{"id":36060,"date":"2024-07-31T12:44:39","date_gmt":"2024-07-31T12:44:39","guid":{"rendered":"https:\/\/science-hub.click\/%E3%82%A8%E3%83%AB%E3%83%9F%E3%83%BC%E3%83%88%E5%A4%9A%E9%A0%85%E5%BC%8F%E3%81%AB%E3%81%A4%E3%81%84%E3%81%A6%E8%A9%B3%E3%81%97%E3%81%8F%E8%A7%A3%E8%AA%AC\/"},"modified":"2024-07-31T12:44:39","modified_gmt":"2024-07-31T12:44:39","slug":"%E3%82%A8%E3%83%AB%E3%83%9F%E3%83%BC%E3%83%88%E5%A4%9A%E9%A0%85%E5%BC%8F%E3%81%AB%E3%81%A4%E3%81%84%E3%81%A6%E8%A9%B3%E3%81%97%E3%81%8F%E8%A7%A3%E8%AA%AC","status":"publish","type":"post","link":"https:\/\/science-hub.click\/?p=36060","title":{"rendered":"\u30a8\u30eb\u30df\u30fc\u30c8\u591a\u9805\u5f0f\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac"},"content":{"rendered":"

\u6570\u5b66\u306b\u304a\u3051\u308b\u30a8\u30eb\u30df\u30fc\u30c8\u591a\u9805\u5f0f<\/strong>\u306f\u3001\u30c1\u30e3\u30fc\u30eb\u30ba \u30a8\u30eb\u30df\u30fc\u30c8\u306b\u3061\u306a\u3093\u3067\u540d\u4ed8\u3051\u3089\u308c\u305f\u4e00\u9023\u306e\u591a\u9805\u5f0f\u3067\u3059\u3002\u305d\u308c\u3089\u306f\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002 <\/p>

$$ {H_n(x)=(-1)^n e^{x^2\/2}\\frac{d^n}{dx^n}e^{-x^2\/2}} $$<\/div> (\u3044\u308f\u3086\u308b\u78ba\u7387<\/i>\u5f62\u5f0f) <\/dd><\/dl>
$$ {H_n(x)=(-1)^n e^{x^2}\\frac{d^n}{dx^n}e^{-x^2}} $$<\/div> \uff08\u3044\u308f\u3086\u308b\u7269\u7406\u7684\u306a<\/i>\u5f62\uff09<\/dd><\/dl>

2 \u3064\u306e\u5b9a\u7fa9\u306f\u5b8c\u5168\u306b<\/a><\/span>\u540c\u7b49\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002 1 \u3064\u306e\u5b9a\u7fa9<\/a><\/span>\u306e\u591a\u9805\u5f0f\u306f\u3001\u4ed6\u306e\u5b9a\u7fa9\u306b\u6bd4\u3079\u3066\u300c\u5727\u7e2e\u300d\u307e\u305f\u306f\u300c\u62e1\u5f35\u300d\u3055\u308c\u307e\u3059\u3002<\/p>

\u6b21\u306e\u95a2\u4fc2\u3092\u4f7f\u7528\u3057\u3066\u3001\u3042\u308b\u30d5\u30a9\u30fc\u30e0\u304b\u3089\u5225\u306e\u30d5\u30a9\u30fc\u30e0\u306b\u79fb\u884c\u3067\u304d\u307e\u3059\u3002

$$ {H_n^{phys}(x) = 2^{n\/2}H_n^{proba}(\\sqrt{2}\\,x)\\,\\!} $$<\/div> \u3002<\/p>

\u6700\u521d\u306e\u30a8\u30eb\u30df\u30fc\u30c8\u591a\u9805\u5f0f\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059 (\u300c\u78ba\u7387\u7684\u300d\u5f62\u5f0f)\u3002 <\/p>

$$ {H_0(x)=1~} $$<\/div><\/dd>
$$ {H_1(x)=x~} $$<\/div><\/dd>
$$ {H_2(x)=x^2-1~} $$<\/div><\/dd>
$$ {H_3(x)=x^3-3x~} $$<\/div><\/dd>
$$ {H_4(x)=x^4-6x^2+3~} $$<\/div><\/dd>
$$ {H_5(x)=x^5-10x^3+15x~} $$<\/div><\/dd>
$$ {H_6(x)=x^6-15x^4+45x^2-15~} $$<\/div><\/dd><\/dl>

\u5b9f\u969b\u306b\u3001\u4fc2\u6570<\/a><\/span>\u304c

$$ {x^{p-2}~} $$<\/div>\u306e
$$ {{H_p}~} $$<\/div>\u306f -p(p-1)\/4 \u306e\u4fa1\u5024\u304c\u3042\u308a\u3001\u3082\u3061\u308d\u3093\u4fc2\u6570\u3067\u3059
$$ {x^{p-1}~} $$<\/div>\u306e
$$ {{H_p}~} $$<\/div>\u306f\u5e38\u306b\u30bc\u30ed\u3067\u3059\u3002<\/p>

\u300c\u7269\u7406\u7684\u300d\u5f62\u5f0f\u3067\u306f\u3001\u6700\u521d\u306e\u591a\u9805\u5f0f\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059\u3002 <\/p>

$$ {H_0(x)=1~} $$<\/div><\/dd>
$$ {H_1(x)=2x~} $$<\/div><\/dd>
$$ {H_2(x)=4x^2-2~} $$<\/div><\/dd>
$$ {H_3(x)=8x^3-12x~} $$<\/div><\/dd>
$$ {H_4(x)=16x^4-48x^2+12~} $$<\/div><\/dd>
$$ {H_5(x)=32x^5-160x^3+120x~} $$<\/div><\/dd>
$$ {H_6(x)=64x^6-480x^4+720x^2-120~} $$<\/div><\/dd><\/dl>

\u76f4\u4ea4\u6027<\/a><\/span><\/span><\/h2>

\u30b7\u30fc\u30b1\u30f3\u30b9\u306en<\/i>\u756a\u76ee\u306e\u95a2\u6570\u306f\u3001\u6b21\u6570<\/span>n<\/i>\u306e\u591a\u9805\u5f0f<\/a><\/span>\u3067\u3059\u3002\u3053\u308c\u3089\u306e\u591a\u9805\u5f0f\u306f\u6e2c\u5b9a\u306b\u5bfe\u3057\u3066\u76f4\u4ea4\u3057\u307e\u3059\u3002 <\/p>

$$ {e^{-x^2\/2}\\,dx,} $$<\/div><\/dd><\/dl>

\u3064\u307e\u308a: <\/p>

$$ {\\int_{-\\infty}^{+\\infty} H_n(x)H_m(x)\\,e^{-x^2\/2}\\,dx=n!\\sqrt{2\\pi}~\\delta_{nm}} $$<\/div><\/dd><\/dl>

\u3053\u3053\u3067\u3001\u03b4 nm \u306f<\/i><\/sub>\u30af\u30ed\u30cd\u30c3\u30ab\u30fc\u8a18\u53f7<\/a><\/span>\u3067\u3042\u308a\u3001 n<\/i> = m<\/i>\u306e\u5834\u5408\u306f 1\u3001\u305d\u308c\u4ee5\u5916\u306e\u5834\u5408\u306f 0 \u3067\u3059\u3002\u3053\u308c\u3089\u306e\u95a2\u6570\u306f\u3001\u6b21\u306e\u7279\u6027\u3092\u6e80\u305f\u3059\u30d2\u30eb\u30d9\u30eb\u30c8\u7a7a\u9593<\/a><\/span>\u306e\u76f4\u4ea4\u57fa\u5e95\u3092\u5f62\u6210\u3057\u307e\u3059\u3002 <\/p>

$$ {\\int_{-\\infty}^{+\\infty}\\left|f(x)\\right|^2\\,e^{-x^2\/2}\\,dx< +\\infty,} $$<\/div><\/dd><\/dl>

\u3053\u3053\u3067\u3001\u30b9\u30ab\u30e9\u30fc\u7a4d\u306f<\/a><\/span>\u7a4d\u5206<\/a><\/span>\u306b\u3088\u3063\u3066\u4e0e\u3048\u3089\u308c\u307e\u3059\u3002 <\/p>

$$ {\\langle f,g\\rangle=\\int_{-\\infty}^{+\\infty} f(x)\\overline{g(x)}\\,e^{-x^2\/2}\\,dx.} $$<\/div><\/dd><\/dl>
\n\"\u30a8\u30eb\u30df\u30fc\u30c8\u591a\u9805\u5f0f\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac\"<\/figure>

\u3055\u307e\u3056\u307e\u306a\u7279\u6027<\/span><\/h2>

n<\/i>\u6b21\u30a8\u30eb\u30df\u30fc\u30c8\u591a\u9805\u5f0f\u306f\u6b21\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f<\/a><\/span>\u3092\u6e80\u305f\u3057\u307e\u3059\u3002 <\/p>

$$ {H_n”(x)-2xH_n'(x)+2nH_n(x)=0.\\,} $$<\/div><\/dd><\/dl>

\u6b21\u306e\u3088\u3046\u306a\u7e70\u308a\u8fd4\u3057\u30b7\u30fc\u30b1\u30f3\u30b9\u3082\u3042\u308a\u307e\u3059\u3002 <\/p>

$$ {H_{n+1}(x)- xH_{n}=-(n(n-1)\/4)H_{n-2}(x).\\,} $$<\/div><\/dd><\/dl>

\u591a\u9805\u5f0f\u306f\u6b21\u306e\u6027\u8cea\u3092\u6e80\u305f\u3057\u307e\u3059<\/p>

$$ {H_n'(x)=nH_{n-1}(x),\\,} $$<\/div><\/dd><\/dl>

\u3053\u308c\u306f\u6b21\u306e\u3088\u3046\u306b\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059<\/p>

$$ {H_n(x+y)=\\sum_{k=0}^n{n \\choose k}x^k H_{n-k}(y)} $$<\/div><\/dd><\/dl>
\n\"\u30a8\u30eb\u30df\u30fc\u30c8\u591a\u9805\u5f0f\"<\/figure>
\u30a8\u30eb\u30df\u30fc\u30c8\u591a\u9805\u5f0f<\/div><\/div><\/div><\/div><\/div>
\n\"\u30a8\u30eb\u30df\u30fc\u30c8\u591a\u9805\u5f0f\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac\"<\/figure>

\u53c2\u8003\u8cc7\u6599<\/h2>
  1. \u0645\u062a\u0639\u062f\u062f\u0627\u062a \u0627\u0644\u062d\u062f\u0648\u062f \u0644\u0647\u064a\u0631\u0645\u062a \u2013 arabe<\/a><\/li>
  2. \u0415\u0440\u043c\u0438\u0442\u043e\u0432 \u043f\u043e\u043b\u0438\u043d\u043e\u043c \u2013 bulgare<\/a><\/li>
  3. Polinomis d’Hermite \u2013 catalan<\/a><\/li>
  4. Hermitovy polynomy \u2013 tch\u00e8que<\/a><\/li>
  5. Hermitesches Polynom \u2013 allemand<\/a><\/li>
  6. Hermite polynomials \u2013 anglais<\/a><\/li><\/ol><\/div>\n
    \n

    \n \u30a8\u30eb\u30df\u30fc\u30c8\u591a\u9805\u5f0f\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac\u30fb\u95a2\u9023\u52d5\u753b\n <\/h2>\n
    \n \n
    \n
    \n