|
In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates. There are two kinds: the ''regular solid harmonics'' , which vanish at the origin and the ''irregular solid harmonics'' , which are singular at the origin. Both sets of functions play an important role in potential theory, and are obtained by rescaling spherical harmonics appropriately: : R^m_(\mathbf) \equiv \sqrt}\; r^\ell Y^m_(\theta,\varphi) : I^m_(\mathbf) \equiv \sqrt} \; \frac{r^{\ell+1}} == Derivation, relation to spherical harmonics == Introducing ''r'', θ, and φ for the spherical polar coordinates of the 3-vector r, we can write the Laplace equation in the following form : where ''l''2 is the square of the nondimensional angular momentum operator, : It is known that spherical harmonics Yml are eigenfunctions of ''l''2: : \hat l^2 Y^m_\equiv \left(^2 +\hat l^2_y+\hat l^2_z\right )Y^m_ = \ell(\ell+1) Y^m_. Substitution of Φ(r) = ''F''(''r'') Yml into the Laplace equation gives, after dividing out the spherical harmonic function, the following radial equation and its general solution, : \frac\fracr F(r) = \frac F(r) \Longrightarrow F(r) = A r^\ell + B r^. The particular solutions of the total Laplace equation are regular solid harmonics: : R^m_(\mathbf) \equiv \sqrt}\; r^\ell Y^m_(\theta,\varphi), and irregular solid harmonics: : I^m_(\mathbf) \equiv \sqrt} \; \frac{r^{\ell+1}} . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「solid harmonics」の詳細全文を読む スポンサード リンク
|