在球坐標系下，拉普拉斯算符 作用在一三維向量場 上可以寫為
∇
2
A
→
(
r
,
θ
,
ϕ
)
=
0
{\displaystyle \nabla ^{2}{\vec {A}}(r,\theta ,\phi )=0}
利用分離變數法 可以將此一方程式的解分解為一系列本徵函數 的線性組合
A
→
=
R
l
(
r
)
Y
m
,
l
(
n
)
(
θ
,
ϕ
)
,
n
=
1
,
2
,
3
{\displaystyle {\vec {A}}=R_{l}(r)\mathbf {Y} _{m,l}^{(n)}(\theta ,\phi ),n=1,2,3}
其中的徑向解
R
l
{\displaystyle R_{l}}
純量球諧函數 相同，而
Y
m
,
l
(
n
)
{\displaystyle \mathbf {Y} _{m,l}^{(n)}}
向量球諧函數 。
[ 1] [ 2] [ 3] [ 4] [ 5] 球諧函數 Yℓm (θ , φ )
Y
l
m
=
Y
l
m
r
^
{\displaystyle \mathbf {Y} _{lm}=Y_{lm}{\hat {\mathbf {r} }}}
Ψ
l
m
=
r
∇
Y
l
m
{\displaystyle \mathbf {\Psi } _{lm}=r\nabla Y_{lm}}
Φ
l
m
=
r
×
∇
Y
l
m
{\displaystyle \mathbf {\Phi } _{lm}=\mathbf {r} \times \nabla Y_{lm}}
這邊
r
{\displaystyle \mathbf {r} }
(r , θ , φ )
r
^
{\displaystyle {\hat {\mathbf {r} }}}
單位向量 。
依照上述 Barrera 的定義，向量球諧函數有以下特性：
與球諧函數相同，向量球諧函數有對稱性
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 \mathbf {Y} _{l,-m}=(-1)^{m}\mathbf {Y} _{lm}^{*}\qquad \mathbf {\Psi } _{l,-m}=(-1)^{m}\mathbf {\Psi } _{lm}^{*}\qquad \mathbf {\Phi } _{l,-m}=(-1)^{m}\mathbf {\Phi } _{lm}^{*}}
星號 * 代表共軛函數 。
三種向量球諧函數彼此兩兩正交
Y
l
m
⋅
Ψ
l
m
=
0
Y
l
m
⋅
Φ
l
m
=
0
Ψ
l
m
⋅
Φ
l
m
=
0
{\displaystyle \mathbf {Y} _{lm}\cdot \mathbf {\Psi } _{lm}=0\qquad \mathbf {Y} _{lm}\cdot \mathbf {\Phi } _{lm}=0\qquad \mathbf {\Psi } _{lm}\cdot \mathbf {\Phi } _{lm}=0}
另外同種類的球諧函數的內積為：
∫
Y
l
m
⋅
Y
l
′
m
′
∗
d
Ω
=
δ
l
l
′
δ
m
m
′
{\displaystyle \int \mathbf {Y} _{lm}\cdot \mathbf {Y} _{l'm'}^{*}\,\mathrm {d} \Omega =\delta _{ll'}\delta _{mm'}}
∫
Ψ
l
m
⋅
Ψ
l
′
m
′
∗
d
Ω
=
l
(
l
+
1
)
δ
l
l
′
δ
m
m
′
{\displaystyle \int \mathbf {\Psi } _{lm}\cdot \mathbf {\Psi } _{l'm'}^{*}\,\mathrm {d} \Omega =l(l+1)\delta _{ll'}\delta _{mm'}}
∫
Φ
l
m
⋅
Φ
l
′
m
′
∗
d
Ω
=
l
(
l
+
1
)
δ
l
l
′
δ
m
m
′
{\displaystyle \int \mathbf {\Phi } _{lm}\cdot \mathbf {\Phi } _{l'm'}^{*}\,\mathrm {d} \Omega =l(l+1)\delta _{ll'}\delta _{mm'}}
對一個純量場
ϕ
{\displaystyle \phi }
多極展開 可表示為：
ϕ
=
∑
l
=
0
∞
∑
m
=
−
l
l
ϕ
l
m
(
r
)
Y
l
m
(
θ
,
ϕ
)
{\displaystyle \phi =\sum _{l=0}^{\infty }\sum _{m=-l}^{l}\phi _{lm}(r)Y_{lm}(\theta ,\phi )}
則其梯度 可以向量球諧函數表示為：
∇
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 _{l=0}^{\infty }\sum _{m=-l}^{l}\left({\frac {\mathrm {d} \phi _{lm}}{\mathrm {d} r}}\mathbf {Y} _{lm}+{\frac {\phi _{lm}}{r}}\mathbf {\Psi } _{lm}\right)}
三種向量球諧函數之散度 分別為：
∇
⋅
(
f
(
r
)
Y
l
m
)
=
(
d
f
d
r
+
2
r
f
)
Y
l
m
{\displaystyle \nabla \cdot \left(f(r)\mathbf {Y} _{lm}\right)=\left({\frac {\mathrm {d} f}{\mathrm {d} r}}+{\frac {2}{r}}f\right)Y_{lm}}
∇
⋅
(
f
(
r
)
Ψ
l
m
)
=
−
l
(
l
+
1
)
r
f
Y
l
m
{\displaystyle \nabla \cdot \left(f(r)\mathbf {\Psi } _{lm}\right)=-{\frac {l(l+1)}{r}}fY_{lm}}
∇
⋅
(
f
(
r
)
Φ
l
m
)
=
0
{\displaystyle \nabla \cdot \left(f(r)\mathbf {\Phi } _{lm}\right)=0}
其中
f
(
r
)
{\textstyle f(r)}
Y
l
m
{\displaystyle Y_{lm}}
球諧函數 。
三種向量球諧函數之旋度 分別為：
∇
×
(
f
(
r
)
Y
l
m
)
=
−
1
r
f
Φ
l
m
{\displaystyle \nabla \times \left(f(r)\mathbf {Y} _{lm}\right)=-{\frac {1}{r}}f\mathbf {\Phi } _{lm}}
∇
×
(
f
(
r
)
Ψ
l
m
)
=
(
d
f
d
r
+
1
r
f
)
Φ
l
m
{\displaystyle \nabla \times \left(f(r)\mathbf {\Psi } _{lm}\right)=\left({\frac {\mathrm {d} f}{\mathrm {d} r}}+{\frac {1}{r}}f\right)\mathbf {\Phi } _{lm}}
∇
×
(
f
(
r
)
Φ
l
m
)
=
−
l
(
l
+
1
)
r
f
Y
l
m
−
(
d
f
d
r
+
1
r
f
)
Ψ
l
m
{\displaystyle \nabla \times \left(f(r)\mathbf {\Phi } _{lm}\right)=-{\frac {l(l+1)}{r}}f\mathbf {Y} _{lm}-\left({\frac {\mathrm {d} f}{\mathrm {d} r}}+{\frac {1}{r}}f\right)\mathbf {\Psi } _{lm}}
其中
f
(
r
)
{\textstyle f(r)}
^ R.G. Barrera, G.A. Estévez and J. Giraldo, Vector spherical harmonics and their application to magnetostatics , Eur. J. Phys. 6 287-294 (1985)
^ B. Carrascal, G.A. Estevez, P. Lee and V. Lorenzo Vector spherical harmonics and their application to classical electrodynamics , Eur. J. Phys., 12 , 184-191 (1991)
^ E. L. Hill, The theory of Vector Spherical Harmonics , Am. J. Phys. 22 , 211-214 (1954)
^ E. J. Weinberg, Monopole vector spherical harmonics , Phys. Rev. D. 49 , 1086-1092 (1994)
^ P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II , New York: McGraw-Hill, 1898-1901 (1953)
