# 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 elements of the D-matrix with integral indices are eigenfunctions of the Hamiltonian of spherical and symmetric rigid rotors. The matrix was introduced by E. Wigner in 1927[1]

### Definition Wigner D-matrix

Let ${\displaystyle j_{x}}$, ${\displaystyle j_{y}}$, ${\displaystyle j_{z}}$ 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,

${\displaystyle [j_{x},j_{y}]=ij_{z},\quad [j_{z},j_{x}]=ij_{y},\quad [j_{y},j_{z}]=ij_{x},}$

where i is the purely imaginary number and Planck's constant ${\displaystyle \hbar }$ has been put equal to one. The operator

${\displaystyle j^{2}=j_{x}^{2}+j_{y}^{2}+j_{z}^{2}}$

is a Casimir operator of SU(2) (or SO(3) as the case may be). It may be diagonalized together with ${\displaystyle j_{z}}$ (the choice of this operator is a convention), which commutes with ${\displaystyle j^{2}}$. That is, it can be shown that there is a complete set of kets with

${\displaystyle j^{2}|jm\rangle =j(j+1)|jm\rangle ,\quad j_{z}|jm\rangle =m|jm\rangle ,}$

where ${\displaystyle j=0,{\tfrac {1}{2}},1,{\tfrac {3}{2}},2,\dots }$ and ${\displaystyle m=-j,-j+1,\ldots ,j}$. (For SO(3) the quantum number ${\displaystyle j}$ is integer.)

A rotation operator can be written as

${\displaystyle {\mathcal {R}}(\alpha ,\beta ,\gamma )=e^{-i\alpha j_{z}}e^{-i\beta j_{y}}e^{-i\gamma j_{z}},}$

where ${\displaystyle \alpha ,\;\beta ,}$ and ${\displaystyle \gamma \;}$ 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 ${\displaystyle 2j+1}$ with general element

${\displaystyle D_{m'm}^{j}(\alpha ,\beta ,\gamma )\ {\stackrel {\mathrm {def} }{=}}\ \langle jm'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|jm\rangle =e^{-im'\alpha }d_{m'm}^{j}(\beta )e^{-im\gamma }.}$

The matrix with general element

${\displaystyle d_{m'm}^{j}(\beta )=\langle jm'|e^{-i\beta j_{y}}|jm\rangle }$

is known as Wigner's (small) d-matrix.

### Wigner (small) d-matrix

Wigner[2] gave the following expression

${\displaystyle {\begin{array}{lcl}d_{m'm}^{j}(\beta )&=&[(j+m')!(j-m')!(j+m)!(j-m)!]^{1/2}\sum _{s}{\frac {(-1)^{m'-m+s}}{(j+m-s)!s!(m'-m+s)!(j-m'-s)!}}\\&&\times \left(\cos {\frac {\beta }{2}}\right)^{2j+m-m'-2s}\left(\sin {\frac {\beta }{2}}\right)^{m'-m+2s}\end{array}}}$

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 ${\displaystyle (-1)^{m'-m+s}}$ in this formula is replaced by ${\displaystyle (-1)^{s}\,i^{m-m'}}$, 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 ${\displaystyle P_{k}^{(a,b)}(\cos \beta )}$ with nonnegative ${\displaystyle a\,}$ and ${\displaystyle b\,}$. [3] Let

${\displaystyle k=\min(j+m,\,j-m,\,j+m',\,j-m').}$

${\displaystyle {\hbox{If}}\quad k={\begin{cases}j+m:&\quad a=m'-m;\quad \lambda =m'-m\\j-m:&\quad a=m-m';\quad \lambda =0\\j+m':&\quad a=m-m';\quad \lambda =0\\j-m':&\quad a=m'-m;\quad \lambda =m'-m\\\end{cases}}}$

Then, with ${\displaystyle b=2j-2k-a\,}$, the relation is

${\displaystyle d_{m'm}^{j}(\beta )=(-1)^{\lambda }{\binom {2j-k}{k+a}}^{1/2}{\binom {k+b}{b}}^{-1/2}\left(\sin {\frac {\beta }{2}}\right)^{a}\left(\cos {\frac {\beta }{2}}\right)^{b}P_{k}^{(a,b)}(\cos \beta ),}$

where ${\displaystyle a,b\geq 0.\,}$

### Properties of 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 ${\displaystyle (x,\,y,\,z)=(1,\,2,\,3)}$,

${\displaystyle {\begin{array}{lcl}{\hat {\mathcal {J}}}_{1}&=&i\left(\cos \alpha \cot \beta \,{\partial \over \partial \alpha }\,+\sin \alpha \,{\partial \over \partial \beta }\,-{\cos \alpha \over \sin \beta }\,{\partial \over \partial \gamma }\,\right)\\{\hat {\mathcal {J}}}_{2}&=&i\left(\sin \alpha \cot \beta \,{\partial \over \partial \alpha }\,-\cos \alpha \;{\partial \over \partial \beta }\,-{\sin \alpha \over \sin \beta }\,{\partial \over \partial \gamma }\,\right)\\{\hat {\mathcal {J}}}_{3}&=&-i\;{\partial \over \partial \alpha },\end{array}}}$

which have quantum mechanical meaning: they are space-fixed rigid rotor angular momentum operators.

Further,

${\displaystyle {\begin{array}{lcl}{\hat {\mathcal {P}}}_{1}&=&\,i\left({\cos \gamma \over \sin \beta }{\partial \over \partial \alpha }-\sin \gamma {\partial \over \partial \beta }-\cot \beta \cos \gamma {\partial \over \partial \gamma }\right)\\{\hat {\mathcal {P}}}_{2}&=&\,i\left(-{\sin \gamma \over \sin \beta }{\partial \over \partial \alpha }-\cos \gamma {\partial \over \partial \beta }+\cot \beta \sin \gamma {\partial \over \partial \gamma }\right)\\{\hat {\mathcal {P}}}_{3}&=&-i{\partial \over \partial \gamma },\\\end{array}}}$

which have quantum mechanical meaning: they are body-fixed rigid rotor angular momentum operators.

The operators satisfy the commutation relations

${\displaystyle \left[{\mathcal {J}}_{1},\,{\mathcal {J}}_{2}\right]=i{\mathcal {J}}_{3},\qquad {\hbox{and}}\qquad \left[{\mathcal {P}}_{1},\,{\mathcal {P}}_{2}\right]=-i{\mathcal {P}}_{3}}$

and the corresponding relations with the indices permuted cyclically. The ${\displaystyle {\mathcal {P}}_{i}}$ satisfy anomalous commutation relations (have a minus sign on the right hand side).

The two sets mutually commute,

${\displaystyle \left[{\mathcal {P}}_{i},\,{\mathcal {J}}_{j}\right]=0,\quad i,\,j=1,\,2,\,3,}$

and the total operators squared are equal,

${\displaystyle {\mathcal {J}}^{2}\equiv {\mathcal {J}}_{1}^{2}+{\mathcal {J}}_{2}^{2}+{\mathcal {J}}_{3}^{2}={\mathcal {P}}^{2}\equiv {\mathcal {P}}_{1}^{2}+{\mathcal {P}}_{2}^{2}+{\mathcal {P}}_{3}^{2}.}$

Their explicit form is,

${\displaystyle {\mathcal {J}}^{2}={\mathcal {P}}^{2}=-{\frac {1}{\sin ^{2}\beta }}\left({\frac {\partial ^{2}}{\partial \alpha ^{2}}}+{\frac {\partial ^{2}}{\partial \gamma ^{2}}}-2\cos \beta {\frac {\partial ^{2}}{\partial \alpha \partial \gamma }}\right)-{\frac {\partial ^{2}}{\partial \beta ^{2}}}-\cot \beta {\frac {\partial }{\partial \beta }}.}$

The operators ${\displaystyle {\mathcal {J}}_{i}}$ act on the first (row) index of the D-matrix,

${\displaystyle {\mathcal {J}}_{3}\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}=m'\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*},}$

and

${\displaystyle ({\mathcal {J}}_{1}\pm i{\mathcal {J}}_{2})\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}={\sqrt {j(j+1)-m'(m'\pm 1)}}\,D_{m'\pm 1,m}^{j}(\alpha ,\beta ,\gamma )^{*}.}$

The operators ${\displaystyle {\mathcal {P}}_{i}}$ act on the second (column) index of the D-matrix

${\displaystyle {\mathcal {P}}_{3}\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}=m\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*},}$

and because of the anomalous commutation relation the raising/lowering operators are defined with reversed signs,

${\displaystyle ({\mathcal {P}}_{1}\mp i{\mathcal {P}}_{2})\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}={\sqrt {j(j+1)-m(m\pm 1)}}\,D_{m',m\pm 1}^{j}(\alpha ,\beta ,\gamma )^{*}.}$

Finally,

${\displaystyle {\mathcal {J}}^{2}\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}={\mathcal {P}}^{2}\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}=j(j+1)D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}.}$

In other words, the rows and columns of the (complex conjugate) Wigner D-matrix span irreducible representations of the isomorphic Lie algebra's generated by ${\displaystyle \{{\mathcal {J}}_{i}\}}$ and ${\displaystyle \{-{\mathcal {P}}_{i}\}}$.

An important property of the Wigner D-matrix follows from the commutation of ${\displaystyle {\mathcal {R}}(\alpha ,\beta ,\gamma )}$ with the time reversal operator ${\displaystyle T\,}$,

${\displaystyle \langle jm'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|jm\rangle =\langle jm'|T^{\,\dagger }{\mathcal {R}}(\alpha ,\beta ,\gamma )T|jm\rangle =(-1)^{m'-m}\langle j,-m'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|j,-m\rangle ^{*},}$

or

${\displaystyle D_{m'm}^{j}(\alpha ,\beta ,\gamma )=(-1)^{m'-m}D_{-m',-m}^{j}(\alpha ,\beta ,\gamma )^{*}.}$

Here we used that ${\displaystyle T\,}$ is anti-unitary (hence the complex conjugation after moving ${\displaystyle T^{\dagger }\,}$ from ket to bra), ${\displaystyle T|jm\rangle =(-1)^{j-m}|j,-m\rangle }$ and ${\displaystyle (-1)^{2j-m'-m}=(-1)^{m'-m}}$.

### Relation with spherical harmonic functions

The D-matrix elements with second index equal to zero, are proportional to spherical harmonics, normalized to unity and with Condon and Shortley phase convention,

${\displaystyle D_{m0}^{\ell }(\alpha ,\beta ,\gamma )^{*}={\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell }^{m}(\beta ,\alpha ).}$

The left-hand side contains γ in the expression exp[−i⋅0⋅γ] and hence does not depend on γ. The angle γ does not appear in the right-hand side either, so that the expression is valid for any γ. In the present convention of Euler angles, ${\displaystyle \alpha }$ is a longitudinal angle and ${\displaystyle \beta }$ 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

${\displaystyle \left(Y_{\ell }^{m}\right)^{*}=(-1)^{m}Y_{\ell }^{-m}.}$

## References

Cited references

1. E. Wigner, Einige Folgerungen aus der Schrödingerschen Theorie für die Termstrukturen [Some consequences from Schödinger's theory for term structures], Zeitschrift für Physik vol. 43, pp. 601–623 (1927). Reprinted in: L. C. Biedenharn and H. van Dam, Quantum Theory of Angular Momentum, Academic Press, New York (1965)
2. 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).
3. L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, Addison-Wesley, Reading, (1981).