球面波
r = x 2 + y 2 + z 2 {\displaystyle r={\sqrt {x^{2}+y^{2}+z^{2}}}} このとき ∂ r ∂ x = 2 x 2 x 2 + y 2 + z 2 = x r {\displaystyle {\dfrac {\partial r}{\partial x}}={\dfrac {2x}{2{\sqrt {x^{2}+y^{2}+z^{2}}}}}={\dfrac {x}{r}}} したがって ∂ ψ ∂ x = ∂ ψ ∂ r ∂ r ∂ x = x r ∂ ψ ∂ r , ∂ 2 ψ ∂ x 2 = ∂ ∂ x ( x r ∂ ψ ∂ r ) = 1 r ∂ ψ ∂ r + x r ∂ ∂ r ( ∂ r ∂ x ∂ ψ ∂ r ) = 1 r ∂ ψ ∂ r + x r ∂ ∂ r ( x r ∂ ψ ∂ r ) = 1 r ∂ ψ ∂ r + x r ( − x r 2 ∂ ψ ∂ r + x r ∂ 2 ψ ∂ r 2 ) = r 2 − x 2 r 3 ∂ ψ ∂ r + x 2 r 2 ∂ 2 ψ ∂ r 2 {\displaystyle {\begin{aligned}{\frac {\partial \psi }{\partial x}}&={\frac {\partial \psi }{\partial r}}{\frac {\partial r}{\partial x}}={\frac {x}{r}}{\frac {\partial \psi }{\partial r}},\\{\frac {\partial ^{2}\psi }{\partial x^{2}}}&={\frac {\partial }{\partial x}}\left({\frac {x}{r}}{\frac {\partial \psi }{\partial r}}\right)={\frac {1}{r}}{\frac {\partial \psi }{\partial r}}+{\frac {x}{r}}{\frac {\partial }{\partial r}}\left({\frac {\partial r}{\partial x}}{\frac {\partial \psi }{\partial r}}\right)={\frac {1}{r}}{\frac {\partial \psi }{\partial r}}+{\frac {x}{r}}{\frac {\partial }{\partial r}}\left({\frac {x}{r}}{\frac {\partial \psi }{\partial r}}\right)={\frac {1}{r}}{\frac {\partial \psi }{\partial r}}+{\frac {x}{r}}\left(-{\frac {x}{r^{2}}}{\frac {\partial \psi }{\partial r}}+{\frac {x}{r}}{\frac {\partial ^{2}\psi }{\partial r^{2}}}\right)\\&={\frac {r^{2}-x^{2}}{r^{3}}}{\frac {\partial \psi }{\partial r}}+{\frac {x^{2}}{r^{2}}}{\frac {\partial ^{2}\psi }{\partial r^{2}}}\end{aligned}}} 同様に ∂ 2 ψ ∂ y 2 = r 2 − y 2 r 3 ∂ ψ ∂ r + y 2 r 2 ∂ 2 ψ ∂ r 2 , ∂ 2 ψ ∂ z 2 = r 2 − z 2 r 3 ∂ ψ ∂ r + z 2 r 2 ∂ 2 ψ ∂ r 2 {\displaystyle {\begin{aligned}{\frac {\partial ^{2}\psi }{\partial y^{2}}}&={\frac {r^{2}-y^{2}}{r^{3}}}{\frac {\partial \psi }{\partial r}}+{\frac {y^{2}}{r^{2}}}{\frac {\partial ^{2}\psi }{\partial r^{2}}},\\{\frac {\partial ^{2}\psi }{\partial z^{2}}}&={\frac {r^{2}-z^{2}}{r^{3}}}{\frac {\partial \psi }{\partial r}}+{\frac {z^{2}}{r^{2}}}{\frac {\partial ^{2}\psi }{\partial r^{2}}}\end{aligned}}} したがって Δ ψ = 3 r 2 − ( x 2 + y 2 + z 2 ) r 3 ∂ ψ ∂ r + x 2 + y 2 + z 2 r 2 ∂ 2 ψ ∂ r 2 = 2 r ∂ ψ ∂ r + ∂ 2 ψ ∂ r 2 {\displaystyle {\begin{aligned}\Delta \psi &={\frac {3r^{2}-(x^{2}+y^{2}+z^{2})}{r^{3}}}{\frac {\partial \psi }{\partial r}}+{\frac {x^{2}+y^{2}+z^{2}}{r^{2}}}{\frac {\partial ^{2}\psi }{\partial r^{2}}}\\&={\frac {2}{r}}{\frac {\partial \psi }{\partial r}}+{\frac {\partial ^{2}\psi }{\partial r^{2}}}\end{aligned}}}
次ページ記事の検索おまかせリスト▼オプションを表示暇つぶしWikipedia
Size:37 KB
出典: フリー百科事典『ウィキペディア（Wikipedia）』
担当:undef