Wigner D-Matrices

Wigner D-matrices are the matrix elements of rotation operators in an angular momentum basis. They make the representation-theoretic statement

$$SO(3)\ \text{or}\ SU(2) \quad\longrightarrow\quad \text{unitary matrices on }V_j$$

explicit in the basis $\{\lvert j,m\rangle\}_{m=-j}^{j}$.

This page records the mathematical convention used across angular momentum calculations. The underlying Lie algebra is Angular Momentum Algebra, the spinor double-cover issue is SU(2) versus SO(3), and the physical rotation convention is summarized in Notation and Conventions.

Definition

Let $V_j$ be the irreducible spin-$j$ representation space with basis vectors

$$J^2\lvert j,m\rangle = \hbar^2 j(j+1)\lvert j,m\rangle, \qquad J_z\lvert j,m\rangle = \hbar m\lvert j,m\rangle.$$

For an active rotation $R$, let $U_j(R)$ be the corresponding unitary operator on $V_j$. The Wigner D-matrix entries are

$$D^j_{m'm}(R) = \langle j,m'\vert U_j(R)\vert j,m\rangle.$$

The row label is the final $J_z$ label $m'$ and the column label is the initial $J_z$ label $m$. With this convention,

$$U_j(R)\lvert j,m\rangle = \sum_{m'=-j}^{j} \lvert j,m'\rangle D^j_{m'm}(R).$$

Because $U_j(R)$ is unitary, $D^j(R)$ is a unitary matrix:

$$\sum_{m''=-j}^{j} D^j_{m''m}(R)^*D^j_{m''n}(R) = \delta_{mn}.$$

Euler-Angle Convention

The most common angular momentum convention uses a $z$-$y$-$z$ Euler-angle decomposition:

$$R(\alpha,\beta,\gamma) = R_z(\alpha)R_y(\beta)R_z(\gamma).$$

For active quantum rotations this page uses

$$U_j(\alpha,\beta,\gamma) = \exp\left(-\frac{i}{\hbar}\alpha J_z\right) \exp\left(-\frac{i}{\hbar}\beta J_y\right) \exp\left(-\frac{i}{\hbar}\gamma J_z\right).$$

Since the first and last factors are diagonal in the $\lvert j,m\rangle$ basis,

$$D^j_{m'm}(\alpha,\beta,\gamma) = e^{-im'\alpha} d^j_{m'm}(\beta) e^{-im\gamma}.$$

The reduced Wigner matrix, or small d-matrix, is

$$d^j_{m'm}(\beta) = \langle j,m'\vert \exp\left(-\frac{i}{\hbar}\beta J_y\right) \vert j,m\rangle.$$

Different books sometimes reverse the order of active rotations, use passive coordinate rotations, or place the complex conjugate on the spherical harmonics relation. Always compare the definition of $D^j_{m'm}$ before importing signs from a table.

Small d-Matrix Formula

With the Condon–Shortley phase convention, a standard finite-sum expression is

$$\begin{aligned} d^j_{m'm}(\beta) &= \sqrt{ (j+m')!(j-m')!(j+m)!(j-m)! } \\ &\quad\times \sum_k \frac{ (-1)^{k-m'+m} \left(\cos\frac{\beta}{2}\right)^{2j+m-m'-2k} \left(\sin\frac{\beta}{2}\right)^{m'-m+2k} }{ (j+m-k)!\,k!\,(m'-m+k)!\,(j-m'-k)! }. \end{aligned}$$

The sum runs over the integers $k$ for which every factorial argument is a nonnegative integer. This condition is the compact way to encode all allowed terms.

For many calculations, the defining matrix element or a recurrence relation is safer than memorizing the finite sum. The finite sum is best treated as a convention-sensitive table generator.

Low-Spin Examples

For $j=0$,

$$D^0(R)=1.$$

For $j=1/2$, use the ordered basis

$$\left( \lvert\tfrac12,\tfrac12\rangle, \lvert\tfrac12,-\tfrac12\rangle \right).$$

The small d-matrix is

$$d^{1/2}(\beta) = \begin{pmatrix} \cos(\beta/2) & -\sin(\beta/2)\\ \sin(\beta/2) & \cos(\beta/2) \end{pmatrix}.$$

The full Wigner matrix is obtained by multiplying the first row by $e^{-i\alpha/2}$ or $e^{i\alpha/2}$ according to $m'$, and the first or second column by $e^{-i\gamma/2}$ or $e^{i\gamma/2}$ according to $m$.

For $j=1$, use the ordered basis

$$\left( \lvert1,1\rangle, \lvert1,0\rangle, \lvert1,-1\rangle \right).$$

Then

$$d^1(\beta) = \begin{pmatrix} \frac{1+\cos\beta}{2} & -\frac{\sin\beta}{\sqrt2} & \frac{1-\cos\beta}{2} \\ \frac{\sin\beta}{\sqrt2} & \cos\beta & -\frac{\sin\beta}{\sqrt2} \\ \frac{1-\cos\beta}{2} & \frac{\sin\beta}{\sqrt2} & \frac{1+\cos\beta}{2} \end{pmatrix}.$$

This is the spin-$1$ representation of the same rotation, expressed in the spherical basis rather than the Cartesian basis.

Representation Identities

Wigner D-matrices are representation matrices. If $R_1$ and $R_2$ are rotations, then

$$D^j_{m'm}(R_1R_2) = \sum_{n=-j}^{j} D^j_{m'n}(R_1)D^j_{nm}(R_2).$$

The inverse is the Hermitian adjoint:

$$D^j_{m'm}(R^{-1}) = D^j_{mm'}(R)^*.$$

The character of the spin-$j$ representation depends only on the rotation angle $\omega$:

$$\chi_j(R) = \sum_{m=-j}^{j}D^j_{mm}(R) = \frac{ \sin\left((j+\frac12)\omega\right) }{ \sin(\omega/2) }.$$

This character is useful when decomposing representations, checking degeneracies, or comparing $SU(2)$ and $SO(3)$ representations.

Relation to Spherical Harmonics

Spherical harmonics are closely related to Wigner D-matrices. With the Condon–Shortley convention used here,

$$Y_\ell^m(\theta,\phi) = \sqrt{\frac{2\ell+1}{4\pi}}\, D^{\ell}_{m0}(\phi,\theta,0)^*.$$

Equivalently,

$$D^{\ell}_{m0}(\phi,\theta,0) = \sqrt{\frac{4\pi}{2\ell+1}}\, Y_\ell^m(\theta,\phi)^*.$$

This identity is one reason D-matrices appear in orbital angular momentum, rigid rotor problems, molecular rotations, multipole expansions, and angular-momentum coupling.

Rotation of Operators

The same matrices control spherical tensor operators. If $T^k_q$ is a rank-$k$ spherical tensor component, then

$$U(R)T^k_qU(R)^\dagger = \sum_{q'=-k}^{k} D^k_{q'q}(R)T^k_{q'}.$$

This transformation law is the representation-theoretic input behind the Wigner–Eckart theorem and selection rules. The matrix elements of $T^k_q$ are constrained because states and operators transform in compatible $SU(2)$ representations.

Applications

Wigner D-matrices appear whenever a calculation needs the explicit action of rotations on angular momentum states:

• rotating spin states or preparing spin coherent states;
• transforming spherical harmonics under a rotated coordinate axis;
• solving rigid rotor and molecular spectroscopy problems;
• expressing angular distributions in scattering and decay;
• deriving selection rules for tensor operators;
• relating laboratory-frame and body-frame angular momentum bases;
• building numerical rotation matrices for finite-dimensional spin systems.

They are not new dynamical laws. They are the representation matrices for rotations in a chosen angular momentum basis.

Common Mistakes

• Mixing active and passive rotation conventions without taking an inverse or complex conjugate.
• Reading $D^j_{m'm}$ with the row and column labels reversed.
• Forgetting the phase factors $e^{-im'\alpha}$ and $e^{-im\gamma}$ around the small d-matrix.
• Importing a table that uses a different Euler-angle order.
• Treating the $j=1$ spherical-basis matrix as the same array as the Cartesian $3\times3$ rotation matrix.
• Forgetting that half-integer $j$ is a representation of $SU(2)$ and only projectively represents $SO(3)$.

Cross-Links

• SU(2)
• SO(3)
• SU(2) versus SO(3)
• Angular Momentum Algebra
• Ladder Operators as Lie Algebra Tools
• Clebsch–Gordan Coefficients
• Wigner 3j, 6j, and 9j Symbols
• Wigner–Eckart Theorem
• Tensor Product Representations
• Spherical Harmonics
• Spin Rotations
• Notation and Conventions

References

• A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.
• D. M. Brink and G. R. Satchler, Angular Momentum, 3rd ed., Oxford University Press, 1993.
• D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.
• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.

Exercises

1. Show that the Euler-angle factorization gives the phase form of $D^j_{m'm}(\alpha,\beta,\gamma)$.

Solution

Insert the identity in the $\lvert j,n\rangle$ basis between the three exponential factors:

$$\begin{aligned} D^j_{m'm} &= \langle j,m'\vert e^{-i\alpha J_z/\hbar} e^{-i\beta J_y/\hbar} e^{-i\gamma J_z/\hbar} \vert j,m\rangle \\ &= e^{-im'\alpha} \langle j,m'\vert e^{-i\beta J_y/\hbar}\vert j,m\rangle e^{-im\gamma}. \end{aligned}$$

The middle matrix element is $d^j_{m'm}(\beta)$.

2. Verify that the $j=1/2$ small d-matrix is unitary.

Solution

Let $c=\cos(\beta/2)$ and $s=\sin(\beta/2)$. Then

$$d^{1/2}(\beta) = \begin{pmatrix} c & -s\\ s & c \end{pmatrix}.$$

The columns have norms $c^2+s^2=1$ and their inner product is $-cs+sc=0$. Hence the matrix is orthogonal and therefore unitary.

3. Use the spherical harmonic relation to express $D^\ell_{00}(\phi,\theta,0)$.

Solution

For $m=0$,

$$Y_\ell^0(\theta,\phi) = \sqrt{\frac{2\ell+1}{4\pi}}\, D^\ell_{00}(\phi,\theta,0)^*.$$

The $m=0$ spherical harmonic is real in the standard convention, so

$$D^\ell_{00}(\phi,\theta,0) = \sqrt{\frac{4\pi}{2\ell+1}}\, Y_\ell^0(\theta,\phi) = P_\ell(\cos\theta).$$

4. Explain why a spin-$1/2$ D-matrix changes sign under a $2\pi$ rotation while a spin-$1$ D-matrix does not.

Solution

For a rotation about $z$,

$$D^j_{m'm}(0,0,\gamma) = \delta_{m'm}e^{-im\gamma}.$$

If $\gamma=2\pi$ and $j=1/2$, then $m=\pm1/2$ and both diagonal entries equal $-1$. If $j=1$, then $m=1,0,-1$ and all diagonal entries equal $1$. This reflects the fact that half-integer spin is a true representation of $SU(2)$ but only a projective representation of $SO(3)$.