3次元調和振動子/メモ の履歴(No.2)
更新$n=2$ の場合†
であり、
であるから、
&math(\varphi_{002}\propto \left(-1+\frac{2}{3}\frac{r^2}{r_2^2}\right)Y_0^0+ \frac{4}{3\sqrt 5}\frac{r^2}{r_2^2}Y_2^0);
&math(\varphi_{200}\propto \left(-1+\frac{2}{3}\frac{r^2}{r_2^2}\right)Y_0^0+ \sqrt{\frac{2}{15}}\frac{r^2}{r_2^2}\left(Y_2^2+Y_2^{-2}-\frac{\sqrt 6}{3}Y_2^0\right));
&math(\varphi_{020}\propto \left(-1+\frac{2}{3}\frac{r^2}{r_2^2}\right)Y_0^0- \sqrt{\frac{2}{15}}\frac{r^2}{r_2^2}\left(Y_2^2+Y_2^{-2}+\frac{\sqrt 6}{3}Y_2^0\right));
これを逆に解けば を で表すこともできる。
$n=3$ の場合†
&math( &\varphi_{300}\propto \frac{r}{r_3}\cos\phi\sin\theta-\frac{2}{3}\left(\frac{r}{r_3}\cos\phi\sin\theta\right)^3\\ &\varphi_{030}\propto \frac{r}{r_3}\sin\phi\sin\theta-\frac{2}{3}\left(\frac{r}{r_3}\sin\phi\sin\theta\right)^3\\ &\varphi_{003}\propto \frac{r}{r_3}\cos\theta-\frac{2}{3}\left(\frac{r}{r_3}\cos\theta\right)^3\\ &\varphi_{210}\propto \left\{1-2\left(\frac{r}{r_2}\cos\phi\sin\theta\right)^2\right\}\sin\phi\sin\theta\\ &\varphi_{021}\propto \left\{1-2\left(\frac{r}{r_2}\sin\phi\sin\theta\right)^2\right\}\cos\theta\\ &\varphi_{102}\propto \left\{1-2\left(\frac{r}{r_2}\cos\theta\right)^2\right\}\cos\phi\sin\theta\\ &\varphi_{120}\propto \left\{1-2\left(\frac{r}{r_2}\sin\phi\sin\theta\right)^2\right\}\cos\phi\sin\theta\\ &\varphi_{012}\propto \left\{1-2\left(\frac{r}{r_2}\cos\theta\right)^2\right\}\sin\phi\sin\theta\\ &\varphi_{201}\propto \left\{1-2\left(\frac{r}{r_2}\cos\phi\sin\theta\right)^2\right\}\cos\theta\\ &\varphi_{111}\propto \cos\phi\sin\theta\sin\phi\sin\theta\cos\theta\\ );
がんばって計算すると、
&math( &\varphi_{300}\propto \sqrt{\frac{2\pi}{3}}\left(1-\frac{2}{5}\frac{r^2}{r_3^2}\right)\frac{r}{r_3}(-Y_1^1+Y_1^{-1})
- \frac{2}{5}\sqrt{\frac{\pi }{21}}\frac{r^3}{r_3^3}(Y_3^1+Y_3^{-1})
- \frac{2}{3}\sqrt{\frac{\pi }{35}}\frac{r^3}{r_3^3}(-Y_3^3+Y_3^{-3})\\ &\varphi_{030}\propto i\sqrt{\frac{2\pi}{3}}\left(1-\frac{2}{5}\frac{r^2}{r_3^2}\right)\frac{r}{r_3}(-Y_1^1+Y_1^{-1})
- i\frac{2}{5}\sqrt{\frac{\pi }{21}}\frac{r^3}{r_3^3}(Y_3^1+Y_3^{-1})
- i\frac{2}{3}\sqrt{\frac{\pi }{35}}\frac{r^3}{r_3^3}(Y_3^3+Y_3^{-3})\\ &\varphi_{003}\propto -2 \sqrt{\frac{\pi }{3}}\left(1-\frac{2}{5}\frac{r^2}{r_3^2}\right)\frac{r}{r_3}Y_1^0-\frac{8}{15}\sqrt{\frac{\pi }{7}}\frac{r^3}{r_3^3}Y_3^0\\ &\varphi_{210}\propto i\sqrt{\frac{2\pi}{3}}\left(1-\frac{2}{5}\frac{r^2}{r_3^2}\right)(Y_1^1+Y_1^{-1})
- i\frac{2}{5}\sqrt{\frac{\pi }{21}}(Y_3^1+Y_3^{-1})
- i2 \sqrt{\frac{\pi }{35}}(Y_3^3+Y_3^{-3})\\ &\varphi_{021}\propto 2\sqrt{\frac{\pi}{3}}\left(1-\frac{2}{5}\frac{r^2}{r_3^2}\right)Y_1^0
- \frac{4}{5}\sqrt{\frac{\pi }{7}}Y_3^0
- 2 \sqrt{\frac{2 \pi }{105}}(Y_3^2+Y_3^{-2})\\ &\varphi_{102}\propto \sqrt{\frac{2\pi}{3}}\left(1-\frac{2}{5}\frac{r^2}{r_3^2}\right)(-Y_1^1+Y_1^{-1})
- \frac{8}{5}\sqrt{\frac{\pi }{21}}\frac{r^2}{r_3^2}(-Y_3^3+Y_3^{-3})\\ &\varphi_{120}\propto \sqrt{\frac{2\pi}{3}}\left(1-\frac{2}{5}\frac{r^2}{r_3^2}\right)(-Y_1^1+Y_1^{-1})
- \frac{2}{5}\sqrt{\frac{\pi }{21}}(-Y_3^1+Y_3^{-1})
- 2 \sqrt{\frac{\pi }{35}}(-Y_3^3+Y_3^{-3})\\ &\varphi_{012}\propto i\sqrt{\frac{2\pi}{3}}\left(1-\frac{2}{5}\frac{r^2}{r_3^2}\right)(Y_1^1+Y_1^{-1})
- i\frac{8}{5}\sqrt{\frac{\pi }{21}}\frac{r^2}{r_3^2}(Y_3^3+Y_3^{-3})\\ &\varphi_{201}\propto 2\sqrt{\frac{\pi}{3}}\left(1-\frac{2}{5}\frac{r^2}{r_3^2}\right)Y_1^0
- \frac{4}{5}\sqrt{\frac{\pi }{7}}Y_3^0
- 2 \sqrt{\frac{2 \pi }{105}}(Y_3^2+Y_3^{-2})\\ &\varphi_{111}\propto i\sqrt{\frac{2 \pi }{105}}(-Y_3^2+Y_3^{-2}) );
こうなる(たぶん)。