Вигнерова D матрица представља матрицу иредуцибилних репрезентација група SU(2) и SO(3) . Вигнерова D матрица је квадратна матрица оператора ротација димензија
2
j
+
1
{\displaystyle 2j+1}
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 }.}
Матрица је добила име по Еугену Вигнеру , који ју је први увео 1927 . године.
уреди
Генератори Лијевих алгебри SU(2) и SO(3) означимо са
J
x
{\displaystyle J_{x}}
J
y
{\displaystyle J_{y}}
J
z
{\displaystyle J_{z}}
[
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},}
Оператор
J
2
=
J
x
2
+
J
y
2
+
J
z
2
{\displaystyle J^{2}=J_{x}^{2}+J_{y}^{2}+J_{z}^{2}}
представља Казимиров оператор од SU(2) (или SO(3) ).
Оператор ротација може да се прикаже као:
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}},}
где су
α
,
β
,
{\displaystyle \alpha ,\;\beta ,}
γ
{\displaystyle \gamma \;}
Ојлерови углови .
Вигнерова D матрица је квадратна матрица димензија
2
j
+
1
{\displaystyle 2j+1}
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 }.}
При томе мала Вигнерова d- матрица означена је са:
d
m
′
m
j
(
β
)
=
⟨
j
m
′
|
e
−
i
β
j
y
|
j
m
⟩
{\displaystyle d_{m'm}^{j}(\beta )=\langle jm'|e^{-i\beta j_{y}}|jm\rangle }
уреди
Мала Вигнерова d- матрица може да се представи као:
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
]
.
{\displaystyle {\begin{array}{lcl}d_{m'm}^{j}(\beta )&=&[(j+m')!(j-m')!(j+m)!(j-m)!]^{1/2}\sum \limits _{s}\left[{\frac {(-1)^{m'-m+s}}{(j+m-s)!s!(m'-m+s)!(j-m'-s)!}}\right.\\&&\left.\cdot \left(\cos {\frac {\beta }{2}}\right)^{2j+m-m'-2s}\left(\sin {\frac {\beta }{2}}\right)^{m'-m+2s}\right].\end{array}}}
Матрични елементи мале d- матрице повезани су са Јакобијевим полиномима
P
k
(
a
,
b
)
(
cos
β
)
{\displaystyle P_{k}^{(a,b)}(\cos \beta )}
a
{\displaystyle a\,}
b
{\displaystyle b\,}
k
=
min
(
j
+
m
,
j
−
m
,
j
+
m
′
,
j
−
m
′
)
.
{\displaystyle k=\min(j+m,\,j-m,\,j+m',\,j-m').}
Онда је:
Ako je
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 {\hbox{Ako je}}\quad k={\begin{cases}j+m:&\quad a=m'-m;\quad \lambda =m'-m\\j-m:&\quad a=m-m';\quad \lambda =0\\j+m':&\quad a=m-m';\quad \lambda =0\\j-m':&\quad a=m'-m;\quad \lambda =m'-m\\\end{cases}}}
Онда уз услов
b
=
2
j
−
2
k
−
a
{\displaystyle b=2j-2k-a\,}
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}}^{1/2}{\binom {k+b}{b}}^{-1/2}\left(\sin {\frac {\beta }{2}}\right)^{a}\left(\cos {\frac {\beta }{2}}\right)^{b}P_{k}^{(a,b)}(\cos \beta ),}
где су
a
,
b
≥
0.
{\displaystyle a,b\geq 0.\,}
уреди
Следећих шест оператора:
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{array}{lcl}{\hat {\mathcal {J}}}_{1}&=&i\left(\cos \alpha \cot \beta \,{\partial \over \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{array}}}
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{array}{lcl}{\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{array}}}
задовољава комутационе релације:
[
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}}
Уз то два низа узајамно комутирају:
[
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,}
Квадрати тих оператора су једнаки:
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}.}
Експлицитни облик је:
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 }}.}
Дејство оператора
J
i
{\displaystyle {\mathcal {J}}_{i}}
J
3
D
m
′
m
j
(
α
,
β
,
γ
)
∗
=
m
′
D
m
′
m
j
(
α
,
β
,
γ
)
∗
,
{\displaystyle {\mathcal {J}}_{3}\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}=m'\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*},}
(
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 ({\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 )^{*}.}
С друге стране дејство
P
i
{\displaystyle {\mathcal {P}}_{i}}
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 )^{*}=m\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*},}
(
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 )^{*}.}
Коначно добија се:
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 )^{*}.}
уреди
∫
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
.
{\displaystyle \int _{0}^{2\pi }d\alpha \int _{0}^{\pi }\sin \beta d\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}.}
уреди
Кронекеров производ D матрица
D
j
(
α
,
β
,
γ
)
⊗
D
j
′
(
α
,
β
,
γ
)
{\displaystyle \mathbf {D} ^{j}(\alpha ,\beta ,\gamma )\otimes \mathbf {D} ^{j'}(\alpha ,\beta ,\gamma )}
чини редуцибилну матричну репрезентацију специјалних група SO(3) и SU(2) . Редукцијом на иредуцибилне компоненте добија се:
D
m
k
j
(
α
,
β
,
γ
)
D
m
′
k
′
j
′
(
α
,
β
,
γ
)
=
∑
J
=
|
j
−
j
′
|
j
+
j
′
∑
M
=
−
J
J
∑
K
=
−
J
J
⟨
j
m
j
′
m
′
|
J
M
⟩
⟨
j
k
j
′
k
′
|
J
K
⟩
D
M
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'}\sum _{M=-J}^{J}\sum _{K=-J}^{J}\langle jmj'm'|JM\rangle \langle jkj'k'|JK\rangle D_{MK}^{J}(\alpha ,\beta ,\gamma )}
Симболи
⟨
j
m
j
′
m
′
|
J
M
⟩
{\displaystyle \langle jmj'm'|JM\rangle }
Клебш-Горданови коефицијенти .
уреди
За целобројне вредности
l
{\displaystyle l}
сферним хармоницима и придруженим Лежандровим полиномима :
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
{\displaystyle D_{m0}^{\ell }(\alpha ,\beta ,0)={\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 }}
Одатле се добија следећа релација за мале d-матрице:
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 })}
Ако су оба индекса једнака нули тада су матрични елементи пропрционални Лежандровом полиному :
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 ).}
уреди
За j=1/2
d
1
/
2
,
1
/
2
1
/
2
=
cos
(
θ
/
2
)
{\displaystyle d_{1/2,1/2}^{1/2}=\cos(\theta /2)}
d
1
/
2
,
−
1
/
2
1
/
2
=
−
sin
(
θ
/
2
)
{\displaystyle d_{1/2,-1/2}^{1/2}=-\sin(\theta /2)}
За j=1
d
1
,
1
1
=
1
+
cos
θ
2
{\displaystyle d_{1,1}^{1}={\frac {1+\cos \theta }{2}}}
d
1
,
0
1
=
−
sin
θ
2
{\displaystyle d_{1,0}^{1}={\frac {-\sin \theta }{\sqrt {2}}}}
d
1
,
−
1
1
=
1
−
cos
θ
2
{\displaystyle d_{1,-1}^{1}={\frac {1-\cos \theta }{2}}}
d
0
,
0
1
=
cos
θ
{\displaystyle d_{0,0}^{1}=\cos \theta }
d
3
/
2
,
3
/
2
3
/
2
=
1
+
cos
θ
2
cos
θ
2
{\displaystyle d_{3/2,3/2}^{3/2}={\frac {1+\cos \theta }{2}}\cos {\frac {\theta }{2}}}
d
3
/
2
,
1
/
2
3
/
2
=
−
3
1
+
cos
θ
2
sin
θ
2
{\displaystyle d_{3/2,1/2}^{3/2}=-{\sqrt {3}}{\frac {1+\cos \theta }{2}}\sin {\frac {\theta }{2}}}
d
3
/
2
,
−
1
/
2
3
/
2
=
3
1
−
cos
θ
2
cos
θ
2
{\displaystyle d_{3/2,-1/2}^{3/2}={\sqrt {3}}{\frac {1-\cos \theta }{2}}\cos {\frac {\theta }{2}}}
d
3
/
2
,
−
3
/
2
3
/
2
=
−
1
−
cos
θ
2
sin
θ
2
{\displaystyle d_{3/2,-3/2}^{3/2}=-{\frac {1-\cos \theta }{2}}\sin {\frac {\theta }{2}}}
d
1
/
2
,
1
/
2
3
/
2
=
3
cos
θ
−
1
2
cos
θ
2
{\displaystyle d_{1/2,1/2}^{3/2}={\frac {3\cos \theta -1}{2}}\cos {\frac {\theta }{2}}}
d
1
/
2
,
−
1
/
2
3
/
2
=
−
3
cos
θ
+
1
2
sin
θ
2
{\displaystyle d_{1/2,-1/2}^{3/2}=-{\frac {3\cos \theta +1}{2}}\sin {\frac {\theta }{2}}}
За j=2
d
2
,
2
2
=
1
4
(
1
+
cos
θ
)
2
{\displaystyle d_{2,2}^{2}={\frac {1}{4}}\left(1+\cos \theta \right)^{2}}
d
2
,
1
2
=
−
1
2
sin
θ
(
1
+
cos
θ
)
{\displaystyle d_{2,1}^{2}=-{\frac {1}{2}}\sin \theta \left(1+\cos \theta \right)}
d
2
,
0
2
=
3
8
sin
2
θ
{\displaystyle d_{2,0}^{2}={\sqrt {\frac {3}{8}}}\sin ^{2}\theta }
d
2
,
−
1
2
=
−
1
2
sin
θ
(
1
−
cos
θ
)
{\displaystyle d_{2,-1}^{2}=-{\frac {1}{2}}\sin \theta \left(1-\cos \theta \right)}
d
2
,
−
2
2
=
1
4
(
1
−
cos
θ
)
2
{\displaystyle d_{2,-2}^{2}={\frac {1}{4}}\left(1-\cos \theta \right)^{2}}
d
1
,
1
2
=
1
2
(
2
cos
2
θ
+
cos
θ
−
1
)
{\displaystyle d_{1,1}^{2}={\frac {1}{2}}\left(2\cos ^{2}\theta +\cos \theta -1\right)}
d
1
,
0
2
=
−
3
8
sin
2
θ
{\displaystyle d_{1,0}^{2}=-{\sqrt {\frac {3}{8}}}\sin 2\theta }
d
1
,
−
1
2
=
1
2
(
−
2
cos
2
θ
+
cos
θ
+
1
)
{\displaystyle d_{1,-1}^{2}={\frac {1}{2}}\left(-2\cos ^{2}\theta +\cos \theta +1\right)}
d
0
,
0
2
=
1
2
(
3
cos
2
θ
−
1
)
{\displaystyle d_{0,0}^{2}={\frac {1}{2}}\left(3\cos ^{2}\theta -1\right)}
Abramowitz, Milton; Stegun, Irene A., eds. , Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. . New York: Dover. 1965. ISBN 978-0486612720 .
Wigner E. P., Group Theory and its Application to the Quantum Mechanics of Atomic Spectra, New York: Academic Press (1959)
Messiah, Albert, Quantum Mechanics (Volume II) (12th ed.). . New York: North Holland Publishing. 1981. ISBN 978-0-7204-0045-8 . 
![{\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)
![{\displaystyle {\begin{array}{lcl}d_{m'm}^{j}(\beta )&=&[(j+m')!(j-m')!(j+m)!(j-m)!]^{1/2}\sum \limits _{s}\left[{\frac {(-1)^{m'-m+s}}{(j+m-s)!s!(m'-m+s)!(j-m'-s)!}}\right.\\&&\left.\cdot \left(\cos {\frac {\beta }{2}}\right)^{2j+m-m'-2s}\left(\sin {\frac {\beta }{2}}\right)^{m'-m+2s}\right].\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca8ba08176519ca889915a21d33df4ba9bae9b59)
![{\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/00d986047e2b66ba6077837c0231c2bba69ea11e)
![{\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/98334cb50f15cfab709970b0137ab8ac16c484b8)
