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 .
