In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.
Definition Several conventions have been used to define the VSH. We follow that of Barrera et al.. Given a scalar spherical harmonic Yℓm(θ, φ), we define three VSH:
Y ℓ m = Y ℓ m r ^ , {\displaystyle \mathbf {Y} _{\ell m}=Y_{\ell m}{\hat {\mathbf {r} }},}
Ψ ℓ m = r ∇ Y ℓ m , {\displaystyle \mathbf {\Psi } _{\ell m}=r\nabla Y_{\ell m},}
Φ ℓ m = r × ∇ Y ℓ m , {\displaystyle \mathbf {\Phi } _{\ell m}=\mathbf {r} \times \nabla Y_{\ell m},}
with r ^ {\displaystyle {\hat {\mathbf {r} }}} being the unit vector along the radial direction in spherical coordinates and r {\displaystyle \mathbf {r} } the vector along the radial direction with the same norm as the radius, i.e., r = r r ^ {\displaystyle \mathbf {r} =r{\hat {\mathbf {r} }}} . The radial factors are included to guarantee that the dimensions of the VSH are the same as those of the ordinary spherical harmonics and that the VSH do not depend on the radial spherical coordinate. The interest of these new vector fields is to separate the radial dependence from the angular one when using spherical coordinates, so that a vector field admits a multipole expansion
E = ∑ ℓ = 0 ∞ ∑ m = − ℓ ℓ ( E ℓ m r ( r ) Y ℓ m + E ℓ m ( 1 ) ( r ) Ψ ℓ m + E ℓ m ( 2 ) ( r ) Φ ℓ m ) . {\displaystyle \mathbf {E} =\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }\left(E_{\ell m}^{r}(r)\mathbf {Y} _{\ell m}+E_{\ell m}^{(1)}(r)\mathbf {\Psi } _{\ell m}+E_{\ell m}^{(2)}(r)\mathbf {\Phi } _{\ell m}\right).}
The labels on the components reflect that E ℓ m r {\displaystyle E_{\ell m}^{r}} is the radial component of the vector field, while E ℓ m ( 1 ) {\displaystyle E_{\ell m}^{(1)}} and E ℓ m ( 2 ) {\displaystyle E_{\ell m}^{(2)}} are transverse components (with respect to the radius vector r {\displaystyle \mathbf {r} } ).
In physics In physics, the vector spherical harmonics Y j , ℓ , s m j {\displaystyle \mathbf {Y} _{j,\ell ,s}^{m_{j}}} are defined as spin s = 1 {\textstyle s=1} eigenfunctions of the angular momentum operators J 2 , J z , L 2 {\textstyle J^{2},J_{z},L^{2}} , and S 2 {\textstyle S^{2}} , where J = L + S {\textstyle \mathbf {J} =\mathbf {L} +\mathbf {S} } is the total angular momentum. They are written as Y j , ℓ , 1 m j ( k ) = ∑ m ℓ = − ℓ + ℓ ∑ m s = − 1 + 1 ⟨ j m j | ℓ 1 m ℓ m s ⟩ Y ℓ m ℓ ( k ) e ^ m s , {\displaystyle \mathbf {Y} _{j,\ell ,1}^{m_{j}}(\mathbf {k} )=\sum _{m_{\ell }\,=\,-\ell }^{+\ell }~\sum _{m_{s}\,=\,-1}^{+1}\langle j~m_{j}|\ell ~1~m_{\ell }~m_{s}\rangle Y_{\ell }^{m_{\ell }}(\mathbf {k} )\,{\hat {\mathbf {e} }}_{m_{s}},} which are linear combinations of the scalar spherical harmonics Y ℓ m ℓ {\displaystyle Y_{\ell }^{m_{\ell }}} with the vector angular momentum basis e ^ ± 1 = ∓ x ^ ± i y ^ 2 , e ^ 0 = z ^ . {\displaystyle {\hat {\mathbf {e} }}_{\pm 1}=\mp {\frac {{\hat {\mathbf {x} }}\pm i{\hat {\mathbf {y} }}}{\sqrt {2}}},\quad {\hat {\mathbf {e} }}_{0}={\hat {\mathbf {z} }}.} using the Clebsch-Gordan coefficients ⟨ j m j | ℓ 1 m ℓ m s ⟩ {\displaystyle \langle j~m_{j}|\ell ~1~m_{\ell }~m_{s}\rangle } . Because vector bosons such as the photon are spin-one, the vector spherical harmonics are commonly used in physics to describe vector and pseudovector interactions, such as electromagnetic transitions, in atomic and nuclear systems. They are a special ( s = 1 {\textstyle s=1} ) case of the spin spherical harmonics. To derive these relations, one begins with the plane-wave expansion for plane waves with vector polarization.
Main properties
Symmetry Like the scalar spherical harmonics, the VSH satisfy
Y ℓ , − m = ( − 1 ) m Y ℓ m ∗ , Ψ ℓ , − m = ( − 1 ) m Ψ ℓ m ∗ , Φ ℓ , − m = ( − 1 ) m Φ ℓ m ∗ , {\displaystyle {\begin{aligned}\mathbf {Y} _{\ell ,-m}&=(-1)^{m}\mathbf {Y} _{\ell m}^{*},\\\mathbf {\Psi } _{\ell ,-m}&=(-1)^{m}\mathbf {\Psi } _{\ell m}^{*},\\\mathbf {\Phi } _{\ell ,-m}&=(-1)^{m}\mathbf {\Phi } _{\ell m}^{*},\end{aligned}}}
which cuts the number of independent functions roughly in half. The star indicates complex conjugation.
Orthogonality The VSH are orthogonal in the usual three-dimensional way at each point r {\displaystyle \mathbf {r} } :
Y ℓ m ( r ) ⋅ Ψ ℓ m ( r ) = 0 , Y ℓ m ( r ) ⋅ Φ ℓ m ( r ) = 0 , Ψ ℓ m ( r ) ⋅ Φ ℓ m ( r ) = 0. {\displaystyle {\begin{aligned}\mathbf {Y} _{\ell m}(\mathbf {r} )\cdot \mathbf {\Psi } _{\ell m}(\mathbf {r} )&=0,\\\mathbf {Y} _{\ell m}(\mathbf {r} )\cdot \mathbf {\Phi } _{\ell m}(\mathbf {r} )&=0,\\\mathbf {\Psi } _{\ell m}(\mathbf {r} )\cdot \mathbf {\Phi } _{\ell m}(\mathbf {r} )&=0.\end{aligned}}}
They are also orthogonal in Hilbert space:
∫ Y ℓ m ⋅ Y ℓ ′ m ′ ∗ d Ω = δ ℓ ℓ ′ δ m m ′ , ∫ Ψ ℓ m ⋅ Ψ ℓ ′ m ′ ∗ d Ω = ℓ ( ℓ + 1 ) δ ℓ ℓ ′ δ m m ′ , ∫ Φ ℓ m ⋅ Φ ℓ ′ m ′ ∗ d Ω = ℓ ( ℓ + 1 ) δ ℓ ℓ ′ δ m m ′ , ∫ Y ℓ m ⋅ Ψ ℓ ′ m ′ ∗ d Ω = 0 , ∫ Y ℓ m ⋅ Φ ℓ ′ m ′ ∗ d Ω = 0 , ∫ Ψ ℓ m ⋅ Φ ℓ ′ m ′ ∗ d Ω = 0. {\displaystyle {\begin{aligned}\int \mathbf {Y} _{\ell m}\cdot \mathbf {Y} _{\ell 'm'}^{*}\,d\Omega &=\delta _{\ell \ell '}\delta _{mm'},\\\int \mathbf {\Psi } _{\ell m}\cdot \mathbf {\Psi } _{\ell 'm'}^{*}\,d\Omega &=\ell (\ell +1)\delta _{\ell \ell '}\delta _{mm'},\\\int \mathbf {\Phi } _{\ell m}\cdot \mathbf {\Phi } _{\ell 'm'}^{*}\,d\Omega &=\ell (\ell +1)\delta _{\ell \ell '}\delta _{mm'},\\\int \mathbf {Y} _{\ell m}\cdot \mathbf {\Psi } _{\ell 'm'}^{*}\,d\Omega &=0,\\\int \mathbf {Y} _{\ell m}\cdot \mathbf {\Phi } _{\ell 'm'}^{*}\,d\Omega &=0,\\\int \mathbf {\Psi } _{\ell m}\cdot \mathbf {\Phi } _{\ell 'm'}^{*}\,d\Omega &=0.\end{aligned}}}
An additional result at a single point r {\displaystyle \mathbf {r} } (not reported in Barrera et al., 1985) is, for all ℓ , m , ℓ ′ , m ′ {\displaystyle \ell ,m,\ell ',m'} ,
Y ℓ m ( r ) ⋅ Ψ ℓ ′ m ′ ( r ) = 0 , Y ℓ m ( r ) ⋅ Φ ℓ ′ m ′ ( r ) = 0. {\displaystyle {\begin{aligned}\mathbf {Y} _{\ell m}(\mathbf {r} )\cdot \mathbf {\Psi } _{\ell 'm'}(\mathbf {r} )&=0,\\\mathbf {Y} _{\ell m}(\mathbf {r} )\cdot \mathbf {\Phi } _{\ell 'm'}(\mathbf {r} )&=0.\end{aligned}}}
Vector multipole moments The orthogonality relations allow one to compute the spherical multipole moments of a vector field as
E ℓ m r = ∫ E ⋅ Y ℓ m ∗ d Ω , E ℓ m ( 1 ) = 1 ℓ ( ℓ + 1 ) ∫ E ⋅ Ψ ℓ m ∗ d Ω , E ℓ m ( 2 ) = 1 ℓ ( ℓ + 1 ) ∫ E ⋅ Φ ℓ m ∗ d Ω . {\displaystyle {\begin{aligned}E_{\ell m}^{r}&=\int \mathbf {E} \cdot \mathbf {Y} _{\ell m}^{*}\,d\Omega ,\\E_{\ell m}^{(1)}&={\frac {1}{\ell (\ell +1)}}\int \mathbf {E} \cdot \mathbf {\Psi } _{\ell m}^{*}\,d\Omega ,\\E_{\ell m}^{(2)}&={\frac {1}{\ell (\ell +1)}}\int \mathbf {E} \cdot \mathbf {\Phi } _{\ell m}^{*}\,d\Omega .\end{aligned}}}
The gradient of a scalar field Given the multipole expansion of a scalar field
ϕ = ∑ ℓ = 0 ∞ ∑ m = − ℓ ℓ ϕ ℓ m ( r ) Y ℓ m ( θ , ϕ ) , {\displaystyle \phi =\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }\phi _{\ell m}(r)Y_{\ell m}(\theta ,\phi ),}
we can express its gradient in terms of the VSH as
∇ ϕ = ∑ ℓ = 0 ∞ ∑ m = − ℓ ℓ ( d ϕ ℓ m d r Y ℓ m + ϕ ℓ m r Ψ ℓ m ) . {\displaystyle \nabla \phi =\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }\left({\frac {d\phi _{\ell m}}{dr}}\mathbf {Y} _{\ell m}+{\frac {\phi _{\ell m}}{r}}\mathbf {\Psi } _{\ell m}\right).}
Divergence For any multipole field we have
∇ ⋅ ( f ( r ) Y ℓ m ) = ( d f d r + 2 r f ) Y ℓ m , ∇ ⋅ ( f ( r ) Ψ ℓ m ) = − ℓ ( ℓ + 1 ) r f Y ℓ m , ∇ ⋅ ( f ( r ) Φ ℓ m ) = 0. {\displaystyle {\begin{aligned}\nabla \cdot \left(f(r)\mathbf {Y} _{\ell m}\right)&=\left({\frac {df}{dr}}+{\frac {2}{r}}f\right)Y_{\ell m},\\\nabla \cdot \left(f(r)\mathbf {\Psi } _{\ell m}\right)&=-{\frac {\ell (\ell +1)}{r}}fY_{\ell m},\\\nabla \cdot \left(f(r)\mathbf {\Phi } _{\ell m}\right)&=0.\end{aligned}}}
By superposition we obtain the divergence of any vector field:
∇ ⋅ E = ∑ ℓ = 0 ∞ ∑ m = − ℓ ℓ ( d E ℓ m r d r + 2 r E ℓ m r − ℓ ( ℓ + 1 ) r E ℓ m ( 1 ) ) Y ℓ m . {\displaystyle \nabla \cdot \mathbf {E} =\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }\left({\frac {dE_{\ell m}^{r}}{dr}}+{\frac {2}{r}}E_{\ell m}^{r}-{\frac {\ell (\ell +1)}{r}}E_{\ell m}^{(1)}\right)Y_{\ell m}.}
We see that the component on Φℓm is always solenoidal.
Curl For any multipole field we have
∇ × ( f ( r ) Y ℓ m ) = − 1 r f Φ ℓ m , ∇ × ( f ( r ) Ψ ℓ m ) = ( d f d r + 1 r f ) Φ ℓ m , ∇ × ( f ( r ) Φ ℓ m ) = − ℓ ( ℓ + 1 ) r f