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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 rotor]]s. The matrix was introduced in 1927 by [[Eugene Wigner]].
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| == Definition of the Wigner D-matrix ==
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| Let ''J<sub>x</sub>'', ''J<sub>y</sub>'', ''J<sub>z</sub>'' be generators of the [[Lie algebra]] of SU(2) and SO(3). In [[quantum mechanics]] these
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| three operators are the components of a vector operator known as ''angular momentum''. Examples
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| are the [[Angular_momentum#Angular_momentum_in_quantum_mechanics|angular momentum]] of an electron
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| in an atom, [[Spin (physics)|electronic spin]],and the angular momentum
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| of a [[rigid rotor]]. In all cases the three operators satisfy the following [[commutation relations]],
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| :<math> [J_x,J_y] = i J_z,\quad [J_z,J_x] = i J_y,\quad [J_y,J_z] = i J_x, </math>
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| where ''i'' is the purely [[imaginary number]] and Planck's constant <math>\hbar</math> has been put equal to one. The operator
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| :<math> J^2 = J_x^2 + J_y^2 + J_z^2 </math>
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| is a [[Casimir invariant|Casimir operator]] of SU(2) (or SO(3) as the case may be). | |
| It may be diagonalized together with <math>J_z</math> (the choice of this operator
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| is a convention), which commutes with <math>J^ 2</math>. That is, it can be shown that there is a complete set of kets with
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| :<math> J^2 |jm\rangle = j(j+1) |jm\rangle,\quad J_z |jm\rangle = m |jm\rangle,
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| </math>
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| where ''j'' = 0, 1/2, 1, 3/2, 2,... and ''m'' = -j, -j + 1,..., ''j''. For SO(3) the ''quantum number'' ''j'' is integer.
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| | |
| A [[rotation operator]] can be written as
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| :<math> \mathcal{R}(\alpha,\beta,\gamma) = e^{-i\alpha J_z}e^{-i\beta J_y}e^{-i\gamma J_z},
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| </math>
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| where ''α'', ''β'', ''γ'' are [[Euler angles]] (characterized by the keywords: z-y-z convention, right-handed frame, right-hand screw rule, active interpretation).
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| The '''Wigner D-matrix''' is a square matrix of dimension 2''j'' + 1 with general element
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| :<math> D^j_{m'm}(\alpha,\beta,\gamma) \equiv
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| \langle jm' | \mathcal{R}(\alpha,\beta,\gamma)| jm \rangle =
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| e^{-im'\alpha } d^j_{m'm}(\beta)e^{-i m\gamma}.
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| </math>
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| The matrix with general element
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| :<math>
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| d^j_{m'm}(\beta)= \langle jm' |e^{-i\beta J_y} | jm \rangle
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| </math>
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| is known as '''Wigner's (small) d-matrix'''.
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| == Wigner (small) d-matrix ==
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| Wigner<ref>{{cite book |first=E. P. |last=Wigner |title={{lang|de|Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren}} |publisher=Vieweg Verlag |location=Braunschweig |year=1931 }} Translated into English by {{cite book |first=J. J. |last=Griffin |title=Group Theory and its Application to the Quantum Mechanics of Atomic Spectra |publisher=Academic Press |location=New York |year=1959 }}</ref> gave the following expression
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| :<math>
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| \begin{array}{lcl}
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| d^j_{m'm}(\beta) &=& [(j+m')!(j-m')!(j+m)!(j-m)!]^{1/2}
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| \sum\limits_s \left[\frac{(-1)^{m'-m+s}}{(j+m-s)!s!(m'-m+s)!(j-m'-s)!} \right.\\
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| &&\left. \cdot \left(\cos\frac{\beta}{2}\right)^{2j+m-m'-2s}\left(\sin\frac{\beta}{2}\right)^{m'-m+2s} \right].
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| \end{array}
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| </math>
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| The sum over ''s'' is over such values that the factorials are nonnegative.
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| ''Note:'' The d-matrix elements defined here are real. In the often-used z-x-z convention of [[Euler_angles#Conventions|Euler angles]], the factor <math>(-1)^{m'-m+s}</math> in this formula is replaced by <math>(-1)^s\, i^{m-m'}</math>, 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.
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| The d-matrix elements are related to [[Jacobi polynomials]] <math>P^{(a,b)}_k(\cos\beta)</math> with nonnegative <math>a\,</math> and <math>b\,</math>.<ref>{{cite book |first=L. C. |last=Biedenharn |first2=J. D. |last2=Louck |title=Angular Momentum in Quantum Physics |publisher=Addison-Wesley |location=Reading |year=1981 |isbn=0-201-13507-8 }}</ref> Let
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| :<math> k = \min(j+m,\,j-m,\,j+m',\,j-m').
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| </math> | |
|
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| :<math>
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| \hbox{If}\quad k =
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| \begin{cases}
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| j+m: &\quad a=m'-m;\quad \lambda=m'-m\\
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| j-m: &\quad a=m-m';\quad \lambda= 0 \\
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| j+m': &\quad a=m-m';\quad \lambda= 0 \\
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| j-m': &\quad a=m'-m;\quad \lambda=m'-m \\
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| \end{cases}
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| </math>
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| | |
| Then, with <math>b=2j-2k-a\,</math>, the relation is
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| :<math>
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| d^j_{m'm}(\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^{(a,b)}_k(\cos\beta),
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| </math>
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| where <math> a,b \ge 0. \, </math>
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| | |
| == Properties of the Wigner D-matrix ==
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| The complex conjugate of the D-matrix satisfies a number of differential properties
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| that can be formulated concisely by introducing the following operators with <math>(x,\, y,\,z) = (1,\,2,\,3)</math>,
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| :<math>
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| \begin{array}{lcl}
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| \hat{\mathcal{J}}_1 &=& i \left( \cos \alpha \cot \beta \,
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| {\partial \over \partial \alpha} \, + \sin \alpha \,
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| {\partial \over \partial \beta} \, - {\cos \alpha \over \sin \beta} \,
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| {\partial \over \partial \gamma} \, \right) \\
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| \hat{\mathcal{J}}_2 &=& i \left( \sin \alpha \cot \beta \,
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| {\partial \over \partial \alpha} \, - \cos \alpha \;
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| {\partial \over \partial \beta } \, - {\sin \alpha \over \sin \beta} \,
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| {\partial \over \partial \gamma } \, \right) \\
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| \hat{\mathcal{J}}_3 &=& - i \; {\partial \over \partial \alpha} ,
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| \end{array}
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| </math>
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| which have quantum mechanical meaning: they are space-fixed [[rigid rotor]] angular momentum operators.
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| Further,
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| :<math>
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| \begin{array}{lcl}
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| \hat{\mathcal{P}}_1 &=& \, i \left( {\cos \gamma \over \sin \beta}
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| {\partial \over \partial \alpha } - \sin \gamma
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| {\partial \over \partial \beta }
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| - \cot \beta \cos \gamma {\partial \over \partial \gamma} \right)
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| \\
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| \hat{\mathcal{P}}_2 &=& \, i \left( - {\sin \gamma \over \sin \beta}
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| {\partial \over \partial \alpha} - \cos \gamma
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| {\partial \over \partial \beta}
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| + \cot \beta \sin \gamma {\partial \over \partial \gamma} \right)
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| \\
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| \hat{\mathcal{P}}_3 &=& - i {\partial\over \partial \gamma}, \\
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| \end{array}
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| </math>
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| which have quantum mechanical meaning: they are body-fixed [[rigid rotor]] angular momentum operators.
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| The operators satisfy the [[commutation relations]]
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| :<math>
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| \left[\mathcal{J}_1, \, \mathcal{J}_2\right] = i \mathcal{J}_3, \qquad \hbox{and}\qquad
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| \left[\mathcal{P}_1, \, \mathcal{P}_2\right] = -i \mathcal{P}_3
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| </math>
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| and the corresponding relations with the indices permuted cyclically.
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| The <math>\mathcal{P}_i</math> satisfy ''anomalous commutation relations''
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| (have a minus sign on the right hand side).
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|
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| The two sets mutually commute,
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| :<math>
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| \left[\mathcal{P}_i, \, \mathcal{J}_j\right] = 0,\quad i,\,j = 1,\,2,\,3,
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| </math>
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| and the total operators squared are equal,
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| :<math>
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| \mathcal{J}^2 \equiv \mathcal{J}_1^2+ \mathcal{J}_2^2 + \mathcal{J}_3^2 =
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| \mathcal{P}^2 \equiv \mathcal{P}_1^2+ \mathcal{P}_2^2 + \mathcal{P}_3^2 .
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| </math>
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| Their explicit form is,
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| :<math>
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| \mathcal{J}^2= \mathcal{P}^2 =
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| -\frac{1}{\sin^2\beta} \left(
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| \frac{\partial^2}{\partial \alpha^2}
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| +\frac{\partial^2}{\partial \gamma^2}
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| -2\cos\beta\frac{\partial^2}{\partial\alpha\partial \gamma} \right)
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| -\frac{\partial^2}{\partial \beta^2}
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| -\cot\beta\frac{\partial}{\partial \beta}.
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| </math>
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| The operators <math>\mathcal{J}_i</math> act on the first (row) index of the D-matrix,
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| :<math>
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| \mathcal{J}_3 \, D^j_{m'm}(\alpha,\beta,\gamma)^* =
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| m' \, D^j_{m'm}(\alpha,\beta,\gamma)^* ,
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| </math>
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| and
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| :<math>
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| (\mathcal{J}_1 \pm i \mathcal{J}_2)\, D^j_{m'm}(\alpha,\beta,\gamma)^* =
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| \sqrt{j(j+1)-m'(m'\pm 1)} \, D^j_{m'\pm 1, m}(\alpha,\beta,\gamma)^* .
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| </math>
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| The operators <math>\mathcal{P}_i</math> act on the second (column) index of the D-matrix
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| :<math>
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| \mathcal{P}_3 \, D^j_{m'm}(\alpha,\beta,\gamma)^* =
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| m \, D^j_{m'm}(\alpha,\beta,\gamma)^* ,
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| </math>
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| and because of the anomalous commutation relation the raising/lowering operators
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| are defined with reversed signs,
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| :<math>
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| (\mathcal{P}_1 \mp i \mathcal{P}_2)\, D^j_{m'm}(\alpha,\beta,\gamma)^* =
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| \sqrt{j(j+1)-m(m\pm 1)} \, D^j_{m', m\pm1}(\alpha,\beta,\gamma)^* .
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| </math>
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| Finally,
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| :<math>
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| \mathcal{J}^2\, D^j_{m'm}(\alpha,\beta,\gamma)^* =
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| \mathcal{P}^2\, D^j_{m'm}(\alpha,\beta,\gamma)^* = j(j+1) D^j_{m'm}(\alpha,\beta,\gamma)^*.
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| </math>
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| In other words, the rows and columns of the (complex conjugate) Wigner D-matrix span
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| [[irreducible representations]] of the isomorphic [[Lie algebra|Lie algebra's]] generated by <math>\{\mathcal{J}_i\}</math> and <math>\{-\mathcal{P}_i\}</math>.
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| An important property of the Wigner D-matrix follows from the commutation of
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| <math> \mathcal{R}(\alpha,\beta,\gamma) </math> with the [[T-symmetry#Time reversal in quantum mechanics|time reversal operator]]
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| <math>T\,</math>,
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| :<math>
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| \langle jm' | \mathcal{R}(\alpha,\beta,\gamma)| jm \rangle =
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| \langle jm' | T^{\,\dagger} \mathcal{R}(\alpha,\beta,\gamma) T| jm \rangle =
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| (-1)^{m'-m} \langle j,-m' | \mathcal{R}(\alpha,\beta,\gamma)| j,-m \rangle^*,
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| </math>
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| or
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| :<math> | |
| D^j_{m'm}(\alpha,\beta,\gamma) = (-1)^{m'-m} D^j_{-m',-m}(\alpha,\beta,\gamma)^*.
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| </math>
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| Here we used that <math>T\,</math> is anti-unitary (hence the complex conjugation after moving
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| <math>T^\dagger\,</math> from ket to bra), <math> T | jm \rangle = (-1)^{j-m} | j,-m \rangle</math> and <math>(-1)^{2j-m'-m} = (-1)^{m'-m}</math>.
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| == Orthogonality relations ==
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| The Wigner D-matrix elements <math>D^j_{mk}(\alpha,\beta,\gamma)</math> form a complete set
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| of orthogonal functions of the Euler angles <math>\alpha</math>, <math>\beta,</math> and <math>\gamma</math>:
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| :<math>
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| \int_0^{2\pi} d\alpha \int_0^\pi \sin \beta d\beta \int_0^{2\pi} d\gamma \,\,
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| D^{j'}_{m'k'}(\alpha,\beta,\gamma)^\ast D^j_{mk}(\alpha,\beta,\gamma) =
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| \frac{8\pi^2}{2j+1} \delta_{m'm}\delta_{k'k}\delta_{j'j}.
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| </math>
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| This is a special case of the [[Schur orthogonality relations]].
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| == Kronecker product of Wigner D-matrices, Clebsch-Gordan series ==
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| The set of [[Kronecker product]] matrices
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| :<math>
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| \mathbf{D}^j(\alpha,\beta,\gamma)\otimes \mathbf{D}^{j'}(\alpha,\beta,\gamma)
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| </math>
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| forms a reducible matrix representation of the groups SO(3) and SU(2). Reduction into irreducible components is by the following equation:
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| :<math>
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| D^j_{m k}(\alpha,\beta,\gamma) D^{j'}_{m' k'}(\alpha,\beta,\gamma) =
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| \sum_{J=|j-j'|}^{j+j'} \sum_{M=-J}^J \sum_{K=-J}^J \langle j m j' m' | J M \rangle
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| \langle j k j' k' | J K \rangle
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| D^J_{M K}(\alpha,\beta,\gamma)
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| </math>
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| The symbol <math>\langle j m j' m' | J M \rangle</math> is a
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| [[Clebsch-Gordan coefficient]].
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| == Relation to spherical harmonics and Legendre polynomials ==
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| For integer values of <math>l</math>, the D-matrix elements with second index equal to zero are proportional
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| to [[spherical harmonics]] and [[associated Legendre polynomials]], normalized to unity and with Condon and Shortley phase convention:
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| :<math>
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| D^{\ell}_{m 0}(\alpha,\beta,0) = \sqrt{\frac{4\pi}{2\ell+1}} Y_{\ell}^{m*} (\beta, \alpha ) = \sqrt{\frac{(\ell-m)!}{(\ell+m)!}} \, P_\ell^m ( \cos{\beta} ) \, e^{-i m \alpha }
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| </math>
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| This implies the following relationship for the d-matrix:
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| :<math>
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| d^{\ell}_{m 0}(\beta) = \sqrt{\frac{(\ell-m)!}{(\ell+m)!}} \, P_\ell^m ( \cos{\beta} )
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| </math>
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| When both indices are set to zero, the Wigner D-matrix elements are given by ordinary [[Legendre polynomials]]:
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| :<math>
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| D^{\ell}_{0,0}(\alpha,\beta,\gamma) = d^{\ell}_{0,0}(\beta) = P_{\ell}(\cos\beta).
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| </math>
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| In the present convention of Euler angles, <math>\alpha</math> is
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| a longitudinal angle and <math>\beta</math> is a colatitudinal angle (spherical polar angles
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| in the physical definition of such angles). This is one of the reasons that the ''z''-''y''-''z''
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| [[Euler_angles#Conventions|convention]] is used frequently in molecular physics.
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| From the time-reversal property of the Wigner D-matrix follows immediately
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| :<math>
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| \left( Y_{\ell}^m \right) ^* = (-1)^m Y_{\ell}^{-m}.
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| </math>
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| There exists a more general relationship to the [[spin-weighted spherical harmonics]]:
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| :<math>
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| D^{\ell}_{-m s}(\alpha,\beta,-\gamma) =(-1)^m \sqrt\frac{4\pi}{2{\ell}+1} {}_sY_{{\ell}m}(\beta,\alpha) e^{is\gamma}.
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| </math>
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| == Relation to Bessel functions ==
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| In the limit when <math>\ell \gg m, m^\prime</math> we have <math>D^\ell_{mm^\prime}(\alpha,\beta,\gamma) \approx e^{-im\alpha-im^\prime\gamma}J_{m-m^\prime}(\ell\beta)</math> where <math>J_{m-m^\prime}(\ell\beta)</math> is the [[Bessel function]] and <math> \ell\beta</math> is finite.
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| == List of d-matrix elements ==
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| Using sign convention of Wigner, et al. the d-matrix elements for ''j'' = 1/2, 1, 3/2, and 2 are given below.
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| for ''j'' = 1/2
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| *<math>d_{1/2,1/2}^{1/2} = \cos (\theta/2)</math>
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| *<math>d_{1/2,-1/2}^{1/2} = -\sin (\theta/2)</math>
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| for ''j'' = 1
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| *<math>d_{1,1}^{1} = \frac{1+\cos \theta}{2}</math>
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| *<math>d_{1,0}^{1} = \frac{-\sin \theta}{\sqrt{2}}</math>
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| *<math>d_{1,-1}^{1} = \frac{1-\cos \theta}{2}</math>
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| *
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| *<math>d_{0,0}^{1} = \cos \theta</math>
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| for ''j'' = 3/2
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| *<math>d_{3/2,3/2}^{3/2} = \frac{1+\cos \theta}{2} \cos \frac{\theta}{2}</math>
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| *<math>d_{3/2,1/2}^{3/2} = -\sqrt{3} \frac{1+\cos \theta}{2} \sin \frac{\theta}{2}</math>
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| *<math>d_{3/2,-1/2}^{3/2} = \sqrt{3} \frac{1-\cos \theta}{2} \cos \frac{\theta}{2}</math>
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| *<math>d_{3/2,-3/2}^{3/2} = - \frac{1-\cos \theta}{2} \sin \frac{\theta}{2}</math>
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| *
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| *<math>d_{1/2,1/2}^{3/2} = \frac{3\cos \theta - 1}{2} \cos \frac{\theta}{2}</math>
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| *<math>d_{1/2,-1/2}^{3/2} = - \frac{3\cos \theta + 1}{2} \sin \frac{\theta}{2}</math>
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| | |
| for ''j'' = 2 <ref>{{cite journal | doi = 10.1002/cmr.a.10061 | author = Edén, M.
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| | title = Computer simulations in solid-state NMR. I. Spin dynamics theory| journal = Concepts Magn. Reson.| volume=17A| issue=1| pages=117–154| year=2003|}}</ref>
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| *<math>d_{2,2}^{2} = \frac{1}{4}\left(1 +\cos \theta\right)^2</math>
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| *<math>d_{2,1}^{2} = -\frac{1}{2}\sin \theta \left(1 + \cos \theta\right)</math>
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| *<math>d_{2,0}^{2} = \sqrt{\frac{3}{8}}\sin^2 \theta</math>
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| *<math>d_{2,-1}^{2} = -\frac{1}{2}\sin \theta \left(1 - \cos \theta\right)</math>
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| *<math>d_{2,-2}^{2} = \frac{1}{4}\left(1 -\cos \theta\right)^2</math>
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| *
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| *<math>d_{1,1}^{2} = \frac{1}{2}\left(2\cos^2\theta + \cos \theta-1 \right)</math>
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| *<math>d_{1,0}^{2} = -\sqrt{\frac{3}{8}} \sin 2 \theta</math>
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| *<math>d_{1,-1}^{2} = \frac{1}{2}\left(- 2\cos^2\theta + \cos \theta +1 \right)</math>
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| *
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| *<math>d_{0,0}^{2} = \frac{1}{2} \left(3 \cos^2 \theta - 1\right)</math>
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| Wigner d-matrix elements with swapped lower indices are found with the relation:
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| :<math>d_{m', m}^j = (-1)^{m-m'}d_{m, m'}^j = d_{-m,-m'}^j</math>.
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| == See also ==
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| * [[Clebsch–Gordan coefficients]]
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| * [[Tensor operator]]
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| * [[Symmetries in quantum mechanics]]
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| ==References==
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| <!-- ----------------------------------------------------------
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| See http://en.wikipedia.org/wiki/Wikipedia:Footnotes for a
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| discussion of different citation methods and how to generate
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| footnotes using the<ref>, </ref> and <reference /> tags
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| ----------------------------------------------------------- -->
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| {{Reflist}}
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| ==External links==
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| * [http://pdg.lbl.gov/2008/reviews/clebrpp.pdf PDG Table of Clebsch-Gordon Coefficients, Spherical Harmonics, and d-Functions]
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| [[Category:Representation theory of Lie groups]]
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| [[Category:Matrices]]
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| [[Category:Special hypergeometric functions]]
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| [[Category:Rotational symmetry]]
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