WignerD.jl

Definition

The Wigner D matrix is the matrix representation of the rotation operator in the eigenbasis of the angular momentum operator $J_z$. Representing a general eigenvector of $J_z$ as $\left|jm\right\rangle$ and given a rotation operator $U(R)$, we obtain $D_{mm^{\prime}}^{j}\left(R\right)=\left\langle jm\right|U\left(R\right)\left|jm^{\prime}\right\rangle$. If we represent the rotation in terms of Euler angles, we obtain $U\left(R\right)=e^{-i J_z \alpha}e^{-i J_y \beta}e^{-i J_z \gamma}$. In this representation, we obtain

\[\begin{aligned} D_{mm^{\prime}}^{j}\left(\alpha,\beta,\gamma\right)&=e^{-i\left(m\alpha+m^{\prime}\gamma\right)}\left\langle jm\right|e^{-iJ_{y}\beta}\left|jm^{\prime}\right\rangle \\&=e^{-i\left(m\alpha+m^{\prime}\gamma\right)}d_{mm^{\prime}}^{j}\left(\beta\right) \end{aligned}\]

where $d_{mm^{\prime}}^{j}\left(\beta\right)$ is the Wigner d matrix.

We use the phase convention of Varshalovich et al. (1988). In this conevntion, wavefunctions transform under a rotation of coordinate frames as

\[\psi_{jm^{\prime}}\left(\theta^{\prime},\phi^{\prime},\sigma^{\prime}\right)=\sum_{m}\psi_{jm}\left(\theta,\phi,\sigma\right)D_{mm^{\prime}}^{j}\left(\alpha,\beta,\gamma\right).\]

where $\alpha$, $\beta$ and $\gamma$ are the Euler angles corresponding to the rotation, $(\theta,\phi)$ and $(\theta^\prime,\phi^\prime)$ are the polar coordinates in the initial and rotated coordinate systems $S$ and $S^{\prime}$, and $\sigma$ and $\sigma^{\prime}$ are the spin variables in $S$ and $S^{\prime}$ respectively.

In this package we evaluate the Wigner d matrix following Feng (2015). We diagonalize the operator $J_y$ to obtain its eigenbasis $\left|j _{y}n\right\rangle$. In terms of these, we may obtain

\[\begin{aligned} d_{mm^{\prime}}^{\ell}\left(\beta\right)&=\left\langle jm\right|e^{-iJ_{y}\beta}\left|jm^{\prime}\right\rangle \\&=\left\langle jm|j_{y}p\right\rangle \left\langle j_{y}p\right|e^{-iJ_{y}\beta}\left|j_{y}n\right\rangle \left\langle j_{y}n|jm^{\prime}\right\rangle \\&=e^{-in\beta}\left\langle jm|j_{y}n\right\rangle \left\langle j_{y}n|jm^{\prime}\right\rangle . \end{aligned}\]

Rotations of unit vectors

The rotation of a vector may be described in terms of a rotation operator $U(R)$ as $U(R)\,\mathbf{x}=\mathbf{y}$. In particular, for the Cartesian unit vectors $(\mathbf{e}_x,\,\mathbf{e}_y,\mathbf{e}_z)$ we obtain $U(R)\,\mathbf{e}_i = \mathbf{e}^{\prime}_i$, where $\mathbf{e}^{\prime}_i$ are the set of rotated unit vectors. In index notation, we may express this as a matrix relation $R_{ij}\,\mathbf{e}_i = \mathbf{e}^{\prime}_j$. Note that the rows of $R$ are summed over here, and not columns. The matrix elements $R_{ij}=\left\langle \mathbf{e}_{i}\right|U\left(R\right)\left|\mathbf{e}_{j}\right\rangle$ are given by $R_{ij}=\left\langle \mathbf{e}_{i}|\mathbf{e}_{j}^{\prime}\right\rangle$, and the operator $U(R)$ may be expressed in the Cartesian basis as $U\left(R\right)=\left|\mathbf{e}_{j}^{\prime}\right\rangle \left\langle \mathbf{e}_{j}\right|=\left\langle \mathbf{e}_{i}|\mathbf{e}_{j}^{\prime}\right\rangle \left|\mathbf{e}_{i}\right\rangle \left\langle \mathbf{e}_{j}\right|$.

Often in the analysis of rotation and spherical harmonics, it is convenient to switch to a basis that is referred to as the "spherical covariant" basis by Varshalovich, and is related to the Cartesian basis through a unitary transformation:

\[\begin{aligned} \chi_{-1} & =\frac{1}{\sqrt{2}}\left(\mathbf{e}_{x}-i\mathbf{e}_{y}\right),\\ \chi_{0} & =\mathbf{e}_{z},\\ \chi_{1} & =-\frac{1}{\sqrt{2}}\left(\mathbf{e}_{x}+i\mathbf{e}_{y}\right). \end{aligned}\]

These vectors form an eigenbasis of the angular momentum operator $J_z$ for $j=1$. The rotation of these are also carried out by the same operator: $U(R)\,\chi_ν = \chi^{\prime}_ν$. In index notation this may be represented as $D^{1}(R)_{\nu\mu}\,\chi_ν = \chi^{\prime}_\mu$, where, as before, the rows of $D$ are summed over. The Wigner D matrix $D^{1}(R)$ is therefore the matrix representation of the rotation operator in the spherical basis $\chi_ν$.

If we denote the transformation between the Cartesian and spherical covariant bases as a matrix relation $\chi_μ = U_{\mu i} \mathbf{e}_i$, the relation between the Wigner D matrix $D^{1}_{\mu\nu}(R)$ and the Cartesian Rotation matrix $R_{ij}$ is

\[(D^{1}(R))^* = U R U^{\dagger}\]

The conjugation here is a consequence of the fact that $U$ has its columns summed over, whereas $R$ and $D$ have their rows summed over. If we choose the transpose $V=U^T$, we may rewrite the relation as

\[D^{1}(R) = V^\dagger R V\]

We may also derive this using the bra-ket notation as:

\[\begin{aligned} D_{\mu\nu}^{1}\left(R\right) & =\left\langle \chi_{\mu}\right|U\left(R\right)\left|\chi_{\nu}\right\rangle \\ & =\left\langle \chi_{\mu}|\mathbf{e}_{i}\right\rangle \left\langle \mathbf{e}_{i}\right|U\left(R\right)\left|\mathbf{e}_{j}\right\rangle \left\langle \mathbf{e}_{j}|\chi_{\nu}\right\rangle \\ & =U_{i\mu}^{\dagger}R_{ij}U_{\nu j}\\ & =U_{\mu i}^{*}R_{ij}U_{j\nu}^{T}\\ & =\left[U^{*}RU^{T}\right]_{\mu\nu} \end{aligned}\]

Rotation of Spherical harmonics

Spherical harmonics $Y_{ℓm}(\hat{n})$ form an eigenbasis of $J_z$ for $j=\ell$. We assume that a rotation $R$ takes the coordinate frame $S$ to $S^\prime$. We assume that a point has coordinates $\hat{n}=(\theta,\phi)$ in $S$ and $\hat{n}^\prime=(\theta^\prime,\phi^\prime)$ in $S^\prime$, where $\hat{n}^\prime=R^{-1}\hat{n}$. Under this rotation, spherical harmonics transform as

\[\begin{aligned} Y_{\ell m^{\prime}}\left(\hat{n}^{\prime}\right)&=\sum_{m}D_{mm^{\prime}}^{\ell}\left(R\right)Y_{\ell m}\left(\hat{n}\right)\\&=\sum_{m}D_{mm^{\prime}}^{\ell}\left(R\right)Y_{\ell m}\left(R^{-1}\hat{n}^{\prime}\right). \end{aligned}\]

Reference

Equator

NorthPole

SouthPole

wignerD(j, α::Real, β::Real, γ::Real, [Jy = zeros(ComplexF64, 2j+1, 2j+1)])

wignerD!(D, j, α::Real, β::Real, γ::Real, [Jy = zeros(ComplexF64, 2j+1, 2j+1)])

wignerDjmn(j, m, n, α::Real, β::Real, γ::Real, [Jy::AbstractMatrix{ComplexF64}])

wignerd(j, β::Real, [Jy = zeros(ComplexF64, 2j+1, 2j+1)])

wignerd!(d, j, β::Real, [Jy = zeros(ComplexF64, 2j+1, 2j+1)])

wignerdjmn(j, m, n, β::Real, [Jy::AbstractMatrix{ComplexF64}])