3次元調和振動子/メモ のバックアップ差分(No.5)
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[[量子力学Ⅰ/3次元調和振動子]] #mathjax * $n=2$ の場合 [#a34560b6] &math(\varphi_{200}\propto -1+2\frac{r^2}{r_2^2}\cos^2\phi\sin^2\theta); &math(\varphi_{200}\propto \left(-1+2\frac{r^2}{r_0^2}\cos^2\phi\sin^2\theta\right)e^{-\frac{r^2}{2r_0^2}}); &math(\varphi_{020}\propto -1+2\frac{r^2}{r_2^2}\sin^2\phi\sin^2\theta); &math(\varphi_{020}\propto \left(-1+2\frac{r^2}{r_0^2}\sin^2\phi\sin^2\theta\right)e^{-\frac{r^2}{2r_0^2}}); &math(\varphi_{002}\propto -1+2\frac{r^2}{r_2^2}\cos^2\theta); &math(\varphi_{002}\propto \left(-1+2\frac{r^2}{r_0^2}\cos^2\theta\right)e^{-\frac{r^2}{2r_0^2}}); &math(\varphi_{110}\propto \frac{r^2}{r_2^2}\sin\phi\cos\phi\sin^2\theta); &math(\varphi_{110}\propto \left(\frac{r^2}{r_0^2}\sin\phi\cos\phi\sin^2\theta\right)e^{-\frac{r^2}{2r_0^2}}); &math(\varphi_{101}\propto \frac{r^2}{r_2^2}\sin\phi\sin\theta\cos\theta); &math(\varphi_{101}\propto \left(\frac{r^2}{r_0^2}\sin\phi\sin\theta\cos\theta\right)e^{-\frac{r^2}{2r_0^2}}); &math(\varphi_{011}\propto \frac{r^2}{r_2^2}\cos\phi\sin\theta\cos\theta); &math(\varphi_{011}\propto \left(\frac{r^2}{r_0^2}\cos\phi\sin\theta\cos\theta\right)e^{-\frac{r^2}{2r_0^2}}); であり、 &math(Y_0^0\propto 1); &math(Y_3^3\propto 3\cos^2\theta-1); &math(Y_2^{\pm 1}\propto e^{\pm i\phi}\sin\theta\cos\theta); &math(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} +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} +\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}) \\ &\varphi_{200}\propto \left\{ 2 \sqrt{\pi}\left(-1+\frac{2}{3}\frac{r^2}{r_0^2}\right) Y_{0}^{0} -\frac{4}{3} \sqrt{\frac{\pi }{5}} \frac{r^2}{r_0^2} Y_{2}^{0} +2 \sqrt{\frac{2 \pi }{15}} \frac{r^2}{r_0^2} (Y_{2}^{-2}+Y_{2}^{2}) \right\}e^{-\frac{r^2}{2r_0^2}}\\ &\varphi_{020}\propto \left\{ 2 \sqrt{\pi}\left(-1+\frac{2}{3}\frac{r^2}{r_0^2}\right) Y_{0}^{0} -\frac{4}{3} \sqrt{\frac{\pi }{5}} \frac{r^2}{r_0^2} Y_{2}^{0} -2 \sqrt{\frac{2 \pi }{15}} \frac{r^2}{r_0^2} (Y_{2}^{-2}+Y_{2}^{2}) \right\}e^{-\frac{r^2}{2r_0^2}}\\ &\varphi_{002}\propto \left\{ 2 \sqrt{\pi}\left(-1+\frac{2}{3}\frac{r^2}{r_0^2}\right) Y_{0}^{0} +\frac{8}{3} \sqrt{\frac{\pi }{5}} \frac{r^2}{r_0^2} Y_{2}^{0} \right\}e^{-\frac{r^2}{2r_0^2}}\\ &\varphi_{110}\propto \left\{ -i 2 \sqrt{\frac{\pi }{30}} \frac{r^2}{r_0^2}(Y_{2}^{2}-Y_{2}^{-2}) \right\}e^{-\frac{r^2}{2r_0^2}}\\ &\varphi_{101}\propto \left\{ i 2 \sqrt{\frac{\pi }{30}} \frac{r^2}{r_0^2}(Y_{2}^{-1}+Y_{2}^{1}) \right\}e^{-\frac{r^2}{2r_0^2}}\\ &\varphi_{011}\propto \left\{ 2 \sqrt{\frac{\pi }{30}} \frac{r^2}{r_0^2}(Y_{2}^{-1}-Y_{2}^{1}) \right\}e^{-\frac{r^2}{2r_0^2}}\\ ); として、&math(\left(-1+\frac{2}{3}\frac{r^2}{r_2^2}\right) 2 \sqrt{\pi }\,Y_{0}^{0}); と、&math(\frac{r^2}{r_2^2}Y_{2}^{m}); との線形結合で &math(\varphi_{abc}); を表せる。 のように、 &math(\left(-1+\frac{2}{3}\frac{r^2}{r_0^2}\right)e^{-\frac{r^2}{2r_0^2}}\,Y_{0}^{0}); と、 &math(\frac{r^2}{r_0^2}e^{-\frac{r^2}{2r_0^2}}\,Y_{2}^{m}); との線形結合で &math(\varphi_{abc}); を表せる。 これを逆に解けば &math(\left(-1+\frac{2}{3}\frac{r^2}{r_2^2}\right) 2 \sqrt{\pi }\,Y_{0}^{0}); と、&math(\frac{r^2}{r_2^2}Y_{2}^{m}); とを &math(\varphi_{abc}); で表すこともできる。 &math(\left(-1+\frac{2}{3}\frac{r^2}{r_0^2}\right)e^{-\frac{r^2}{2r_0^2}}\,Y_{0}^{0}); と、&math(\frac{r^2}{r_0^2}e^{-\frac{r^2}{2r_0^2}}\,Y_{2}^{m}); とを &math(\varphi_{abc}); で表すこともできる。 #collapsible(検算) 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 Y[l_, m_] := SphericalHarmonicY[l, m, \[Theta], \[Phi]] -1 + 2 ( \[Xi] Cos[\[Phi]] Sin[\[Theta]])^2 == 2 Sqrt[\[Pi]] (-1 + (2 \[Xi]^2)/3) Y[0, 0] + 2 Sqrt[(2 \[Pi])/15] \[Xi]^2 (Y[2, 2] + Y[2, -2]) - 4/3 Sqrt[\[Pi]/5] \[Xi]^2 Y[2, 0] // FullSimplify -1 + 2 ( \[Xi] Sin[\[Phi]] Sin[\[Theta]])^2 == 2 Sqrt[\[Pi]] (-1 + (2 \[Xi]^2)/3) Y[0, 0] - 2 Sqrt[(2 \[Pi])/15] \[Xi]^2 (Y[2, 2] + Y[2, -2]) - 4/3 Sqrt[\[Pi]/5] \[Xi]^2 Y[2, 0] // FullSimplify -1 + 2 ( \[Xi] Cos[\[Theta]])^2 == 2 Sqrt[\[Pi]] (-1 + (2 \[Xi]^2)/3) Y[0, 0] + 8/3 Sqrt[\[Pi]/5] \[Xi]^2 Y[2, 0] // FullSimplify Sin[\[Phi]] Cos[\[Phi]] Sin[\[Theta]]^2 == -I 2 Sqrt[\[Pi]/30] (Y[2, 2] - Y[2, -2]) // FullSimplify Sin[\[Phi]] Sin[\[Theta]] Cos[\[Theta]] == I 2 Sqrt[\[Pi]/30] (Y[2, 1] + Y[2, -1]) // FullSimplify Cos[\[Phi]] Sin[\[Theta]] Cos[\[Theta]] == 2 Sqrt[\[Pi]/30] (-Y[2, 1] + Y[2, -1]) // FullSimplify &math( -1+2 (r \cos \theta)^2 -1+2 (\xi \cos \theta)^2 &= 2 \sqrt{\pi } \left(\frac{2 r^2}{3}-1\right) Y_{0}^{0} +\frac{8}{3} \sqrt{\frac{\pi }{5}} r^2 Y_{2}^{0} 2 \sqrt{\pi } \left(-1+\frac{2 \xi^2}{3}\right) Y_{0}^{0} +\frac{8}{3} \sqrt{\frac{\pi }{5}} \xi^2 Y_{2}^{0} \\ -1+2 (r \sin \phi \sin \theta)^2 -1+2 (\xi \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}) 2 \sqrt{\pi } \left(-1+\frac{2 \xi^2}{3}\right) Y_{0}^{0} -\frac{4}{3} \sqrt{\frac{\pi }{5}} \xi^2 Y_{2}^{0} -2 \sqrt{\frac{2 \pi }{15}} \xi^2 (Y_{2}^{-2}+Y_{2}^{2}) \\ -1+2 (r \cos \phi \sin \theta)^2 -1+2 (\xi \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} +2 \sqrt{\frac{2 \pi }{15}} r^2 (Y_{2}^{-2}+Y_{2}^{2}) 2 \sqrt{\pi } \left(-1+\frac{2 \xi^2}{3}\right) Y_{0}^{0} -\frac{4}{3} \sqrt{\frac{\pi }{5}} \xi^2 Y_{2}^{0} +2 \sqrt{\frac{2 \pi }{15}} \xi^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}) \\ ); #collapsible * $n=3$ の場合 [#t68bfc99] &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\\ &\varphi_{300}\propto e^{-\frac{r^2}{2r_0^2}}\left\{\frac{r}{r_0}\cos\phi\sin\theta -\frac{2}{3}\left(\frac{r}{r_0}\cos\phi\sin\theta\right)^3\right\}\\ &\varphi_{030}\propto e^{-\frac{r^2}{2r_0^2}}\left\{\frac{r}{r_0}\sin\phi\sin\theta -\frac{2}{3}\left(\frac{r}{r_0}\sin\phi\sin\theta\right)^3\right\}\\ &\varphi_{003}\propto e^{-\frac{r^2}{2r_0^2}}\left\{\frac{r}{r_0}\cos\theta -\frac{2}{3}\left(\frac{r}{r_0}\cos\theta\right)^3\right\}\\ &\varphi_{210}\propto e^{-\frac{r^2}{2r_0^2}}\frac{r}{r_0}\left\{1-2\left(\frac{r}{r_0}\cos\phi\sin\theta\right)^2\right\}\sin\phi\sin\theta\\ &\varphi_{021}\propto e^{-\frac{r^2}{2r_0^2}}\frac{r}{r_0}\left\{1-2\left(\frac{r}{r_0}\sin\phi\sin\theta\right)^2\right\}\cos\theta\\ &\varphi_{102}\propto e^{-\frac{r^2}{2r_0^2}}\frac{r}{r_0}\left\{1-2\left(\frac{r}{r_0}\cos\theta\right)^2\right\}\cos\phi\sin\theta\\ &\varphi_{120}\propto e^{-\frac{r^2}{2r_0^2}}\frac{r}{r_0}\left\{1-2\left(\frac{r}{r_0}\sin\phi\sin\theta\right)^2\right\}\cos\phi\sin\theta\\ &\varphi_{012}\propto e^{-\frac{r^2}{2r_0^2}}\frac{r}{r_0}\left\{1-2\left(\frac{r}{r_0}\cos\theta\right)^2\right\}\sin\phi\sin\theta\\ &\varphi_{201}\propto e^{-\frac{r^2}{2r_0^2}}\frac{r}{r_0}\left\{1-2\left(\frac{r}{r_0}\cos\phi\sin\theta\right)^2\right\}\cos\theta\\ &\varphi_{111}\propto e^{-\frac{r^2}{2r_0^2}}\left(\frac{r}{r_0}\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}) +\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}}\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 +\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_{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}) +\frac{2}{5}\sqrt{\frac{\pi }{21}}\frac{r^2}{r_3^2}(-Y_3^1+Y_3^{-1}) +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 +\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}) &\varphi_{300}\propto \left\{\sqrt{\frac{2\pi}{3}}\left(1-\frac{2}{5}\frac{r^2}{r_0^2}\right)\frac{r}{r_0}(-Y_1^1+Y_1^{-1}) +\frac{2}{5}\sqrt{\frac{\pi }{21}}\frac{r^3}{r_0^3}(-Y_3^1+Y_3^{-1}) -\frac{2}{3}\sqrt{\frac{\pi }{35}}\frac{r^3}{r_0^3}(-Y_3^3+Y_3^{-3})\right\}e^{-\frac{r^2}{2r_0^2}}\\ &\varphi_{030}\propto \left\{i\sqrt{\frac{2\pi}{3}}\left(1-\frac{2}{5}\frac{r^2}{r_0^2}\right)\frac{r}{r_0}(Y_1^1+Y_1^{-1}) +i\frac{2}{5}\sqrt{\frac{\pi }{21}}\frac{r^3}{r_0^3}(Y_3^1+Y_3^{-1}) +i\frac{2}{3}\sqrt{\frac{\pi }{35}}\frac{r^3}{r_0^3}(Y_3^3+Y_3^{-3})\right\}e^{-\frac{r^2}{2r_0^2}}\\ &\varphi_{003}\propto \left\{2 \sqrt{\frac{\pi }{3}}\left(1-\frac{2}{5}\frac{r^2}{r_0^2}\right)\frac{r}{r_0}Y_1^0-\frac{8}{15}\sqrt{\frac{\pi }{7}}\frac{r^3}{r_0^3}Y_3^0\right\}e^{-\frac{r^2}{2r_0^2}}\\ &\varphi_{210}\propto \left\{i\sqrt{\frac{2\pi}{3}}\left(1-\frac{2}{5}\frac{r^2}{r_0^2}\right)\frac{r}{r_0}(Y_1^1+Y_1^{-1}) +i\frac{2}{5}\sqrt{\frac{\pi }{21}}\frac{r^3}{r_0^3}(Y_3^1+Y_3^{-1}) -i2 \sqrt{\frac{\pi }{35}}\frac{r^3}{r_0^3}(Y_3^3+Y_3^{-3})\right\}e^{-\frac{r^2}{2r_0^2}}\\ &\varphi_{021}\propto \left\{2\sqrt{\frac{\pi}{3}}\left(1-\frac{2}{5}\frac{r^2}{r_0^2}\right)\frac{r}{r_0}Y_1^0 +\frac{4}{5}\sqrt{\frac{\pi }{7}}\frac{r^3}{r_0^3}Y_3^0 +2 \sqrt{\frac{2 \pi }{105}}\frac{r^3}{r_0^3}(Y_3^2+Y_3^{-2})\right\}e^{-\frac{r^2}{2r_0^2}}\\ &\varphi_{102}\propto \left\{\sqrt{\frac{2\pi}{3}}\left(1-\frac{2}{5}\frac{r^2}{r_0^2}\right)\frac{r}{r_0}(-Y_1^1+Y_1^{-1}) -\frac{8}{5}\sqrt{\frac{\pi }{21}}\frac{r^3}{r_0^3}(-Y_3^1+Y_3^{-1})\right\}e^{-\frac{r^2}{2r_0^2}}\\ &\varphi_{120}\propto \left\{\sqrt{\frac{2\pi}{3}}\left(1-\frac{2}{5}\frac{r^2}{r_0^2}\right)\frac{r}{r_0}(-Y_1^1+Y_1^{-1}) +\frac{2}{5}\sqrt{\frac{\pi }{21}}\frac{r^3}{r_0^3}(-Y_3^1+Y_3^{-1}) +2 \sqrt{\frac{\pi }{35}}\frac{r^3}{r_0^3}(-Y_3^3+Y_3^{-3})\right\}e^{-\frac{r^2}{2r_0^2}}\\ &\varphi_{012}\propto \left\{i\sqrt{\frac{2\pi}{3}}\left(1-\frac{2}{5}\frac{r^2}{r_0^2}\right)\frac{r}{r_0}(Y_1^1+Y_1^{-1}) -i\frac{8}{5}\sqrt{\frac{\pi }{21}}\frac{r^3}{r_0^3}(Y_3^3+Y_3^{-3})\right\}e^{-\frac{r^2}{2r_0^2}}\\ &\varphi_{201}\propto \left\{2\sqrt{\frac{\pi}{3}}\left(1-\frac{2}{5}\frac{r^2}{r_0^2}\right)\frac{r}{r_0}Y_1^0 +\frac{4}{5}\sqrt{\frac{\pi }{7}}\frac{r^3}{r_0^3}Y_3^0 -2 \sqrt{\frac{2 \pi }{105}}\frac{r^3}{r_0^3}(Y_3^2+Y_3^{-2})\right\}e^{-\frac{r^2}{2r_0^2}}\\ &\varphi_{111}\propto \left\{i\sqrt{\frac{2 \pi }{105}}\frac{r^3}{r_0^3}(-Y_3^2+Y_3^{-2})\right\}e^{-\frac{r^2}{2r_0^2}}\\ ); のように、 &math(\left(1-\frac{2}{5}\frac{r^2}{r_3^2}\right)\frac{r}{r_3}Y_1^m); と、 &math(r^3Y_3^m); との線形結合で &math(\varphi_{abc}); を表せる。 &math(\left(1-\frac{2}{5}\frac{r^2}{r_0^2}\right)\frac{r}{r_0}e^{-\frac{r^2}{2r_0^2}}\,Y_1^m); と、 &math(\frac{r^3}{r_0^3}e^{-\frac{r^2}{2r_0^2}}\,Y_3^m); との線形結合で &math(\varphi_{abc}); を表せる。 #collapsible(検算) 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 Y[l_, m_] := SphericalHarmonicY[l, m, \[Theta], \[Phi]] \[Xi] Cos[\[Phi]] Sin[\[Theta]] - 2/3 (\[Xi] Cos[\[Phi]] Sin[\[Theta]])^3 == -(1/ 3) (2 Sqrt[\[Pi]/35] \[Xi]^3 (Y[3, -3] - Y[3, 3])) + 2/5 Sqrt[\[Pi]/21] \[Xi]^3 (Y[3, -1] - Y[3, 1]) + Sqrt[(2 \[Pi])/ 3] (1 - (2 \[Xi]^2)/5) \[Xi] (Y[1, -1] - Y[1, 1]) // FullSimplify \[Xi] Cos[\[Phi]] Sin[\[Theta]] - 2/3 (\[Xi] Cos[\[Phi]] Sin[\[Theta]])^3 == -(1/ 3) (2 Sqrt[\[Pi]/35] \[Xi]^3 (Y[3, -3] - Y[3, 3])) + 2/5 Sqrt[\[Pi]/21] \[Xi]^3 (Y[3, -1] - Y[3, 1]) + Sqrt[(2 \[Pi])/ 3] (1 - (2 \[Xi]^2)/5) \[Xi] (Y[1, -1] - Y[1, 1]) // FullSimplify \[Xi] Sin[\[Phi]] Sin[\[Theta]] - 2/3 (\[Xi] Sin[\[Phi]] Sin[\[Theta]])^3 == I 2/3 Sqrt[\[Pi]/35] \[Xi]^3 (Y[3, -3] + Y[3, 3]) + I 2/5 Sqrt[\[Pi]/21] \[Xi]^3 (Y[3, -1] + Y[3, 1]) + I Sqrt[(2 \[Pi])/ 3] (1 - (2 \[Xi]^2)/5) \[Xi] (Y[1, -1] + Y[1, 1]) // FullSimplify \[Xi] Cos[\[Theta]] - 2/3 (\[Xi] Cos[\[Theta]])^3 == 2 Sqrt[ \[Pi]/3] (1 - (2 \[Xi]^2)/5) \[Xi] (Y[1, 0]) - 8/15 Sqrt[\[Pi]/7] \[Xi]^3 Y[3, 0] // FullSimplify (1 - 2 (\[Xi] Cos[\[Phi]] Sin[\[Theta]])^2) Sin[\[Phi]] Sin[\[Theta]] == I Sqrt[( 2 \[Pi])/3] (1 - (2 \[Xi]^2)/5) (Y[1, 1] + Y[1, -1]) + I 2/5 Sqrt[\[Pi]/21] \[Xi]^2 (Y[3, 1] + Y[3, -1]) - I 2/1 Sqrt[\[Pi]/35] \[Xi]^2 (Y[3, 3] + Y[3, -3]) // FullSimplify (1 - 2 (\[Xi] Sin[\[Phi]] Sin[\[Theta]])^2) Cos[\[Theta]] == 2 Sqrt[ \[Pi]/3] (1 - (2 \[Xi]^2)/5) (Y[1, 0]) + 4/5 Sqrt[\[Pi]/7] \[Xi]^2 (Y[3, 0]) + 2/1 Sqrt[(2 \[Pi])/105] \[Xi]^2 (Y[3, 2] + Y[3, -2]) // FullSimplify (1 - 2 (\[Xi] Cos[\[Theta]])^2) Cos[\[Phi]] Sin[\[Theta]] == Sqrt[(2 \[Pi])/3] (1 - (2 \[Xi]^2)/5) (-Y[1, 1] + Y[1, -1]) - 8/5 Sqrt[\[Pi]/21] \[Xi]^2 (-Y[3, 1] + Y[3, -1]) // FullSimplify (1 - 2 (\[Xi] Sin[\[Phi]] Sin[\[Theta]])^2) Cos[\[Phi]] Sin[\[Theta]] == Sqrt[(2 \[Pi])/3] (1 - (2 \[Xi]^2)/5) (-Y[1, 1] + Y[1, -1]) + 2/5 Sqrt[\[Pi]/21] \[Xi]^2 (-Y[3, 1] + Y[3, -1]) + 2/1 Sqrt[\[Pi]/35] \[Xi]^2 (-Y[3, 3] + Y[3, -3]) // FullSimplify (1 - 2 (\[Xi] Cos[\[Theta]])^2) Sin[\[Phi]] Sin[\[Theta]] == I Sqrt[(2 \[Pi])/3] (1 - (2 \[Xi]^2)/5) (Y[1, 1] + Y[1, -1]) - (I 8)/ 5 Sqrt[\[Pi]/21] \[Xi]^2 (Y[3, 1] + Y[3, -1]) // FullSimplify (1 - 2 (\[Xi] Cos[\[Phi]] Sin[\[Theta]])^2) Cos[\[Theta]] == 2 Sqrt[ \[Pi]/3] (1 - (2 \[Xi]^2)/5) (Y[1, 0]) + 4/5 Sqrt[\[Pi]/7] \[Xi]^2 (Y[3, 0]) - 2/1 Sqrt[(2 \[Pi])/105] \[Xi]^2 (Y[3, 2] + Y[3, -2]) // FullSimplify Cos[\[Phi]] Sin[\[Phi]] Sin[\[Theta]]^2 Cos[\[Theta]] == 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 +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}) \xi \cos \phi \sin \theta-\frac{2}{3} (\xi \cos \phi \sin \theta)^3&= \left(1-\frac{2 \xi^2}{5}\right) \sqrt{\frac{2 \pi }{3}} (Y_{1}^{-1}-Y_{1}^{1}) \xi +\frac{2}{5} \sqrt{\frac{\pi }{21}} (Y_{3}^{-1}-Y_{3}^{1}) \xi^3 -\frac{2}{3} \sqrt{\frac{\pi }{35}} \xi^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 +\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}) \xi \cos \phi \sin \theta-\frac{2}{3} (\xi \cos \phi \sin \theta)^3&= \left(1-\frac{2 \xi^2}{5}\right) \sqrt{\frac{2 \pi }{3}} (Y_{1}^{-1}-Y_{1}^{1}) \xi +\frac{2}{5} \sqrt{\frac{\pi }{21}} (Y_{3}^{-1}-Y_{3}^{1}) \xi^3 -\frac{2}{3} \sqrt{\frac{\pi }{35}} \xi^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 +\frac{2}{5} i \sqrt{\frac{\pi }{21}} (Y_{3}^{-1}+Y_{3}^{1}) r^3 +\frac{2}{3} i \sqrt{\frac{\pi }{35}} (Y_{3}^{-3}+Y_{3}^{3}) r^3 \xi \sin \phi \sin \theta-\frac{2}{3} (\xi \sin \phi \sin \theta)^3&= \left(1-\frac{2 \xi^2}{5}\right) i \sqrt{\frac{2 \pi }{3}} (Y_{1}^{-1}+Y_{1}^{1}) \xi +\frac{2}{5} i \sqrt{\frac{\pi }{21}} (Y_{3}^{-1}+Y_{3}^{1}) \xi^3 +\frac{2}{3} i \sqrt{\frac{\pi }{35}} (Y_{3}^{-3}+Y_{3}^{3}) \xi^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 \xi \cos \theta-\frac{2}{3} (\xi \cos \theta)^3&= \left(1-\frac{2 \xi^2}{5}\right) 2 \sqrt{\frac{\pi }{3}} Y_{1}^{0} \xi -\frac{8}{15} \sqrt{\frac{\pi }{7}} Y_{3}^{0} \xi^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}) +\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 (\xi \cos \phi \sin \theta)^2\right) \sin \phi \sin \theta&= \left(1-\frac{2 \xi^2}{5}\right) i \sqrt{\frac{2 \pi }{3}} (Y_{1}^{-1}+Y_{1}^{1}) +\frac{2}{5} i \sqrt{\frac{\pi }{21}} (Y_{3}^{-1}+Y_{3}^{1}) \xi^2 -i 2 \sqrt{\frac{\pi }{35}} (Y_{3}^{-3}+Y_{3}^{3}) \xi^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} +2 \sqrt{\frac{2 \pi }{105}} (Y_{3}^{-2}+Y_{3}^{2}) r^2 +\frac{4}{5} \sqrt{\frac{\pi }{7}} Y_{3}^{0} r^2 \left(1-2 (\xi \sin \phi \sin \theta)^2\right) \cos \theta&= \left(1-\frac{2 \xi^2}{5}\right) 2 \sqrt{\frac{\pi }{3}} Y_{1}^{0} +2 \sqrt{\frac{2 \pi }{105}} (Y_{3}^{-2}+Y_{3}^{2}) \xi^2 +\frac{4}{5} \sqrt{\frac{\pi }{7}} Y_{3}^{0} \xi^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 (\xi \cos \theta)^2\right) \cos \phi \sin \theta&= \left(1-\frac{2 \xi^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}) \xi^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}) +2 \sqrt{\frac{\pi }{35}} (Y_{3}^{-3}-Y_{3}^{3}) r^2 +\frac{2}{5} \sqrt{\frac{\pi }{21}} (Y_{3}^{-1}-Y_{3}^{1}) r^2 \left(1-2 (\xi \sin \phi \sin \theta)^2\right) \cos \phi \sin \theta&= \left(1-\frac{2 \xi^2}{5}\right) \sqrt{\frac{2 \pi }{3}} (Y_{1}^{-1}-Y_{1}^{1}) +2 \sqrt{\frac{\pi }{35}} (Y_{3}^{-3}-Y_{3}^{3}) \xi^2 +\frac{2}{5} \sqrt{\frac{\pi }{21}} (Y_{3}^{-1}-Y_{3}^{1}) \xi^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 (\xi \cos \theta)^2\right) \sin \phi \sin \theta&= \left(1-\frac{2 \xi^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}) \xi^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 +\frac{4}{5} \sqrt{\frac{\pi }{7}} Y_{3}^{0} r^2 \left(1-2 (\xi \cos \phi \sin \theta)^2\right) \cos \theta&= \left(1-\frac{2 \xi^2}{5}\right) 2 \sqrt{\frac{\pi }{3}} Y_{1}^{0} -2 \sqrt{\frac{2 \pi }{105}} (Y_{3}^{-2}+Y_{3}^{2}) \xi^2 +\frac{4}{5} \sqrt{\frac{\pi }{7}} Y_{3}^{0} \xi^2 \\ \cos \phi \sin \phi \sin ^2\theta \cos \theta&= i \sqrt{\frac{2 \pi }{105}} (Y_{3}^{-2}-Y_{3}^{2}) \\ ); #collapsible * 解答 [#ne558c03] (1) &math( &-\frac{\hbar^2}{2m}\frac{d^2}{dr^2}rR(r)+ \left\{\frac{1}{2}Kr^2+\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}\right\} rR(r)=\varepsilon rR(r)\\ &-\frac{\hbar^2}{2m}\frac{d^2}{dr^2}\underbrace{rR(r)}_{X(\xi)}+ \left\{\frac{1}{2}m\omega^2r^2+\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}\right\} rR(r)=\varepsilon rR(r)\\ &-\frac{\hbar\omega}{2}\frac{d^2}{d\xi^2}X(\xi)+ \left\{\frac{1}{2}\hbar\omega\xi^2+\frac{\hbar\omega}{2}\frac{l(l+1)}{\xi^2}\right\} X(\xi)=\varepsilon X(\xi)\\ &-\frac{d^2}{d\xi^2}X(\xi)+ \left\{\xi^2+\frac{l(l+1)}{\xi^2}\right\} X(\xi)=\frac{2\varepsilon}{\hbar\omega} X(\xi)\\ &-\frac{X''(\xi)}{X(\xi)}+\xi^2+\frac{l(l+1)}{\xi^2}=\frac{2\varepsilon}{\hbar\omega}\\ ); (3) &math( &-\left(\xi^{l+1}-\frac{2}{2(l+1)+1}\xi^{l+3}\right)e^{-\xi^2/2}\\ &=\left\{ \left(-\xi^2+(2l+3)-\frac{l(l+1)}{\xi^2}\right)\xi^{l+1}- \left(-\xi^2+(2l+7)-\frac{(l+2)(l+3)}{\xi^2}\right)\frac{2}{2l+3}\xi^{l+3}\right\}e^{-\xi^2/2}\\ &=\left\{ \left(-\xi^2+(2l+7)-\frac{l(l+1)}{\xi^2}\right)\xi^{l+1}- \left(-\xi^2+(2l+7)-\frac{l^2+5l+6-2(2l+3)}{\xi^2}\right)\frac{2}{2l+3}\xi^{l+3}\right\}e^{-\xi^2/2}\\ &=\left\{ \left(-\xi^2+(2l+7)-\frac{l(l+1)}{\xi^2}\right)\xi^{l+1}- \left(-\xi^2+(2l+7)-\frac{l(l+1)}{\xi^2}\right)\frac{2}{2l+3}\xi^{l+3}\right\}e^{-\xi^2/2}\\ &=\left(-\xi^2+(2l+7)-\frac{l(l+1)}{\xi^2}\right)\left( \xi^{l+1}-\frac{2}{2l+3}\xi^{l+3}\right)e^{-\xi^2/2}\\ );
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