mathcal {R}}(\alpha ,\beta ,\gamma )}
with the time reversal operator
T,
⟨
j
m
′
|
R
(
α
,
β
,
γ
)
|
j
m
⟩
=
⟨
j
m
′
|
T
†
R
(
α
,
β
,
γ
)
T
|
j
m
⟩
=
(
−
1
)
m
′
−
m
⟨
j
,
−
m
′
|
R
(
α
,
β
,
γ
)
|
j
,
−
m
⟩
∗
,
{\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 ^{*},}
or
D
m
′
m
j
(
α
,
β
,
γ
)
=
(
−
1
)
m
′
−
m
D
−
m
′
,
−
m
j
(
α
,
β
,
γ
)
∗
.
{\displaystyle D_{m'm}^{j}(\alpha ,\beta ,\gamma )=(-1)^{m'-m}D_{-m',-m}^{j}(\alpha ,\beta ,\gamma )^{*}.}
Here, we used that
T
{\displaystyle T}
is anti-unitary (hence the complex conjugation after moving
T
†
{\displaystyle T^{\dagger }}
from ket to bra),
T
|
j
m
⟩
=
(
−
1
)
j
−
m
|
j
,
−
m
⟩
{\displaystyle T|jm\rangle =(-1)^{j-m}|j,-m\rangle }
and
(
−
1
)
2
j
−
m
′
−
m
=
(
−
1
)
m
′
−
m
{\displaystyle (-1)^{2j-m'-m}=(-1)^{m'-m}}
.
A further symmetry implies
(
−
1
)
m
′
−
m
D
m
m
′
j
(
α
,
β
,
γ
)
=
D
m
′
m
j
(
γ
,
β
,
α
)
.
{\displaystyle (-1)^{m'-m}D_{mm'}^{j}(\alpha ,\beta ,\gamma )=D_{m'm}^{j}(\gamma ,\beta ,\alpha )~.}
Orthogonality relations
The Wigner D-matrix elements
D
m
k
j
(
α
,
β
,
γ
)
{\displaystyle D_{mk}^{j}(\alpha ,\beta ,\gamma )}
form a set of orthogonal functions of the Euler angles
α
,
β
,
{\displaystyle \alpha ,\beta ,}
and
γ
{\displaystyle \gamma }
:
∫
0
2
π
d
α
∫
0
π
d
β
sin
β
∫
0
2
π
d
γ
D
m
′
k
′
j
′
(
α
,
β
,
γ
)
∗
D
m
k
j
(
α
,
β
,
γ
)
=
8
π
2
2
j
+
1
δ
m
′
m
δ
k
′
k
δ
j
′
j
.
{\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}.}
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
D
m
k
j
(
α
,
β
,
γ
)
{\displaystyle D_{mk}^{j}(\alpha ,\beta ,\gamma )}
are matrix elements of a unitary transformation from one spherical basis
|
l
m
⟩
{\displaystyle |lm\rangle }
to another
R
(
α
,
β
,
γ
)
|
l
m
⟩
{\displaystyle {\mathcal {R}}(\alpha ,\beta ,\gamma )|lm\rangle }
is represented by the relations:
∑
k
D
m
′
k
j
(
α
,
β
,
γ
)
∗
D
m
k
j
(
α
,
β
,
γ
)
=
δ
m
,
m
′
,
{\displaystyle \sum _{k}D_{m'k}^{j}(\alpha ,\beta ,\gamma )^{*}D_{mk}^{j}(\alpha ,\beta ,\gamma )=\delta _{m,m'},}
∑
k
D
k
m
′
j
(
α
,
β
,
γ
)
∗
D
k
m
j
(
α
,
β
,
γ
)
=
δ
m
,
m
′
.
{\displaystyle \sum _{k}D_{km'}^{j}(\alpha ,\beta ,\gamma )^{*}D_{km}^{j}(\alpha ,\beta ,\gamma )=\delta _{m,m'}.}
The group characters for SU(2) only depend on the rotation angle β, being class functions, so, then, independent of the axes of rotation,
χ
j
(
β
)
≡
∑
m
D
m
m
j
(
β
)
=
∑
m
d
m
m
j
(
β
)
=
sin
(
(
2
j
+
1
)
β
2
)
sin
(
β
2
)
,
{\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)}},}
and consequently satisfy simpler orthogonality relations, through the Haar measure of the group,
1
π
∫
0
2
π
d
β
sin
2
(
β
2
)
χ
j
(
β
)
χ
j
′
(
β
)
=
δ
j
′
j
.
{\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}.}
The completeness relation (worked out in the same reference, (3.95)) is
∑
j
χ
j
(
β
)
χ
j
(
β
′
)
=
δ
(
β
−
β
′
)
,
{\displaystyle \sum _{j}\chi ^{j}(\beta )\chi ^{j}(\beta ')=\delta (\beta -\beta '),}
whence, for
β
′
=
0
,
{\displaystyle \beta '=0,}
∑
j
χ
j
(
β
)
(
2
j
+
1
)
=
δ
(
β
)
.
{\displaystyle \sum _{j}\chi ^{j}(\beta )(2j+1)=\delta (\beta ).}
Kronecker product of Wigner D-matrices, Clebsch–Gordan series
The set of Kronecker product matrices
D
j
(
α
,
β
,
γ
)
⊗
D
j
′
(
α
,
β
,
γ
)
{\displaystyle \mathbf {D} ^{j}(\alpha ,\beta ,\gamma )\otimes \mathbf {D} ^{j'}(\alpha ,\beta ,\gamma )}
forms a reducible matrix representation of the groups SO(3) and SU(2). Reduction into irreducible components is by the following equation:
D
m
k
j
(
α
,
β
,
γ
)
D
m
′
k
′
j
′
(
α
,
β
,
γ
)
=
∑
J
=
|
j
−
j
′
|
j
+
j
′
⟨
j
m
j
′
m
′
|
J
(
m
+
m
′
)
⟩
⟨
j
k
j
′
k
′
|
J
(
k
+
k
′
)
⟩
D
(
m
+
m
′
)
(
k
+
k
′
)
J
(
α
,
β
,
γ
)
{\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 )}
The symbol
⟨
j
1
m
1
j
2
m
2
|
j
3
m
3
⟩
{\displaystyle \langle j_{1}m_{1}j_{2}m_{2}|j_{3}m_{3}\rangle }
is a Clebsch–Gordan coefficient.
Relation to spherical harmonics and Legendre polynomials
For integer values of
l
{\displaystyle l}
, 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:
D
m
0
ℓ
(
α
,
β
,
γ
)
=
4
π
2
ℓ
+
1
Y
ℓ
m
∗
(
β
,
α
)
=
(
ℓ
−
m
)
!
(
ℓ
+
m
)
!
P
ℓ
m
(
cos
β
)
e
−
i
m
α
.
{\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 }.}
This implies the following relationship for the d-matrix:
d
m
0
ℓ
(
β
)
=
(
ℓ
−
m
)
!
(
ℓ
+
m
)
!
P
ℓ
m
(
cos
β
)
.
{\displaystyle d_{m0}^{\ell }(\beta )={\sqrt {\frac {(\ell -m)!}{(\ell +m)!}}}\,P_{\ell }^{m}(\cos {\beta }).}
A rotation of spherical harmonics
⟨
θ
,
ϕ
|
ℓ
m
′
⟩
{\displaystyle \langle \theta ,\phi |\ell m'\rangle }
then is effectively a composition of two rotations,
∑
m
′
=
−
ℓ
ℓ
Y
ℓ
m
′
(
θ
,
ϕ
)
D
m
′
m
ℓ
(
α
,
β
,
γ
)
.
{\displaystyle \sum _{m'=-\ell }^{\ell }Y_{\ell }^{m'}(\theta ,\phi )~D_{m'~m}^{\ell }(\alpha ,\beta ,\gamma ).}
When both indices are set to zero, the Wigner D-matrix elements are given by ordinary Legendre polynomials:
D
0
,
0
ℓ
(
α
,
β
,
γ
)
=
d
0
,
0
ℓ
(
β
)
=
P
ℓ
(
cos
β
)
.
{\displaystyle D_{0,0}^{\ell }(\alpha ,\beta ,\gamma )=d_{0,0}^{\ell }(\beta )=P_{\ell }(\cos \beta ).}
In the present convention of Euler angles,
α
{\displaystyle \alpha }
is
a longitudinal angle and
β
{\displaystyle \beta }
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
(
Y
ℓ
m
)
∗
=
(
−
1
)
m
Y
ℓ
−
m
.
{\displaystyle \left(Y_{\ell }^{m}\right)^{*}=(-1)^{m}Y_{\ell }^{-m}.}
There exists a more general relationship to the spin-weighted spherical harmonics:
D
m
s
ℓ
(
α
,
β
,
−
γ
)
=
(
−
1
)
s
4
π
2
ℓ
+
1
s
Y
ℓ
m
(
β
,
α
)
e
i
s
γ
.
{\displaystyle D_{ms}^{\ell }(\alpha ,\beta ,-\gamma )=(-1)^{s}{\sqrt {\frac {4\pi }{2{\ell }+1}}}{}_{s}Y_{\ell }^{m}(\beta ,\alpha )e^{is\gamma }.}
Connection with transition probability under rotations
The absolute square of an element of the D-matrix,
F
m
m
′
(
β
)
=
|
D
m
m
′
j
(
α
,
β
,
γ
)
|
2
,
{\displaystyle F_{mm'}(\beta )=|D_{mm'}^{j}(\alpha ,\beta ,\gamma )|^{2},}
gives the probability that a system with spin
j
{\displaystyle j}
prepared in a state with spin projection
m
{\displaystyle m}
along
some direction will be measured to have a spin projection
m
′
{\displaystyle m'}
along a second direction at an angle
β
{\displaystyle \beta }
to the first direction. The set of quantities
F
m
m
′
{\displaystyle F_{mm'}}
itself forms a real symmetric matrix, that
depends only on the Euler angle
β
{\displaystyle \beta }
, as indicated.
Remarkably, the eigenvalue problem for the
F
{\displaystyle F}
matrix can be solved completely:
∑
m
′
=
−
j
j
F
m
m
′
(
β
)
f
ℓ
j
(
m
′
)
=
P
ℓ
(
cos
β
)
f
ℓ
j
(
m
)
(
ℓ
=
0
,
1
,
…
,
2
j
)
.
{\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).}
Here, the eigenvector,
f
ℓ
j
(
m
)
{\displaystyle f_{\ell }^{j}(m)}
, is a scaled and shifted discrete Chebyshev polynomial, and the corresponding eigenvalue,
P
ℓ
(
cos
β
)
{\displaystyle P_{\ell }(\cos \beta )}
, is the Legendre polynomial.
Relation to Bessel functions
In the limit when
ℓ
≫
m
,
m
′
{\displaystyle \ell \gg m,m^{\prime }}
we have
D
m
m
′
ℓ
(
α
,
β
,
γ
)
≈
e
−
i
m
α
−
i
m
′
γ
J
m
−
m
′
(
ℓ
β
)
{\displaystyle D_{mm'}^{\ell }(\alpha ,\beta ,\gamma )\approx e^{-im\alpha -im'\gamma }J_{m-m'}(\ell \beta )}
where
J
m
−
m
′
(
ℓ
β
)
{\displaystyle J_{m-m'}(\ell \beta )}
is the Bessel function and
ℓ
β
{\displaystyle \ell \beta }
is finite.
List of d-matrix elements
Using sign convention of Wigner, et al. the d-matrix elements
d
m
′
m
j
(
θ
)
{\displaystyle d_{m'm}^{j}(\theta )}
for j = 1/2, 1, 3/2, and 2 are given below.
For j = 1/2
d
1
2
,
1
2
1
2
=
cos
θ
2
d
1
2
,
−
1
2
1
2
=
−
sin
θ
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}}}
For j = 1
d
1
,
1
1
=
1
2
(
1
+
cos
θ
)
d
1
,
0
1
=
−
1
2
sin
θ
d
1
,
−
1
1
=
1
2
(
1
−
cos
θ
)
d
0
,
0
1
=
cos
θ
{\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}}}
For j = 3/2
d
3
2
,
3
2
3
2
=
1
2
(
1
+
cos
θ
)
cos
θ
2
d
3
2
,
1
2
3
2
=
−
3
2
(
1
+
cos
θ
)
sin
θ
2
d
3
2
,
−
1
2
3
2
=
3
2
(
1
−
cos
θ
)
cos
θ
2
d
3
2
,
−
3
2
3
2
=
−
1
2
(
1
−
cos
θ
)
sin
θ
2
d
1
2
,
1
2
3
2
=
1
2
(
3
cos
θ
−
1
)
cos
θ
2
d
1
2
,
−
1
2
3
2
=
−
1
2
(
3
cos
θ
+
1
)
sin
θ
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}}}
For j = 2
d
2
,
2
2
=
1
4
(
1
+
cos
θ
)
2
d
2
,
1
2
=
−
1
2
sin
θ
(
1
+
cos
θ
)
d
2
,
0
2
=
3
8
sin
2
θ
d
2
,
−
1
2
=
−
1
2
sin
θ
(
1
−
cos
θ
)
d
2
,
−
2
2
=
1
4
(
1
−
cos
θ
)
2
d
1
,
1
2
=
1
2
(
2
cos
2
θ
+
cos
θ
−
1
)
d
1
,
0
2
=
−
3
8
sin
2
θ
d
1
,
−
1
2
=
1
2
(
−
2
cos
2
θ
+
cos
θ
+
1
)
d
0
,
0
2
=
1
2
(
3
cos
2
θ
−
1
)
{\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}}}
Wigner d-matrix elements with swapped lower indices are found with the relation:
d
m
′
,
m
j
=
(
−
1
)
m
−
m
′
d
m
,
m
′
j
=
d
−
m
,
−
m
′
j
.
{\displaystyle d_{m',m}^{j}=(-1)^{m-m'}d_{m,m'}^{j}=d_{-m,-m'}^{j}.}
Symmetries and special cases
d
m
′
,
m
j
(
π
)
=
(
−
1
)
j
−
m
δ
m
′
,
−
m
d
m
′
,
m
j
(
π
−
β
)
=
(
−
1
)
j
+
m
′
d
m
′
,
−
m
j
(
β
)
d
m
′
,
m
j
(
π
+
β
)
=
(
−
1
)
j
−
m
d
m
′
,
−
m
j
(
β
)
d
m
′
,
m
j
(
2
π
+
β
)
=
(
−
1
)
2
j
d
m
′
,
m
j
(
β
)
d
m
′
,
m
j
(
−
β
)
=
d
m
,
m
′
j
(
β
)
=
(
−
1
)
m
′
−
m
d
m
′
,
m
j
(
β
)
{\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}}}
See also
Clebsch–Gordan coefficients
Tensor operator
Symmetries in quantum mechanics
References
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.
Kata Kunci Pencarian:
- Eugene Wigner
- Karbon
- Logam alkali
- Barion
- Terence Tao
- Pengantar mekanika kuantum
- Wigner D-matrix
- Wigner distribution
- Jacobi polynomials
- Eugene Wigner
- Wigner quasiprobability distribution
- List of things named after Eugene Wigner
- Random matrix
- Clebsch–Gordan coefficients
- Density matrix
- Spin matrix
