直交曲線座標系一覧
三次元のシュレーディンガー波動方程式が分離可能な直交曲線座標系について、各座標系への変換と座標 $q$ の表式を一覧にして掲載します。よく使われる座標に関しては$\nabla^{2}$と$\mathrm{d} \tau$も併記してあります。
円筒極座標
$$\begin{align}
x &=\rho \cos \varphi \\
y &=\rho \sin \varphi \\
z &=z \\
q_{\rho} &=1, \quad q_{z}=1, \quad q_{\varphi}=\rho \\
\mathrm{d} \tau &=\rho \mathrm{d} \rho \mathrm{d} z \mathrm{d} \varphi \\
\nabla^{2} &=\dfrac{1}{\rho} \dfrac{\partial}{\partial \rho}\left(\rho \dfrac{\partial}{\partial \rho}\right)+\dfrac{1}{\rho^{2}} \dfrac{\partial^{2}}{\partial \varphi^{2}}+\dfrac{\partial^{2}}{\partial z^{2}}
\end{align}$$
球面極座標
$$\begin{align}
&x=r \sin \theta \cos \varphi\\
&y=r \sin \theta \sin \varphi \\
&z=r \cos \theta \\
&q_{r}=1, \, q_{\theta}=r, \, q_{\varphi}=r \sin \theta \\
&\mathrm{d} \tau=r^{2} \sin \theta \mathrm{d} r \mathrm{d} \theta \mathrm{d} \varphi
\end{align}$$ $$\small \nabla^{2}=\dfrac{1}{r^{2}} \dfrac{\partial}{\partial r}\left(r^{2} \dfrac{\partial}{\partial r}\right)+\dfrac{1}{r^{2} \sin \theta} \dfrac{\partial}{\partial \theta}\left(\sin \theta \dfrac{\partial}{\partial \theta}\right)+\dfrac{1}{r^{2} \sin ^{2} \theta} \dfrac{\partial^{2}}{\partial \varphi^{2}}$$
放射線座標
$$\begin{array}{l}
x=\sqrt{\xi \eta} \cos \varphi \\
y=\sqrt{\xi \eta} \sin \varphi \\
z=\dfrac{1}{2}(\xi-\eta)
\end{array}$$ $$\begin{array}{c}
q_{\xi}=\dfrac{1}{2} \sqrt{\dfrac{\xi+\eta}{\xi}}, \quad q_{\eta}=\dfrac{1}{2} \sqrt{\dfrac{\xi+\eta}{\eta}}, \quad q_{\varphi}=\sqrt{\xi \eta} \\
\mathrm{d} \tau=\dfrac{1}{4}(\xi+\eta) \mathrm{d} \xi \mathrm{d} \eta \mathrm{d} \varphi \\
\small \nabla^{2}=\dfrac{4}{\xi+\eta} \dfrac{\partial}{\partial \xi}\left(\xi \dfrac{\partial}{\partial \xi}\right)+\dfrac{4}{\xi+\eta} \dfrac{\partial}{\partial \eta}\left(\eta \dfrac{\partial}{\partial \eta}\right)+\dfrac{1}{\xi \eta} \dfrac{\partial^{2}}{\partial \varphi^{2}}
\end{array}$$
放物円筒座標
$$\begin{array}{c}
x=\dfrac{1}{2}(u-v), \, y=\sqrt{u v}, \, z=z \\
q_{u}=\dfrac{1}{2} \sqrt{\dfrac{u+v}{u}}, \, q_{v}=\dfrac{1}{2} \sqrt{\dfrac{u+v}{v}}, \, q_{z}=1
\end{array}$$
共焦点楕円座標(長球体)
$$\begin{array}{l}
x=a \sqrt{\xi^{2}-1} \sqrt{1-\eta^{2}} \cos \varphi \\
y=a \sqrt{\xi^{2}-1} \sqrt{1-\eta^{2}} \sin \varphi \\
z=a \xi \eta
\end{array}$$点$(0,0,-\alpha)$と$(0,0,\alpha)$からの距離$r_A$とrを使えば、$\xi$と$\eta$は次のように表せる。$$\xi =\dfrac{r_{\mathrm{A}}+r_{\mathrm{B}}}{2 a}, \quad \eta=\dfrac{r_{\mathrm{A}}-r_{\mathrm{B}}}{2 a}$$ $$\begin{array}{l}
q_{\xi}=a \sqrt{\dfrac{\xi^{2}-\eta^{2}}{\xi^{2}-1}}, \\
q_{\eta}=a \sqrt{\dfrac{\xi^{2}-\eta^{2}}{1-\eta^{2}}}, \\
q_{\varphi}=a \sqrt{\left(\xi^{2}-1\right)\left(1-\eta^{2}\right)}
\end{array}$$ $$\mathrm{d} \tau=a^{3}\left(\xi^{2}-\eta^{2}\right) \mathrm{d} \xi \mathrm{d} \eta \mathrm{d} \varphi$$ $$\scriptsize \nabla^{2}=\dfrac{1}{a^{2}\left(\xi^{2}-\eta^{2}\right)}\left[\dfrac{\partial}{\partial \xi}\left\{\left(\xi^{2}-1\right) \dfrac{\partial}{\partial \xi}\right\}+\dfrac{\partial}{\partial \eta}\left\{\left(1-\eta^{2}\right) \dfrac{\partial}{\partial \eta}\right\}+\dfrac{\xi^{2}-\eta^{2}}{\left(\xi^{2}-1\right)\left(1-\eta^{2}\right)} \dfrac{\partial^{2}}{\partial \varphi^{2}}\right]$$
回転楕円体座標(偏平球体)
$$\small \begin{array}{c}
x=a \xi \eta \cos \varphi, \, y=a \xi \eta \sin \varphi, \, z=a \sqrt{\left(\xi^{2}-1\right)\left(1-\eta^{2}\right)} \\
q_{\xi}=a \sqrt{\dfrac{\xi^{2}-\eta^{2}}{\xi^{2}-1}}, \, q_{n}=a \sqrt{\dfrac{\xi^{2}-\eta^{2}}{1-\eta^{2}}}, \, q_{\varphi}=a \xi \eta
\end{array}$$次のような変換も考えられる。$$\begin{array}{c}
x=\dfrac{1}{2}(u-v), \quad y=\sqrt{u v}, \quad z=z \\
q_{u}=\dfrac{1}{2} \sqrt{\dfrac{u+v}{u}}, \quad q_{v}=\dfrac{1}{2} \sqrt{\dfrac{u+v}{v}}, \quad q_{z}=1
\end{array}$$
楕円円筒座標
$$\begin{array}{c}
x=a \sqrt{\left(u^{2}-1\right)\left(1-v^{2}\right)}, \quad y=a u v, \quad z=z \\
q_{u}=a \sqrt{\dfrac{u^{2}-v^{2}}{u^{2}-1}}, \quad q_{v}=a \sqrt{\dfrac{u^{2}-v^{2}}{1-v^{2}}}, \quad q_{z}=1
\end{array}$$
楕円体座標
$$\begin{array}{l}
x^{2}=\dfrac{\left(a^{2}+u\right)\left(a^{2}+v\right)\left(a^{2}+w\right)}{\left(a^{2}-b^{2}\right)\left(a^{2}-c^{2}\right)}, \\ y^{2}=\dfrac{\left(b^{2}+u\right)\left(b^{2}+v\right)\left(b^{2}+w\right)}{\left(b^{2}-c^{2}\right)\left(b^{2}-a^{2}\right)}, \\
z^{2}=\dfrac{\left(c^{2}+u\right)\left(c^{2}+v\right)\left(c^{2}+w\right)}{\left(c^{2}-a^{2}\right)\left(c^{2}-b^{2}\right)} \\ \\
q_{u}^{2}=\dfrac{(u-v)(u-w)}{4\left(a^{2}+u\right)\left(b^{2}+u\right)\left(c^{2}+u\right)}, \\
q_{v}^{2}=\dfrac{(v-w)(v-u)}{4\left(a^{2}+v\right)\left(b^{2}+v\right)\left(c^{2}+v\right)}, \\
q_{w}^{2}=\dfrac{(w-u)(w-v)}{4\left(a^{2}+w\right)\left(b^{2}+w\right)\left(c^{2}+w\right)}
\end{array}$$
共焦点放物線座標
$$\begin{array}{l}
x=\dfrac{1}{2}(u+v+w-a-b), \\
y^{2}=\dfrac{(a-u)(a-v)(a-w)}{b-a}, \\
z^{2}=\dfrac{(b-u)(b-v)(b-w)}{a-b} \\
(u>b>v>a>w) \\ \\
q_{u}^{2}=\dfrac{(u-v)(u-w)}{4(a-u)(b-u)}, \\
q_{v}^{2}=\dfrac{(v-u)(v-w)}{4(a-v)(b-v)}, \\
q_{w}^{2}=\dfrac{(w-u)(w-v)}{4(a-w)(b-w)}
\end{array}$$
楕円関数を含む座標系
$$\begin{array}{c}
x=u \operatorname{dn}(v, k) \operatorname{sn}\left(w, k^{\prime}\right), \\
y=u \operatorname{sn}(v, k) \operatorname{dn}\left(w, k^{\prime}\right) \\
z=u \operatorname{cn}(v, k) \operatorname{cn}\left(w, k^{\prime}\right), \\
(k^{2}+{k^{\prime}}^{2}=1) \\ \\
q_{u}^{2}=1, \\
\small q_{v}^{2}=q_{w}^{2}=u^{2}\left\{k^{2} \operatorname{cn}^{2}(v, k)+{k^{\prime}}^{2}\operatorname{cn}^{2}\left(w, k^{\prime}\right)\right\}
\end{array}$$
※ $\operatorname{sn}$、$\operatorname{dn}$、$\operatorname{cn}$はいずれも楕円関数。