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
,
where i is the purely imaginary number and Planck's constant
ℏ
has been put equal to one. The Casimir operator
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
J
z
. That is, it can be shown that there is a complete set of kets with
J
2
|
j
m
⟩
=
j
(
j
+
1
)
|
j
m
⟩
,
J
z
|
j
m
⟩
=
m
|
j
m
⟩
,
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 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
,
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 square matrix of dimension 2j + 1 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
,
where
d
m
′
m
j
(
β
)
=
⟨
j
m
′
|
e
−
i
β
J
y
|
j
m
⟩
is an element of Wigner's (small) d-matrix.
Wigner gave the following expression
d
m
′
m
j
(
β
)
=
[
(
j
+
m
′
)
!
(
j
−
m
′
)
!
(
j
+
m
)
!
(
j
−
m
)
!
]
1
/
2
∑
s
[
(
−
1
)
m
′
−
m
+
s
(
j
+
m
−
s
)
!
s
!
(
m
′
−
m
+
s
)
!
(
j
−
m
′
−
s
)
!
⋅
(
cos
β
2
)
2
j
+
m
−
m
′
−
2
s
(
sin
β
2
)
m
′
−
m
+
2
s
]
.
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
(
−
1
)
m
′
−
m
+
s
in this formula is replaced by
(
−
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
)
with nonnegative
a
and
b
. Let
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
Then, with
b
=
2
j
−
2
k
−
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
)
,
where
a
,
b
≥
0.
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
)
,
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
,
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
,
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
and the corresponding relations with the indices permuted cyclically. The
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
,
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
.
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
.
The operators
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
)
∗
,
and
(
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
)
∗
.
The operators
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
)
∗
,
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
)
∗
.
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
)
∗
.
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
{
J
i
}
and
{
−
P
i
}
.
An important property of the Wigner D-matrix follows from the commutation of
R
(
α
,
β
,
γ
)
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
⟩
∗
,
or
D
m
′
m
j
(
α
,
β
,
γ
)
=
(
−
1
)
m
′
−
m
D
−
m
′
,
−
m
j
(
α
,
β
,
γ
)
∗
.
Here we used that
T
is anti-unitary (hence the complex conjugation after moving
T
†
from ket to bra),
T
|
j
m
⟩
=
(
−
1
)
j
−
m
|
j
,
−
m
⟩
and
(
−
1
)
2
j
−
m
′
−
m
=
(
−
1
)
m
′
−
m
.
The Wigner D-matrix elements
D
m
k
j
(
α
,
β
,
γ
)
form a complete set of orthogonal functions of the Euler angles
α
,
β
,
and
γ
:
∫
0
2
π
d
α
∫
0
π
sin
β
d
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
.
This is a special case of the Schur orthogonality relations.
The set of Kronecker product matrices
D
j
(
α
,
β
,
γ
)
⊗
D
j
′
(
α
,
β
,
γ
)
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
⟩
⟨
j
k
j
′
k
′
|
J
K
⟩
D
m
+
m
′
k
+
k
′
J
(
α
,
β
,
γ
)
The symbol
⟨
j
m
j
′
m
′
|
J
M
⟩
is a Clebsch-Gordan coefficient.
Relation to spherical harmonics and Legendre polynomials
For integer values of
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
,
0
)
=
4
π
2
ℓ
+
1
Y
ℓ
m
∗
(
β
,
α
)
=
(
ℓ
−
m
)
!
(
ℓ
+
m
)
!
P
ℓ
m
(
cos
β
)
e
−
i
m
α
This implies the following relationship for the d-matrix:
d
m
0
ℓ
(
β
)
=
(
ℓ
−
m
)
!
(
ℓ
+
m
)
!
P
ℓ
m
(
cos
β
)
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
)
.
In the present convention of Euler angles,
α
is a longitudinal angle and
β
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
.
There exists a more general relationship to the spin-weighted spherical harmonics:
D
−
m
s
ℓ
(
α
,
β
,
−
γ
)
=
(
−
1
)
m
4
π
2
ℓ
+
1
s
Y
ℓ
m
(
β
,
α
)
e
i
s
γ
.
In the limit when
ℓ
≫
m
,
m
′
we have
D
m
m
′
ℓ
(
α
,
β
,
γ
)
≈
e
−
i
m
α
−
i
m
′
γ
J
m
−
m
′
(
ℓ
β
)
where
J
m
−
m
′
(
ℓ
β
)
is the Bessel function and
ℓ
β
is finite.
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
d
1
/
2
,
1
/
2
1
/
2
=
cos
(
θ
/
2
)
d
1
/
2
,
−
1
/
2
1
/
2
=
−
sin
(
θ
/
2
)
for j = 1
d
1
,
1
1
=
1
+
cos
θ
2
d
1
,
0
1
=
−
sin
θ
2
d
1
,
−
1
1
=
1
−
cos
θ
2
d
0
,
0
1
=
cos
θ
for j = 3/2
d
3
/
2
,
3
/
2
3
/
2
=
1
+
cos
θ
2
cos
θ
2
d
3
/
2
,
1
/
2
3
/
2
=
−
3
1
+
cos
θ
2
sin
θ
2
d
3
/
2
,
−
1
/
2
3
/
2
=
3
1
−
cos
θ
2
cos
θ
2
d
3
/
2
,
−
3
/
2
3
/
2
=
−
1
−
cos
θ
2
sin
θ
2
d
1
/
2
,
1
/
2
3
/
2
=
3
cos
θ
−
1
2
cos
θ
2
d
1
/
2
,
−
1
/
2
3
/
2
=
−
3
cos
θ
+
1
2
sin
θ
2
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
)
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
.