The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3) . The complex conjugate of the D-matrix is an eigenfunction of the Hamiltonian of spherical and symmetric rigid rotors . The matrix was introduced in 1927 by Eugene Wigner . 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 Planck's 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, in this basis, there is a complete set of kets 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:[1]
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.} [2] 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.}
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 , {\displaystyle 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}} .
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 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,[3]
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:[4]
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 }.} [5]
Relation to Bessel functions
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[6]
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
References ^ Wigner, E. P. (1931). Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren . Braunschweig: Vieweg Verlag. Translated into English by Griffin, J. J. (1959). Group Theory and its Application to the Quantum Mechanics of Atomic Spectra . New York: Academic Press. ^ Biedenharn, L. C.; Louck, J. D. (1981). Angular Momentum in Quantum Physics . Reading: Addison-Wesley. ISBN 0-201-13507-8 . ^ Schwinger, J. "On Angular Momentum", Harvard University, Nuclear Development Associates, Inc., United States Department of Energy (through predecessor agency the Atomic Energy Commission) (January 26, 1952) ^ Rose, M. E. Elementary Theory of Angular Momentum. New York, JOHN WILEY & SONS, 1957. ^ https://link.springer.com/content/pdf/bbm%3A978-4-431-54180-6%2F1.pdf ^ 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 PDG Table of Clebsch-Gordan Coefficients, Spherical Harmonics, and d-Functions