Girish Mahajan (Editor)

Wigner D matrix

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The Wigner D-matrix is a 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.

Contents

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 ,

where i is the purely imaginary number and Planck's constant 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 (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.

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 ) ,

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 .

Orthogonality relations

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.

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 )

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 ( α , β , 0 ) = 4 π 2 + 1 Y m ( β , α ) = ( m ) ! ( + m ) ! P m ( cos β ) e i m α

This implies the following relationship for the d-matrix:

d m 0 ( β ) = ( m ) ! ( + m ) ! P m ( cos β )

When both indices are set to zero, the Wigner D-matrix elements are given by ordinary Legendre polynomials:

D 0 , 0 ( α , β , γ ) = d 0 , 0 ( β ) = P ( 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 m ) = ( 1 ) m Y m .

There exists a more general relationship to the spin-weighted spherical harmonics:

D m s ( α , β , γ ) = ( 1 ) m 4 π 2 + 1 s Y m ( β , α ) e i s γ .

Relation to Bessel functions

In the limit when m , m we have D m m ( α , β , γ ) e i m α i m γ J m m ( β ) where J m m ( β ) is the Bessel function and β is finite.

List of d-matrix elements

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 .

    References

    Wigner D-matrix Wikipedia