3次元調和振動子/メモ のバックアップ(No.4)

更新


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 \frac{r^2}{r_2^2}\sin\phi\cos\phi\sin^2\theta

  \varphi_{101}\propto \frac{r^2}{r_2^2}\sin\phi\sin\theta\cos\theta

  \varphi_{011}\propto \frac{r^2}{r_2^2}\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、

 &math( &\varphi_{200}\propto \left(-1+\frac{2}{3}\frac{r^2}{r_2^2}\right) 2 \sqrt{\pi}\,Y_{0}^{0}

  • \frac{4}{3} \sqrt{\frac{\pi }{5}} \frac{r^2}{r_2^2} Y_{2}^{0}
  1. 2 \sqrt{\frac{2 \pi }{15}} \frac{r^2}{r_2^2} (Y_{2}^{-2}+Y_{2}^{2}) \\ &\varphi_{020}\propto \left(-1+\frac{2}{3}\frac{r^2}{r_2^2}\right) 2 \sqrt{\pi }\,Y_{0}^{0}
  • \frac{4}{3} \sqrt{\frac{\pi }{5}} \frac{r^2}{r_2^2} Y_{2}^{0}
  • 2 \sqrt{\frac{2 \pi }{15}} \frac{r^2}{r_2^2} (Y_{2}^{-2}+Y_{2}^{2}) \\ &\varphi_{002}\propto \left(-1+\frac{2}{3}\frac{r^2}{r_2^2}\right) 2 \sqrt{\pi }\,Y_{0}^{0}
  1. \frac{8}{3} \sqrt{\frac{\pi }{5}} \frac{r^2}{r_2^2} Y_{2}^{0} \\ &\varphi_{110}\propto
  • i 2 \sqrt{\frac{\pi }{30}} \frac{r^2}{r_2^2}(Y_{2}^{2}-Y_{2}^{-2}) \\ &\varphi_{101}\propto i 2 \sqrt{\frac{\pi }{30}} \frac{r^2}{r_2^2}(Y_{2}^{-1}+Y_{2}^{1}) \\ &\varphi_{011}\propto 2 \sqrt{\frac{\pi }{30}} \frac{r^2}{r_2^2}(Y_{2}^{-1}-Y_{2}^{1}) \\ );

として、 \left(-1+\frac{2}{3}\frac{r^2}{r_2^2}\right) 2 \sqrt{\pi }\,Y_{0}^{0} と、 \frac{r^2}{r_2^2}Y_{2}^{m} との線形結合で \varphi_{abc} を表せる。

これを逆に解けば \left(-1+\frac{2}{3}\frac{r^2}{r_2^2}\right) 2 \sqrt{\pi }\,Y_{0}^{0} と、 \frac{r^2}{r_2^2}Y_{2}^{m} とを \varphi_{abc} で表すこともできる。

LANG:mathematica
-1 + 2 ( r Cos[p] Sin[t])^2 == 2 Sqrt[\[Pi]] (-1 + (2 r^2)/3) Y[0, 0] + 2 Sqrt[(2 \[Pi])/15] r^2 (Y[2, 2] + Y[2, -2]) - 4/3 Sqrt[\[Pi]/5] r^2 Y[2, 0] // FullSimplify
-1 + 2 ( r Sin[p] Sin[t])^2 == 2 Sqrt[\[Pi]] (-1 + (2 r^2)/3) Y[0, 0] - 2 Sqrt[(2 \[Pi])/15] r^2 (Y[2, 2] + Y[2, -2]) - 4/3 Sqrt[\[Pi]/5] r^2 Y[2, 0] // FullSimplify
-1 + 2 ( r Cos[t])^2 == 2 Sqrt[\[Pi]] (-1 + (2 r^2)/3) Y[0, 0] + 8/3 Sqrt[\[Pi]/5] r^2 Y[2, 0] // FullSimplify
Sin[p] Cos[p] Sin[t]^2 == -I 2 Sqrt[\[Pi]/30] (Y[2, 2] - Y[2, -2]) // FullSimplify
Sin[p] Sin[t] Cos[t] == I 2 Sqrt[\[Pi]/30] (Y[2, 1] + Y[2, -1]) // FullSimplify
Cos[p] Sin[t] Cos[t] == 2 Sqrt[\[Pi]/30] (-Y[2, 1] + Y[2, -1]) // FullSimplify

&math(

  • 1+2 (r \cos \theta)^2 &= 2 \sqrt{\pi } \left(\frac{2 r^2}{3}-1\right) Y_{0}^{0}
  1. \frac{8}{3} \sqrt{\frac{\pi }{5}} r^2 Y_{2}^{0} \\
  • 1+2 (r \sin \phi \sin \theta)^2 &= 2 \sqrt{\pi } \left(\frac{2 r^2}{3}-1\right) Y_{0}^{0}
  • \frac{4}{3} \sqrt{\frac{\pi }{5}} r^2 Y_{2}^{0}
  • 2 \sqrt{\frac{2 \pi }{15}} r^2 (Y_{2}^{-2}+Y_{2}^{2}) \\
  • 1+2 (r \cos \phi \sin \theta)^2 &= 2 \sqrt{\pi } \left(\frac{2 r^2}{3}-1\right) Y_{0}^{0}
  • \frac{4}{3} \sqrt{\frac{\pi }{5}} r^2 Y_{2}^{0}
  1. 2 \sqrt{\frac{2 \pi }{15}} r^2 (Y_{2}^{-2}+Y_{2}^{2}) \\ \sin \phi \cos \phi \sin ^2\theta &=-i 2 \sqrt{\frac{\pi }{30}} (Y_{2}^{2}-Y_{2}^{-2}) \\ \sin \phi \sin \theta \cos \theta &=i 2 \sqrt{\frac{\pi }{30}} (Y_{2}^{-1}+Y_{2}^{1}) \\ \cos \phi \sin \theta \cos \theta &=2 \sqrt{\frac{\pi }{30}} (Y_{2}^{-1}-Y_{2}^{1}) \\ );

$n=3$ 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(\frac{r}{r_3}\right)^3\left\{1-2\left(\frac{r}{r_2}\cos\phi\sin\theta\right)^2\right\}\sin\phi\sin\theta\\ &\varphi_{021}\propto \left(\frac{r}{r_3}\right)^3\left\{1-2\left(\frac{r}{r_2}\sin\phi\sin\theta\right)^2\right\}\cos\theta\\ &\varphi_{102}\propto \left(\frac{r}{r_3}\right)^3\left\{1-2\left(\frac{r}{r_2}\cos\theta\right)^2\right\}\cos\phi\sin\theta\\ &\varphi_{120}\propto \left(\frac{r}{r_3}\right)^3\left\{1-2\left(\frac{r}{r_2}\sin\phi\sin\theta\right)^2\right\}\cos\phi\sin\theta\\ &\varphi_{012}\propto \left(\frac{r}{r_3}\right)^3\left\{1-2\left(\frac{r}{r_2}\cos\theta\right)^2\right\}\sin\phi\sin\theta\\ &\varphi_{201}\propto \left(\frac{r}{r_3}\right)^3\left\{1-2\left(\frac{r}{r_2}\cos\phi\sin\theta\right)^2\right\}\cos\theta\\ &\varphi_{111}\propto \left(\frac{r}{r_3}\right)^3\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}}\frac{r^2}{r_3^2}(Y_3^1+Y_3^{-1})
  • i2 \sqrt{\frac{\pi }{35}}\frac{r^2}{r_3^2}(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}}\frac{r^2}{r_3^2}Y_3^0
  2. 2 \sqrt{\frac{2 \pi }{105}}\frac{r^2}{r_3^2}(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_3^1+Y_3^{-1})\\ &\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})
  1. \frac{2}{5}\sqrt{\frac{\pi }{21}}\frac{r^2}{r_3^2}(-Y_3^1+Y_3^{-1})
  2. 2 \sqrt{\frac{\pi }{35}}\frac{r^2}{r_3^2}(-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}\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}}\frac{r^2}{r_3^2}Y_3^0
  • 2 \sqrt{\frac{2 \pi }{105}}\frac{r^2}{r_3^2}(Y_3^2+Y_3^{-2})\\ &\varphi_{111}\propto i\sqrt{\frac{2 \pi }{105}}\frac{r^2}{r_3^2}(-Y_3^2+Y_3^{-2}) );

のように、 \left(1-\frac{2}{5}\frac{r^2}{r_3^2}\right)\frac{r}{r_3}Y_1^m と、 r^3Y_3^m との線形結合で \varphi_{abc} を表せる。

LANG:mathematica
r Cos[p] Sin[t] - 2/3 (r Cos[p] Sin[t])^3 == -(1/ 3) (2 Sqrt[\[Pi]/35] r^3 (Y[3, -3] - Y[3, 3])) + 2/5 Sqrt[\[Pi]/21] r^3 (Y[3, -1] - Y[3, 1]) + Sqrt[(2 \[Pi])/ 3] (1 - (2 r^2)/5) r (Y[1, -1] - Y[1, 1]) // FullSimplify
r Cos[p] Sin[t] - 2/3 (r Cos[p] Sin[t])^3 == -(1/ 3) (2 Sqrt[\[Pi]/35] r^3 (Y[3, -3] - Y[3, 3])) + 2/5 Sqrt[\[Pi]/21] r^3 (Y[3, -1] - Y[3, 1]) + Sqrt[(2 \[Pi])/ 3] (1 - (2 r^2)/5) r (Y[1, -1] - Y[1, 1]) // FullSimplify
r Sin[p] Sin[t] - 2/3 (r Sin[p] Sin[t])^3 == I 2/3 Sqrt[\[Pi]/35] r^3 (Y[3, -3] + Y[3, 3]) + I 2/5 Sqrt[\[Pi]/21] r^3 (Y[3, -1] + Y[3, 1]) + I Sqrt[(2 \[Pi])/ 3] (1 - (2 r^2)/5) r (Y[1, -1] + Y[1, 1]) // FullSimplify
r Cos[t] - 2/3 (r Cos[t])^3 == 2 Sqrt[ \[Pi]/3] (1 - (2 r^2)/5) r (Y[1, 0]) - 8/15 Sqrt[\[Pi]/7] r^3 Y[3, 0] // FullSimplify
(1 - 2 (r Cos[p] Sin[t])^2) Sin[p] Sin[ t] == I Sqrt[( 2 \[Pi])/3] (1 - (2 r^2)/5) (Y[1, 1] + Y[1, -1]) + I 2/5 Sqrt[\[Pi]/21] r^2 (Y[3, 1] + Y[3, -1]) - I 2/1 Sqrt[\[Pi]/35] r^2 (Y[3, 3] + Y[3, -3]) // FullSimplify
(1 - 2 (r Sin[p] Sin[t])^2) Cos[t] == 2 Sqrt[ \[Pi]/3] (1 - (2 r^2)/5) (Y[1, 0]) + 4/5 Sqrt[\[Pi]/7] r^2 (Y[3, 0]) + 2/1 Sqrt[(2 \[Pi])/105] r^2 (Y[3, 2] + Y[3, -2]) // FullSimplify
(1 - 2 (r Cos[t])^2) Cos[p] Sin[t] == Sqrt[(2 \[Pi])/3] (1 - (2 r^2)/5) (-Y[1, 1] + Y[1, -1]) - 8/5 Sqrt[\[Pi]/21] r^2 (-Y[3, 1] + Y[3, -1]) // FullSimplify
(1 - 2 (r Sin[p] Sin[t])^2) Cos[ p] Sin[t] == Sqrt[(2 \[Pi])/3] (1 - (2 r^2)/5) (-Y[1, 1] + Y[1, -1]) + 2/5 Sqrt[\[Pi]/21] r^2 (-Y[3, 1] + Y[3, -1]) + 2/1 Sqrt[\[Pi]/35] r^2 (-Y[3, 3] + Y[3, -3]) // FullSimplify
(1 - 2 (r Cos[t])^2) Sin[p] Sin[t] == I Sqrt[(2 \[Pi])/3] (1 - (2 r^2)/5) (Y[1, 1] + Y[1, -1]) - (I 8)/ 5 Sqrt[\[Pi]/21] r^2 (Y[3, 1] + Y[3, -1]) // FullSimplify
(1 - 2 (r Cos[p] Sin[t])^2) Cos[t] == 2 Sqrt[ \[Pi]/3] (1 - (2 r^2)/5) (Y[1, 0]) + 4/5 Sqrt[\[Pi]/7] r^2 (Y[3, 0]) - 2/1 Sqrt[(2 \[Pi])/105] r^2 (Y[3, 2] + Y[3, -2]) // FullSimplify
Cos[p] Sin[p] Sin[t]^2 Cos[t] == I Sqrt[( 2 \[Pi])/105] (-Y[3, 2] + Y[3, -2]) // FullSimplify

&math( r \cos \phi \sin \theta-\frac{2}{3} (r \cos \phi \sin \theta)^3&= \left(1-\frac{2 r^2}{5}\right) \sqrt{\frac{2 \pi }{3}} (Y_{1}^{-1}-Y_{1}^{1}) r

  1. frac{2}{5} \sqrt{\frac{\pi }{21}} (Y_{3}^{-1}-Y_{3}^{1}) r^3
  • \frac{2}{3} \sqrt{\frac{\pi }{35}} r^3 (Y_{3}^{-3}-Y_{3}^{3}) \\ r \cos \phi \sin \theta-\frac{2}{3} (r \cos \phi \sin \theta)^3&= \left(1-\frac{2 r^2}{5}\right) \sqrt{\frac{2 \pi }{3}} (Y_{1}^{-1}-Y_{1}^{1}) r
  1. \frac{2}{5} \sqrt{\frac{\pi }{21}} (Y_{3}^{-1}-Y_{3}^{1}) r^3
  • \frac{2}{3} \sqrt{\frac{\pi }{35}} r^3 (Y_{3}^{-3}-Y_{3}^{3}) \\ r \sin \phi \sin \theta-\frac{2}{3} (r \sin \phi \sin \theta)^3&= \left(1-\frac{2 r^2}{5}\right) i \sqrt{\frac{2 \pi }{3}} (Y_{1}^{-1}+Y_{1}^{1}) r
  1. \frac{2}{5} i \sqrt{\frac{\pi }{21}} (Y_{3}^{-1}+Y_{3}^{1}) r^3
  2. \frac{2}{3} i \sqrt{\frac{\pi }{35}} (Y_{3}^{-3}+Y_{3}^{3}) r^3 \\ r \cos \theta-\frac{2}{3} (r \cos \theta)^3&= \left(1-\frac{2 r^2}{5}\right) 2 \sqrt{\frac{\pi }{3}} Y_{1}^{0} r
  • \frac{8}{15} \sqrt{\frac{\pi }{7}} Y_{3}^{0} r^3 \\ \left(1-2 (r \cos \phi \sin \theta)^2\right) \sin \phi \sin \theta&= \left(1-\frac{2 r^2}{5}\right) i \sqrt{\frac{2 \pi }{3}} (Y_{1}^{-1}+Y_{1}^{1})
  1. \frac{2}{5} i \sqrt{\frac{\pi }{21}} (Y_{3}^{-1}+Y_{3}^{1}) r^2
  • i 2 \sqrt{\frac{\pi }{35}} (Y_{3}^{-3}+Y_{3}^{3}) r^2 \\ \left(1-2 (r \sin \phi \sin \theta)^2\right) \cos \theta&= \left(1-\frac{2 r^2}{5}\right) 2 \sqrt{\frac{\pi }{3}} Y_{1}^{0}
  1. 2 \sqrt{\frac{2 \pi }{105}} (Y_{3}^{-2}+Y_{3}^{2}) r^2
  2. \frac{4}{5} \sqrt{\frac{\pi }{7}} Y_{3}^{0} r^2 \\ \left(1-2 (r \cos \theta)^2\right) \cos \phi \sin \theta&= \left(1-\frac{2 r^2}{5}\right) \sqrt{\frac{2 \pi }{3}} (Y_{1}^{-1}-Y_{1}^{1})
  • \frac{8}{5} \sqrt{\frac{\pi }{21}} (Y_{3}^{-1}-Y_{3}^{1}) r^2 \\ \left(1-2 (r \sin \phi \sin \theta)^2\right) \cos \phi \sin \theta&= \left(1-\frac{2 r^2}{5}\right) \sqrt{\frac{2 \pi }{3}} (Y_{1}^{-1}-Y_{1}^{1})
  1. 2 \sqrt{\frac{\pi }{35}} (Y_{3}^{-3}-Y_{3}^{3}) r^2
  2. \frac{2}{5} \sqrt{\frac{\pi }{21}} (Y_{3}^{-1}-Y_{3}^{1}) r^2 \\ \left(1-2 (r \cos \theta)^2\right) \sin \phi \sin \theta&= \left(1-\frac{2 r^2}{5}\right) i \sqrt{\frac{2 \pi }{3}} (Y_{1}^{-1}+Y_{1}^{1})
  • i\frac{8}{5} \sqrt{\frac{\pi }{21}} (Y_{3}^{-1}+Y_{3}^{1}) r^2 \\ \left(1-2 (r \cos \phi \sin \theta)^2\right) \cos \theta&= \left(1-\frac{2 r^2}{5}\right) 2 \sqrt{\frac{\pi }{3}} Y_{1}^{0}
  • 2 \sqrt{\frac{2 \pi }{105}} (Y_{3}^{-2}+Y_{3}^{2}) r^2
  1. \frac{4}{5} \sqrt{\frac{\pi }{7}} Y_{3}^{0} r^2 \\ \cos \phi \sin \phi \sin ^2\theta \cos \theta&= i \sqrt{\frac{2 \pi }{105}} (Y_{3}^{-2}-Y_{3}^{2}) \\ );

Counter: 5678 (from 2010/06/03), today: 3, yesterday: 0