物理量の固有関数/メモ の変更点

更新 
* 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]
&katex();
(1) $n\ne m$ のとき、

$$
\begin{aligned}
\int_{-\infty}^\infty\varphi_n^*(x)\varphi_m(x)dx
&=\frac{2}{\,a\,}\int_0^a\sin(n\pi x/a)\sin(m\pi x/a)dx\\
&=\frac{1}{\,a\,}\int_0^a\bigg[-\cos\Big((n+m)\pi x/a\Big)+\cos\Big((n-m)\pi x/a\Big)\bigg]dx\\
&=\frac{1}{\,a\,}\bigg[-\frac{a}{(n+m)\pi}\sin\Big((n+m)\pi x/a\Big)+\\
&\hspace{17.6mm}\frac{a}{(n-m)\pi}\sin\Big((n-m)\pi x/a\Big)\bigg]_0^a\\
&=0
\end{aligned}
$$

(2) $n=m$ 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 $1$ になって、

$$
\int_{-\infty}^\infty\varphi_n^*(x)\varphi_m(x)dx=\frac{1}{\,a\,}\int_0^a 1\,dx=1
$$

(1) と合わせれば、

$$
\int_{-\infty}^\infty\varphi_n^*(x)\varphi_m(x)dx=\delta_{nm}
$$

(3)

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

したがって、
したがって $0<x<1$ において

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

*** 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"})
 ]

 LANG:mathematica
 Plot[
   Table[Sin[a k]/(Pi k), {a, {200, 50, 10}}] // Evaluate, 
   {k, -1, 1}, PlotRange -> {Full, {-15, 65}}, PlotStyle -> Thick, 
   PlotLegends -> {"a=200", "a=50", "a=10"}, PlotPoints -> 200,
   PlotLabel -> "\!\(\*SubscriptBox[\(\[Delta]\), \(a\)]\)(k)"
 ]
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