３次元調和振動子/メモ のバックアップ(No.2)

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量子力学Ⅰ/３次元調和振動子

$n=2$ の場合 †

　 \varphi_{200}\propto -1+2\frac{r^2}{r_2^2}\cos^2\phi\sin^2\theta

　 \varphi_{020}\propto -1+2\frac{r^2}{r_2^2}\sin^2\phi\sin^2\theta

　 \varphi_{002}\propto -1+2\frac{r^2}{r_2^2}\cos^2\theta

　 \varphi_{110}\propto \sin\phi\cos\phi\sin^2\theta

　 \varphi_{101}\propto \sin\phi\sin\theta\cos\theta

　 \varphi_{011}\propto \cos\phi\sin\theta\cos\theta

であり、

　 Y_0^0\propto 1

　 Y_3^3\propto 3\cos^2\theta-1

　 Y_2^{\pm 1}\propto e^{\pm i\phi}\sin\theta\cos\theta

　 Y_2^{\pm 2}\propto e^{\pm 2i\phi}\sin^2\theta

であるから、

　 \varphi_{110}\propto \sin 2\phi\sin^2\theta\propto Y_2^{2}-Y_2^{-2}

　 \varphi_{011}\propto Y_2^{1}-Y_2^{-1}

　 \varphi_{101}\propto Y_2^{1}+Y_2^{-1}

　&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 Y_l^m を \varphi_{abc} 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})

  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})
  1. i\frac{2}{5}\sqrt{\frac{\pi }{21}}\frac{r^3}{r_3^3}(Y_3^1+Y_3^{-1})
  2. 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})
  3. 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
  1. \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})
  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
  1. \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）。

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