Wigner D-matrix

Wigner D-matrix

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.

Definition Wigner D-matrix

Let j_x, j_y, j_z be generators of the Lie algebra of SU(2) and SO(3). In quantum mechanics thesethree operators are the components of a vector operator known as "angular momentum". Examplesare the angular momentum of an electronin an atom, electronic spin,and the angular momentumof a rigid rotor. In all cases the three operators satisfy the following commutation relations,: [j_x,j_y] = i j_z,quad [j_z,j_x] = i j_y,quad [j_y,j_z] = i j_x, where "i" is the purely imaginary number and Planck's constant hbar has been put equal to one. The operator: j^2 = j_x^2 + j_y^2 + j_z^2 is a Casimir operator of SU(2) (or SO(3) as the case may be).It may be diagonalized together with j_z (the choice of this operatoris a convention), which commutes with j^2. That is, it can be shown that there is a complete set of kets with: j^2 |jm angle = j(j+1) |jm angle,quad j_z |jm angle = m |jm angle,where j=0, frac{1}{2}, 1, frac{3}{2}, 2, dots and m=-j, -j+1, ldots, j. (For SO(3) the "quantum number" j is integer.)

A rotation operator can be written as: mathcal{R}(alpha,eta,gamma) = e^{-ialpha j_z}e^{-ieta j_y}e^{-igamma j_z},where alpha, ; eta, and gamma; 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 withgeneral element: D^j_{m'm}(alpha,eta,gamma) equivlangle jm' | mathcal{R}(alpha,eta,gamma)| jm angle = e^{-im'alpha } d^j_{m'm}(eta)e^{-i mgamma}.The matrix with general element :d^j_{m'm}(eta)= langle jm' |e^{-ieta j_y} | jm angle is known as Wigner's (small) d-matrix.

Wigner (small) d-matrix

Wigner [E. P. Wigner, "Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren", Vieweg Verlag, Braunschweig (1931). Translated into English: J. J. Griffin, "Group Theory and its Application to the Quantum Mechanics of Atomic Spectra", Academic Press, New York (1959).] gave the following expression:egin{array}{lcl}d^j_{m'm}(eta) &=& [(j+m')!(j-m')!(j+m)!(j-m)!] ^{1/2}sum_s frac{(-1)^{m'-m+s{(j+m-s)!s!(m'-m+s)!(j-m'-s)!} \&& imes left(cosfrac{eta}{2} ight)^{2j+m-m'-2s}left(sinfrac{eta}{2} ight)^{m'-m+2s}end{array} 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^{(a,b)}_k(coseta) with nonnegative a, and b,. [L. C. Biedenharn and J. D. Louck, "Angular Momentum in Quantum Physics", Addison-Wesley, Reading, (1981).] Let : k = min(j+m,,j-m,,j+m',,j-m').

:hbox{If}quad k = egin{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}

Then, with b=2j-2k-a,, the relation is

:d^j_{m'm}(eta) = (-1)^{lambda} inom{2j-k}{k+a}^{1/2} inom{k+b}{b}^{-1/2} left(sinfrac{eta}{2} ight)^a left(cosfrac{eta}{2} ight)^b P^{(a,b)}_k(coseta),where a,b ge 0. ,

Properties of Wigner D-matrix

The complex conjugate of the D-matrix satisfies a number of differential propertiesthat can be formulated concisely by introducing the following operators with (x,, y,,z) = (1,,2,,3),:egin{array}{lcl}hat{mathcal{J_1 &=& i left( cos alpha cot eta ,{partial over partial alpha} , + sin alpha ,{partial over partial eta} , - {cos alpha over sin eta} ,{partial over partial gamma} , ight) \hat{mathcal{J_2 &=& i left( sin alpha cot eta ,{partial over partial alpha} , - cos alpha ;{partial over partial eta } , - {sin alpha over sin eta} ,{partial over partial gamma } , ight) \hat{mathcal{J_3 &=& - i ; {partial over partial alpha} ,end{array}which have quantum mechanical meaning: they are space-fixed rigid rotor angular momentum operators.

Further,:egin{array}{lcl}hat{mathcal{P_1 &=& , i left( {cos gamma over sin eta} {partial over partial alpha } - sin gamma {partial over partial eta } - cot eta cos gamma {partial over partial gamma} ight) \hat{mathcal{P_2 &=& , i left( - {sin gamma over sin eta} {partial over partial alpha} - cos gamma {partial over partial eta} + cot eta sin gamma {partial over partial gamma} ight) \hat{mathcal{P_3 &=& - i {partialover partial gamma}, \end{array}which have quantum mechanical meaning: they are body-fixed rigid rotor angular momentum operators.

The operators satisfy the commutation relations:left [mathcal{J}_1, , mathcal{J}_2 ight] = i mathcal{J}_3, qquad hbox{and}qquadleft [mathcal{P}_1, , mathcal{P}_2 ight] = -i mathcal{P}_3 and the corresponding relations with the indices permuted cyclically. The mathcal{P}_i satisfy "anomalous commutation relations" (have a minus sign on the right hand side). The two sets mutually commute,:left [mathcal{P}_i, , mathcal{J}_j ight] = 0,quad i,,j = 1,,2,,3,and the total operators squared are equal,: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,:mathcal{J}^2= mathcal{P}^2 =-frac{1}{sin^2eta} left( frac{partial^2}{partial alpha^2}+frac{partial^2}{partial gamma^2}-2cosetafrac{partial^2}{partialalphapartial gamma} ight)-frac{partial^2}{partial eta^2}-cotetafrac{partial}{partial eta}.

The operators mathcal{J}_i act on the first (row) index of the D-matrix,:mathcal{J}_3 , D^j_{m'm}(alpha,eta,gamma)^* = m' , D^j_{m'm}(alpha,eta,gamma)^* ,and:(mathcal{J}_1 pm i mathcal{J}_2), D^j_{m'm}(alpha,eta,gamma)^* = sqrt{j(j+1)-m'(m'pm 1)} , D^j_{m'pm 1, m}(alpha,eta,gamma)^* .

The operators mathcal{P}_i act on the second (column) index of the D-matrix:mathcal{P}_3 , D^j_{m'm}(alpha,eta,gamma)^* = m , D^j_{m'm}(alpha,eta,gamma)^* ,and because of the anomalous commutation relation the raising/lowering operatorsare defined with reversed signs,:(mathcal{P}_1 mp i mathcal{P}_2), D^j_{m'm}(alpha,eta,gamma)^* = sqrt{j(j+1)-m(mpm 1)} , D^j_{m', mpm1}(alpha,eta,gamma)^* .

Finally,:mathcal{J}^2, D^j_{m'm}(alpha,eta,gamma)^* =mathcal{P}^2, D^j_{m'm}(alpha,eta,gamma)^* = j(j+1) D^j_{m'm}(alpha,eta,gamma)^*.

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 {mathcal{J}_i} and {-mathcal{P}_i}.

An important property of the Wigner D-matrix follows from the commutation of mathcal{R}(alpha,eta,gamma) with the time reversal operator T,,:langle jm' | mathcal{R}(alpha,eta,gamma)| jm angle =langle jm' | T^{,dagger} mathcal{R}(alpha,eta,gamma) T| jm angle =(-1)^{m'-m} langle j,-m' | mathcal{R}(alpha,eta,gamma)| j,-m angle^*,or:D^j_{m'm}(alpha,eta,gamma) = (-1)^{m'-m} D^j_{-m',-m}(alpha,eta,gamma)^*. Here we used that T, is anti-unitary (hence the complex conjugation after movingT^dagger, from ket to bra), T | jm angle = (-1)^{j-m} | j,-m angle and (-1)^{2j-m'-m} = (-1)^{m'-m}.

Orthogonality relations

The Wigner D-matrix elements D^j_{mk}(alpha,eta,gamma) form a complete setof orthogonal functions of the Euler angles alpha, eta, and gamma:: int_0^{2pi} dalpha int_0^pi sin eta deta int_0^{2pi} dgamma ,, D^{j'}_{m'k'}(alpha,eta,gamma)^ast D^j_{mk}(alpha,eta,gamma) = frac{8pi^2}{2j+1} delta_{m'm}delta_{k'k}delta_{j'j}.

This is a special case of the Schur orthogonality relations.

Relation with spherical harmonic functions

The D-matrix elements with second index equal to zero, are proportionalto spherical harmonics, normalized to unity and with Condon and Shortley phase convention, :D^{ell}_{m 0}(alpha,eta,gamma)^* = sqrt{frac{4pi}{2ell+1 Y_{ell}^m (eta, alpha ) .In the present convention of Euler angles, alpha is a longitudinal angle and eta is a colatitudinal angle (spherical polar anglesin 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:left( Y_{ell}^m ight) ^* = (-1)^m Y_{ell}^{-m}.There exists a more general relationship to the spin-weighted spherical harmonics::D^{ell}_{-m s}(alpha,eta,-gamma) =(-1)^m sqrtfrac{4pi}{2{ell}+1} {}_sY_ell}m}(eta,alpha) e^{isgamma}.

Relation with Legendre polynomials

The Wigner small d-matrix elements with both indices set to zero are relatedto Legendre polynomials: d^{ell}_{0,0}(eta) = P_{ell}(coseta).

See also

* Clebsch-Gordan coefficients
* Eugene Paul Wigner

References

Cited references

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