物理量の固有関数/メモ のバックアップソース(No.3)

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* ハミルトニアン [#sd3d4059]
** 演習:箱の中の自由粒子 = 実フーリエ級数 [#i5516d37]

*** 解答 [#a60e8522]

(1) &math(n\ne m); のとき、

 &math(
\int_0^a\varphi_n^*(x)\varphi_m(x)dx
&=\frac{2}{\,a\,}\int_0^a\sin(n\pi x/a)\sin(m\pi x/a)dx\\
&=\frac{1}{\,a\,}\int_0^a\sin\Big((n+m)\pi x/a\Big)+\sin\Big((n-m)\pi x/a\Big)dx\\
&=\frac{1}{\,a\,}\bigg[\frac{a}{(n+m)\pi}\cos\Big((n+m)\pi x/a\Big)+\\
&\hspace{12mm}\frac{a}{(n-m)\pi}\cos\Big((n-m)\pi x/a\Big)\bigg]_0^a\\
&=0
);

(2) &math(n=m); のとき、上式の右辺第一項はやはりゼロになるが、第二項は積分内が &math(1); になって、

 &math(
\int_0^a\varphi_n^*(x)\varphi_m(x)dx=\frac{1}{\,a\,}\int_0^a 1\,dx=1
);

(1) と合わせれば、

 &math(
\int_0^a\varphi_n^*(x)\varphi_m(x)dx=\delta_{nm}
);

(3)

 &math(c_n&=\int_0^1\sqrt 2\sin(n\pi x)f(x)dx\\
&=\int_0^{1/2}\sqrt 2\sin(n\pi x)\,x\,dx+\int_{1/2}^1\sqrt 2\sin(n\pi x)\,(x-1)\,dx\\
&=\int_0^{1}\sqrt 2\sin(n\pi x)\,x\,dx-\int_{1/2}^1\sqrt 2\sin(n\pi x)\,dx\\
&=\Big[-\frac{\sqrt 2}{n\pi}\cos(n\pi x)x\Big]_0^1+\int_0^{1}\frac{\sqrt 2}{n\pi}\cos(n\pi x)\,dx-\Big[-\frac{\sqrt 2}{n\pi}\cos(n\pi x)\Big]_{1/2}^1\\
&=-\frac{\sqrt 2}{n\pi}\cos(n\pi)+\Big[\frac{\sqrt 2}{n^2\pi^2}\sin n\pi\Big]_0^1+
\frac{\sqrt 2}{n\pi}\cos(n\pi)-\frac{\sqrt 2}{n\pi}\cos(n\pi/2)\\
&=-\frac{\sqrt 2}{n\pi}\cos(n\pi/2)\\
&=\begin{cases}
0&(n=2m+1)\\
(-1)^m&(n=2m)\\
\end{cases});

したがって、

 &math(f(x)&=-\sum_{n=1}^\infty\frac{2}{n\pi}\cos(n\pi/2)\sin(n\pi x)\\
&=\sum_{m=1}^\infty \frac{(-1)^m}{m\pi}\sin(2m\pi x)\\
&=-\frac{1}{\pi}\sin(2\pi x)+\frac{1}{2\pi}\sin(4\pi x)-\frac{1}{3\pi}\sin(6\pi x)+\dots);

*** 解説 [#w020a327]

Mathematica ソース

 LANG:mathematica
 FourierSinCoefficient[
   If[x < 1/2, x, x - 1], x, n, 
   FourierParameters -> {1, Pi}
 ]
 c[n_Integer] =
   Integrate[
     Sqrt[2] Sin[n Pi xx] If[xx < 1/2, xx, -1 + xx],
     {xx, 0, 1}
   ]
 approx = 
   Table[
     Sum[
       c[n] Sqrt[2] Sin[n Pi x],
       {n, 1, nmax}
     ],
     {nmax, {4, 16, 64, 256}}
 ];
 Plot[
   {approx, If[x < 1/2, x, -1 + x]} // Flatten // Evaluate,
   {x, 0, 1}, ImageSize -> Large, PlotStyle -> {Thick}, 
   BaseStyle -> {FontSize -> 20}, 
   PlotLegends -> (Style[#, FontSize -> 20] & /@ { 
     "n \[LessEqual] 4", "n \[LessEqual] 16", "n \[LessEqual] 64", "n \[LessEqual] 256", 
     "Target"})
 ]

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