{"id":2370,"date":"2022-02-26T10:56:50","date_gmt":"2022-02-26T01:56:50","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2370"},"modified":"2025-08-06T11:49:30","modified_gmt":"2025-08-06T02:49:30","slug":"%e3%83%95%e3%83%bc%e3%83%aa%e3%82%a8%e7%a9%8d%e5%88%86%e3%83%bb%e3%83%95%e3%83%bc%e3%83%aa%e3%82%a8%e5%a4%89%e6%8f%9b","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6c\/%e3%83%95%e3%83%bc%e3%83%aa%e3%82%a8%e8%a7%a3%e6%9e%90\/%e3%83%95%e3%83%bc%e3%83%aa%e3%82%a8%e7%a9%8d%e5%88%86%e3%83%bb%e3%83%95%e3%83%bc%e3%83%aa%e3%82%a8%e5%a4%89%e6%8f%9b\/","title":{"rendered":"\u30d5\u30fc\u30ea\u30a8\u7a4d\u5206\u30fb\u30d5\u30fc\u30ea\u30a8\u5909\u63db"},"content":{"rendered":"

\u4efb\u610f\u306e\u5468\u671f \\(2L\\) \u3092\u3082\u3064\u95a2\u6570\u306e\u8907\u7d20\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b\u3092\uff0c\u975e\u5468\u671f\u7684\u73fe\u8c61\u306b\u307e\u3067\u62e1\u5f35\u3057\u305f\u3082\u306e\u304c\u300c\u30d5\u30fc\u30ea\u30a8\u7a4d\u5206<\/strong><\/span>\u300d\u3067\u3042\u308a\uff0c\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570\u306e\u62e1\u5f35\u304c\u300c\u30d5\u30fc\u30ea\u30a8\u5909\u63db<\/strong><\/span>\u300d\u3002
\n<\/p>\n

\n

\u4efb\u610f\u306e\u5468\u671f \\(2L\\) \u3092\u6301\u3064\u95a2\u6570\u306e\u8907\u7d20\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306e\u307e\u3068\u3081<\/h3>\n

\u533a\u9593 \\( -L \\le x \\le L\\) \u3067\u5b9a\u7fa9\u3055\u308c\u305f\u95a2\u6570 \\(f(x)\\) \u304c\u533a\u9593\u5916\u3067\u306f\u5468\u671f \\(2 L\\) \u306e\u5468\u671f\u95a2\u6570\u3067\u3042\u308b\u5834\u5408\uff0c\u305d\u306e\u8907\u7d20\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570<\/b><\/span>\u306f
\n$$f(x) = \\sum_{n = -\\infty}^{\\infty} c_n \\,\\exp\\left({i \\frac{n\\pi x}{L}}\\right)$$ \u3067\u3042\u308a\uff0c\u305d\u306e\u8907\u7d20\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570<\/b><\/span>\u306f
\n$$
\nc_n =\u00a0 \\frac{1}{2L} \\int_{-L}^{L} f(x)\\,\\biggl(\\exp\\left(i\\frac{n\\pi x}{L}\\right)\\biggr)^* \\, dx
\n$$ \u3068\u66f8\u3051\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n

\u5468\u671f\u6027\u3092\u6301\u305f\u306a\u3044\u95a2\u6570\u3078\u306e\u62e1\u5f35\u306e\u305f\u3081\u306e\u6e96\u5099<\/h3>\n

\\(c_n\\) \u3092\u3042\u3089\u305f\u3081\u3066\u5143\u306e \\(f(x)\\) \u306e\u5f0f\u306b\u4ee3\u5165\u3057\u3066<\/p>\n

\\begin{eqnarray}
\nf(x) &=& \\sum_{n = -\\infty}^{\\infty} \\left\\{ \\frac{1}{2L}
\n\\int_{-L}^{L} f(x’)\\, e^{-i\\frac{n\\pi}{L}x’} \\, dx’ \\right\\} \\ e^{i \\frac{n\\pi}{L}x}\\\\
\n&=& \\frac{1}{2L} \\sum_{n = -\\infty}^{\\infty}\\int_{-L}^{L} f(x’) \\,e^{i \\frac{n\\pi}{L}(x-x’)}\\ dx’ \\\\
\n&=& \\frac{1}{2\\pi} \\sum_{n = -\\infty}^{\\infty} \\frac{\\pi}{L} \\int_{-L}^{L} f(x’) \\,e^{i \\frac{n\\pi}{L}(x-x’)}\\ dx’
\n\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067\uff0c
\n$$ k_n \\equiv \\frac{ n\\pi}{L}, \\quad \\Delta k \\equiv k_{n+1} -k_n = \\frac{\\pi}{L}$$
\n\u3068\u3059\u308b\u3068\uff0c
\n$$ f(x) = \\frac{1}{2\\pi} \\sum_{n = -\\infty}^{\\infty} \\Delta k \\int_{-L}^{L} f(x’) \\,e^{i k_n \\,(x-x’)}\\ dx’$$<\/p>\n

\\( L \\rightarrow \\infty\\) \u3068\u3057\u3066\u5468\u671f\u6027\u306e\u306a\u3044\u95a2\u6570\u3078…<\/h3>\n

\\(L \\rightarrow \\infty \\) \u3068\u3059\u308c\u3070\uff0c\u5468\u671f\u304c\u7121\u9650\u5927\uff0c\u3064\u307e\u308a\u5468\u671f\u6027\u306e\u306a\u3044\u95a2\u6570\u3082\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306e\u62e1\u5f35\u3068\u3057\u3066\u3042\u3089\u308f\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u306f\u305a\u3002\u3053\u306e\u6975\u9650\u3067\u96e2\u6563\u7684\u306a \\(k_n\\) \u306f\u9023\u7d9a\u5909\u6570 \\(k\\) \u306b\u306a\u308a\uff0c
\n$$ {\\color{red}{L \\rightarrow \\infty}} \\ \\ \\Rightarrow \\ \\ {\\color{green}{k_n \\rightarrow k}}, \\ {\\color{blue}{\\Delta k \\rightarrow dk}}, \\ {\\color{blue}{\\sum_{n = -\\infty}^{\\infty} \\Delta k \\rightarrow \\int_{-\\infty}^{\\infty} dk}} $$ \u3068\u3059\u308c\u3070\uff0c\\(L \\rightarrow \\infty \\) \u306e\u6975\u9650\u3067 \\(f(x)\\) \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3042\u3089\u308f\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u306f\u3059\u3067\u3042\u308b\u3002<\/p>\n

\\begin{eqnarray}
\nf(x) &=& \\frac{1}{2\\pi} {\\color{blue}{\\sum_{n = -\\infty}^{\\infty} \\Delta k}} \\int_{-{\\color{red}{L}}}^{{\\color{red}{L}}} f(x’) \\,e^{i {\\color{green}{k_n}} (x-x’)}\\ dx’\\\\
\n&\\Rightarrow& \\frac{1}{2\\pi} {\\color{blue}{\\int_{-\\infty}^{\\infty} dk}} \\int_{-\\color{red}{\\infty}}^{\\color{red}{\\infty}}f(x’) \\,e^{i {\\color{green}{k}} (x-x’)}\\ dx’\\\\
\n&=& \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} \\ \\int_{-\\infty}^{\\infty} \\ f(x’) \\ e^{i k \\,(x-x’)}\\,dx’ \\, dk \\tag{1}
\n\\end{eqnarray}<\/p>\n

\u30d5\u30fc\u30ea\u30a8\u7a4d\u5206\uff0c\u30d5\u30fc\u30ea\u30a8\u5909\u63db\uff0c\u30d1\u30ef\u30fc\u30b9\u30da\u30af\u30c8\u30eb<\/h4>\n

\u4e0a\u3067\u5f97\u3089\u308c\u305f $(1)$ \u5f0f\u3092\u3042\u3089\u305f\u3081\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\uff082\u91cd\u7a4d\u5206\u306f $x’$ \u3067\u5148\u306b\u7a4d\u5206\u3059\u308b\u3053\u3068\u306b\u3059\u308b\uff09\uff1a<\/p>\n

\\begin{eqnarray}
\nf(x) &=& \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} \\ \\int_{-\\infty}^{\\infty} \\ f(x’) \\ e^{i k \\,(x-x’)}\\,dx’ \\, dk \\\\
\n&=& \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} \\left\\{ \\int_{-\\infty}^{\\infty} \\ f(x’) \\ e^{-i k x’}\\,dx’\\right\\} \\, e^{i k x}\\, dk\\\\
\n&\\equiv&\u00a0 \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} \\ F(k)\\ e^{i k x} dk
\n\\end{eqnarray}<\/p>\n

\u3053\u308c\u3092\uff08\u8907\u7d20\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306e\u5468\u671f\u7121\u9650\u5927\u6975\u9650\u3068\u3057\u3066\uff0c\u5468\u671f\u6027\u306e\u306a\u3044\uff09\u95a2\u6570 $f(x)$ \u306e\u30d5\u30fc\u30ea\u30a8\u7a4d\u5206<\/strong><\/span>\u3068\u547c\u3076\u3002\u307e\u305f\uff0c<\/p>\n

$$ F(k) \\equiv \\int_{-\\infty}^{\\infty}\u00a0 \\ f(x’) \\ e^{-i k x’} \\,dx’$$ \u3067\u5b9a\u7fa9\u3055\u308c\u308b \\(F(k)\\) \u3092\uff08\u8907\u7d20\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570\u306e\u5468\u671f\u7121\u9650\u5927\u6975\u9650\u3068\u3057\u3066\uff0c\u5468\u671f\u6027\u306e\u306a\u3044\uff09\u95a2\u6570 $f(x)$ \u306e\u30d5\u30fc\u30ea\u30a8\u5909\u63db<\/b><\/span>\u3068\u547c\u3076\u3002<\/p>\n

\u30d5\u30fc\u30ea\u30a8\u7a4d\u5206\u306e\u5f62\u3092\u3064\u3089\u3064\u3089\u3068\u773a\u3081\u308b\u3068\uff0c$f(x)$ \u306e\u4e2d\u306b\u3042\u308b\u300c\u6ce2\u300d\u306e\u6210\u5206 $e^{i k x}$ \u306e\u632f\u5e45\u3092\u8868\u3059\u306e\u304c $F(k)$ \u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002\u632f\u5e45\u306e\uff08\u7d76\u5bfe\u5024\u306e\uff09\u4e8c\u4e57\u306f\u300c\u6ce2\u300d\u306e\u5f37\u5ea6\u3092\u3042\u3089\u308f\u3059\u304b\u3089\uff0c<\/p>\n

$$ S(k) \\equiv |F(k)|^2 = F(k)\\,\\left\\{F(k)\\right\\}^{*}$$<\/p>\n

\u3068\u3057\u3066\uff0c$S(k)$ \u3092 $f(x)$ \u306e\u30d1\u30ef\u30fc\u30b9\u30da\u30af\u30c8\u30eb<\/strong><\/span>\u3068\u547c\u3093\u3067\u3044\u308b\u3002\uff08$\\left\\{F(k)\\right\\}^{*}$ \u306f $F(k)$ \u306e\u8907\u7d20\u5171\u5f79\u3002\uff09<\/p>\n

\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u3068\u30d5\u30fc\u30ea\u30a8\u7a4d\u5206\u3092\u4e26\u3079\u3066\u6bd4\u8f03<\/h3>\n

\u6bd4\u8f03\u306e\u305f\u3081\u306b\uff0c\u4e26\u3079\u3066\u307f\u308b\u3002\u304b\u305f\u3084\u8907\u7d20\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570\u306b $\\dfrac{1}{2L}$ \u304c\u3064\u304d\uff0c\u304b\u305f\u3084\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u3067\u306f\u306a\u304f\u3066\u30d5\u30fc\u30ea\u30a8\u7a4d\u5206\u306e\u307b\u3046\u306b $\\dfrac{1}{2\\pi}$ \u304c\u3064\u3044\u3066\u3044\u3066\uff0c\u5bfe\u5fdc\u304c\u30a4\u30de\u30a4\u30c1\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u304c\uff0c\u3054\u5bb9\u8d66\u3002<\/p>\n

\u5468\u671f \\(2L\\) \u306e\u5468\u671f\u95a2\u6570\u306b\u5bfe\u3059\u308b\uff08\u96e2\u6563\u7684\u306a\uff09\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b<\/h4>\n

\\( k_n \\equiv \\frac{n\\pi}{L} \\) \u3068\u3057\u3066<\/p>\n

$${\\color{blue}{ f(x) = \\sum_{n = -\\infty}^{\\infty} c_n \\ e^{i k_n x}}}, \\quad {\\color{red}{c_n = \\frac{1}{2L} \\int_{-L}^{L} f(x) \\ e^{- i k_n x}\\ dx }}$$<\/p>\n

\u5468\u671f\u6027\u306e\u306a\u3044\u95a2\u6570\u306b\u5bfe\u3059\u308b\uff08\u9023\u7d9a\u6975\u9650\u3068\u3057\u3066\u306e\uff09\u30d5\u30fc\u30ea\u30a8\u7a4d\u5206<\/h4>\n

$$ {\\color{blue}{f(x) = \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty}\\ F(k)\\ e^{i k x}\\ dk}}, \\quad\u00a0 {\\color{red}{F(k) \\equiv \\int_{-\\infty}^{\\infty} f(x) \\ e^{-i k x} \\ dx }}$$<\/p>\n

\u95a2\u6570 \\(\\exp\\left(i\\frac{n\\pi x}{L}\\right) \\) \u306e\u76f4\u4ea4\u6027\u306e\u9023\u7d9a\u6975\u9650\uff1a\u30c7\u30a3\u30e9\u30c3\u30af\u306e\u30c7\u30eb\u30bf\u95a2\u6570<\/h3>\n

\u5468\u671f\u95a2\u6570\u306b\u5bfe\u3059\u308b\u8907\u7d20\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306b\u95a2\u3059\u308b\u96e2\u6563\u7684\u95a2\u6570\\(\\exp\\left(i\\dfrac{n\\pi x}{L}\\right) \\) \u306e\u76f4\u4ea4\u6027\u306f\uff0c
\n\\begin{eqnarray}
\n\\int_{-L}^L\\exp\\left(i\\frac{(n-n’)\\pi}{L}x\\right)\\,dx
\n&=& \\begin{cases}
\n2L & (n = n’) \\\\
\n0\u00a0 & (n \\neq n’)
\n\\end{cases} \\\\ &=& 2L \\,\\delta_{n n’}
\n\\end{eqnarray}
\n\u3067\u3042\u3063\u305f\u3002
\n$$ k_n \\equiv \\frac{n\\pi}{L}, \\ \\quad k_{n’} \\equiv \\frac{n’ \\pi}{L}, \\quad \\Delta k = k_{n+1} -k_n = \\frac{\\pi}{L} $$ \u3068\u3057\u3066\u66f8\u304d\u76f4\u3059\u3068
\n\\begin{eqnarray}
\n\\int_{-L}^L e^{i(k_n -k_{n’}) x}\\ dx &=& 2L \\,\\delta_{n\u00a0 n’}\\\\
\n&=& 2\\pi \\frac{\\delta_{n\u00a0 n’}}{ \\frac{\\pi}{L} }
\n\\end{eqnarray}<\/p>\n

$$\\therefore\\ \\frac{1}{2\\pi} \\int_{-L}^L e^{i(k_n -k_{n’}) x}\\ dx = \\frac{\\delta_{n\u00a0 n’}}{ \\Delta k }$$<\/p>\n

\\(L \\rightarrow \\infty \\) \u306e\u3068\u304d\uff0c\u96e2\u6563\u7684\u306a \\(k_n\\) \u3082\u9023\u7d9a\u7684\u306a \\(k\\) \u306b\u306a\u308b\u306e\u3067 \\(k_n \\rightarrow k, \\ k_{n’} \\rightarrow k’\\) \u3068\u3057\uff0c\u3053\u306e\u7a4d\u5206\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b
\n$$\\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} e^{i (k -k’) x}\\ dx \\equiv \\delta(k -k’) \\tag{2}$$ \u3068\u3057\u3066\uff0c\u30c7\u30a3\u30e9\u30c3\u30af\u306e\u30c7\u30eb\u30bf\u95a2\u6570 \\(\\delta(k -k’)\\) <\/b><\/span>\u3092\u5b9a\u7fa9\u3059\u308b\u3002<\/p>\n

\u3053\u306e\u30c7\u30a3\u30e9\u30c3\u30af\u306e\u30c7\u30eb\u30bf\u95a2\u6570 \\(\\delta(k -k’)\\) \u306f\uff0c\u30af\u30ed\u30cd\u30c3\u30ab\u30fc\u306e\u30c7\u30eb\u30bf \\(\\delta_{n n’}\\) \u3092 \\(\\Delta k = \\frac{\\pi}{L}\\) \u3067\u5272\u3063\u305f\u91cf\u306e \\(L \\rightarrow \\infty\\) \u6975\u9650\u3067\u5b9a\u7fa9\u3055\u308c\u305f\u95a2\u6570
\n$$
\n\\delta(k -k’) \\equiv \\lim_{L \\rightarrow \\infty} \\frac{\\delta_{n\u00a0 n’}}{\\Delta k},\\\u00a0 \\quad k \\equiv \\frac{n\\pi}{L}, \\ \\ k’ \\equiv \\frac{n’ \\pi}{L}
\n$$\u3067\u3042\u308b\u305f\u3081\uff0c\\( k \\neq k’\\) \u306e\u3068\u304d\u306f \\(\\delta(k -k’) = 0\\) \u3067\u3042\u308b\u3053\u3068\u306f\u306a\u3093\u3068\u304b\u308f\u304b\u308b\u306b\u3057\u3066\u3082\uff0c\\( k = k’\\) \u306e\u3068\u304d\u306f\uff0c \\(L \\rightarrow 0\\) \u3067\u5206\u6bcd\u304c\\(\\Delta k \\rightarrow 0\\) \u3068\u306a\u3063\u3066\u3057\u307e\u3046\u305f\u3081\uff0c\\( \\delta(0) \\rightarrow \\infty !?\\) \u3068\u306a\u3063\u3066\u3057\u307e\u3046\u3088\u3046\u3067\u3042\u308a\uff0c\u3068\u3066\u3082\u666e\u901a\u306e\u95a2\u6570\u3068\u306f\u601d\u3048\u306a\u3044\u306e\u3067\uff0c\u300c\u8d85\u95a2\u6570\u300d\u3068\u547c\u3070\u308c\u308b\u3082\u306e\u306e1\u3064\u3067\u3042\u308b\u3002<\/p>\n

\u3053\u306e\u30c7\u30eb\u30bf\u95a2\u6570\u3092\u4f7f\u3063\u3066\u3042\u3089\u305f\u3081\u3066\u30d5\u30fc\u30ea\u30a8\u5909\u63db $F(k^{\\prime})$ \u3092\u66f8\u304f\u3068<\/p>\n

\\begin{eqnarray}
\nF(k^{\\prime}) &=& \\int_{-\\infty}^{\\infty} \\, f(x) \\,e^{-i k^{\\prime} x} \\,dx\\\\
\n&=& \\int_{-\\infty}^{\\infty} \\left\\{ \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} \\, F(k)\\, e^{i k x} \\,dk \\right\\} \\,e^{-i k^{\\prime} x}\\,dx\\\\
\n&=& \\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} \\, F(k)\\, e^{i (k-k^{\\prime}) x} \\,dx \\,dk\\\\
\n&=& \\int_{-\\infty}^{\\infty} \\, F(k) \\left\\{\\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} \\, e^{i (k-k^{\\prime}) x} \\,dx\\right\\}\\,dk \\\\
\n&=& \\int_{-\\infty}^{\\infty} \\, F(k)\\, \\delta(k -k^{\\prime}) \\,dk
\n\\end{eqnarray}<\/p>\n

\u3053\u306e\u6027\u8cea\u306f\uff0c\u96fb\u78c1\u6c17\u5b66\u3067\u3082\u51fa\u3066\u304d\u3066\u307e\u3057\u305f\u3002$k$\u00a0 \u3067\u306e\u7a4d\u5206\u306e\u969b\uff0c\u30c7\u30eb\u30bf\u95a2\u6570\u306e\u5f15\u6570 $k-k^{\\prime}$ \u304c\u30bc\u30ed\u306e\u6642\u306e\u5024\u3060\u3051\uff08\u3059\u306a\u308f\u3061 $k = k^{\\prime}$ \u306e\u3068\u304d\u306e $F(k^{\\prime})$ \u3060\u3051\uff09\u304c\u6b8b\u308a\u307e\u3059\u3002<\/p>\n