# 向量球諧函數

## 定義

${\displaystyle \nabla ^{2}{\vec {A}}(r,\theta ,\phi )=0}$

${\displaystyle {\vec {A}}=R_{l}(r)\mathbf {Y} _{m,l}^{(n)}(\theta ,\phi ),n=1,2,3}$

• ${\displaystyle \mathbf {Y} _{lm}=Y_{lm}{\hat {\mathbf {r} }}}$
• ${\displaystyle \mathbf {\Psi } _{lm}=r\nabla Y_{lm}}$
• ${\displaystyle \mathbf {\Phi } _{lm}=\mathbf {r} \times \nabla Y_{lm}}$

## 主要特性

### 對稱性

${\displaystyle \mathbf {Y} _{l,-m}=(-1)^{m}\mathbf {Y} _{lm}^{*}\qquad \mathbf {\Psi } _{l,-m}=(-1)^{m}\mathbf {\Psi } _{lm}^{*}\qquad \mathbf {\Phi } _{l,-m}=(-1)^{m}\mathbf {\Phi } _{lm}^{*}}$

### 正交性

${\displaystyle \mathbf {Y} _{lm}\cdot \mathbf {\Psi } _{lm}=0\qquad \mathbf {Y} _{lm}\cdot \mathbf {\Phi } _{lm}=0\qquad \mathbf {\Psi } _{lm}\cdot \mathbf {\Phi } _{lm}=0}$

${\displaystyle \int \mathbf {Y} _{lm}\cdot \mathbf {Y} _{l'm'}^{*}\,\mathrm {d} \Omega =\delta _{ll'}\delta _{mm'}}$
${\displaystyle \int \mathbf {\Psi } _{lm}\cdot \mathbf {\Psi } _{l'm'}^{*}\,\mathrm {d} \Omega =l(l+1)\delta _{ll'}\delta _{mm'}}$
${\displaystyle \int \mathbf {\Phi } _{lm}\cdot \mathbf {\Phi } _{l'm'}^{*}\,\mathrm {d} \Omega =l(l+1)\delta _{ll'}\delta _{mm'}}$

### 純量場的梯度

${\displaystyle \phi =\sum _{l=0}^{\infty }\sum _{m=-l}^{l}\phi _{lm}(r)Y_{lm}(\theta ,\phi )}$

${\displaystyle \nabla \phi =\sum _{l=0}^{\infty }\sum _{m=-l}^{l}\left({\frac {\mathrm {d} \phi _{lm}}{\mathrm {d} r}}\mathbf {Y} _{lm}+{\frac {\phi _{lm}}{r}}\mathbf {\Psi } _{lm}\right)}$

### 散度

${\displaystyle \nabla \cdot \left(f(r)\mathbf {Y} _{lm}\right)=\left({\frac {\mathrm {d} f}{\mathrm {d} r}}+{\frac {2}{r}}f\right)Y_{lm}}$
${\displaystyle \nabla \cdot \left(f(r)\mathbf {\Psi } _{lm}\right)=-{\frac {l(l+1)}{r}}fY_{lm}}$
${\displaystyle \nabla \cdot \left(f(r)\mathbf {\Phi } _{lm}\right)=0}$

### 旋度

${\displaystyle \nabla \times \left(f(r)\mathbf {Y} _{lm}\right)=-{\frac {1}{r}}f\mathbf {\Phi } _{lm}}$
${\displaystyle \nabla \times \left(f(r)\mathbf {\Psi } _{lm}\right)=\left({\frac {\mathrm {d} f}{\mathrm {d} r}}+{\frac {1}{r}}f\right)\mathbf {\Phi } _{lm}}$
${\displaystyle \nabla \times \left(f(r)\mathbf {\Phi } _{lm}\right)=-{\frac {l(l+1)}{r}}f\mathbf {Y} _{lm}-\left({\frac {\mathrm {d} f}{\mathrm {d} r}}+{\frac {1}{r}}f\right)\mathbf {\Psi } _{lm}}$

## 運用

### 電動力學

${\displaystyle \triangledown ^{2}\mathbf {E} +k_{m}^{2}\mathbf {E} =0}$
${\displaystyle \triangledown ^{2}\mathbf {H} +k_{m}^{2}\mathbf {H} =0}$

## 參考資料

1. R.G. Barrera, G.A. Estévez and J. Giraldo, Vector spherical harmonics and their application to magnetostatics, Eur. J. Phys. 6 287-294 (1985)
2. 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)
3. E. L. Hill, The theory of Vector Spherical Harmonics, Am. J. Phys. 22, 211-214 (1954)
4. E. J. Weinberg, Monopole vector spherical harmonics, Phys. Rev. D. 49, 1086-1092 (1994)
5. P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II, New York: McGraw-Hill, 1898-1901 (1953)