{"id":461,"date":"2021-11-22T01:03:58","date_gmt":"2021-11-21T16:03:58","guid":{"rendered":"https:\/\/naru.jpn.com\/wordpress\/?p=461"},"modified":"2021-11-22T12:07:04","modified_gmt":"2021-11-22T03:07:04","slug":"%e8%aa%bf%e5%92%8c%e6%8c%af%e5%8b%95%e5%ad%90%e3%81%ab%e5%af%be%e3%81%99%e3%82%8b%e5%a4%9a%e9%87%8d%e5%ba%a6%e9%96%a2%e6%95%b0","status":"publish","type":"post","link":"https:\/\/naru.jpn.com\/wordpress\/?p=461","title":{"rendered":"\u8abf\u548c\u632f\u52d5\u5b50\u306b\u5bfe\u3059\u308b\u591a\u91cd\u5ea6\u95a2\u6570"},"content":{"rendered":"

\u306a\u3093\u3068\u306a\u304f\u30ad\u30c3\u30c6\u30eb\u306e\u71b1\u7269\u7406\u5b66\u7b2c2\u7248\u3092\u518d\u8aad\u3057\u3066\u307f\u3066\u3044\u307e\u3059. \u305b\u3063\u304b\u304f\u306a\u306e\u3067\u5f0f\u3092\u8ffd\u3044\u306a\u304c\u3089\u8aad\u3093\u3067\u3044\u308b\u306e\u3067\u3059\u304c, 1\u7ae0\u306e\u8abf\u548c\u632f\u52d5\u5b50\u306e\u591a\u91cd\u5ea6\u95a2\u6570\u3092\u6c42\u3081\u308b\u4f8b\u984c(p.20)\u3067\u6c17\u6301\u3061\u304c\u5206\u304b\u308b\u307e\u3067\u5c11\u3057\u8003\u3048\u308b\u5fc5\u8981\u304c\u3042\u3063\u305f\u306e\u3067, \u88dc\u5b8c\u3057\u305f\u60c5\u5831\u3092\u5099\u5fd8\u9332\u3068\u3057\u3066\u6b8b\u3057\u3066\u304a\u304d\u307e\u3059.<\/p>\n

<\/p>\n

\u524d\u63d0\u3068\u6c42\u3081\u305f\u3044\u8868\u5f0f<\/h2>\n

\u91cf\u5b50\u6570 \\(s\\) \u3092\u6b63\u306e\u6574\u6570\u304b\u30bc\u30ed\u3068\u3057, \u8abf\u548c\u632f\u52d5\u5b50\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u56fa\u6709\u5024\u3092 \\(\\epsilon_{s} = s \\hbar \\omega\\) \u3068\u3059\u308b. \\(N\\)\u500b\u306e\u632f\u52d5\u5b50\u3092\u8003\u3048, \\(n = \\sum_{i = 1}^{N} s_{i}\\) \u3068\u3057\u305f\u6642, \u5168\u52b1\u8d77\u30a8\u30cd\u30eb\u30ae\u30fc<\/p>\n

\\[ \\epsilon = \\sum_{i = 1}^{N} s_{i} \\hbar \\omega = n \\hbar \\omega \\]<\/p>\n

\u304c\u632f\u52d5\u5b50\u306b\u5206\u5e03\u3059\u308b\u4ed5\u65b9\u306e\u6570, \u3059\u306a\u308f\u3061\u591a\u91cd\u5ea6\u95a2\u6570 \\(g\\left(N, n\\right)\\) \u3092\u77e5\u308a\u305f\u3044. \u7d50\u679c\u306f,<\/p>\n

\\[ g\\left(N, n\\right) = \\frac{\\left(N + n – 1\\right)!}{n!\\left(N – 1\\right)!} \\]<\/p>\n

\u3067\u3042\u308b.<\/p>\n

\u591a\u91cd\u5ea6\u95a2\u6570\u3092 \\(t\\) \u3092\u7528\u3044\u305f\u8868\u5f0f\u3067\u8868\u3059<\/h2>\n

\u307e\u305a\u306f\u304a\u3082\u3080\u308d\u306b \\(t \\left(|t| < 0\\right)\\) \u3092\u5c0e\u5165\u3057, \\(g\\left(N, n\\right)\\) \u3092 \\(t\\) \u3092\u7528\u3044\u305f\u8868\u5f0f\u3067\u8868\u3059\u3068\u3053\u308d\u3092\u8003\u3048\u308b. \u3053\u306e \\(t\\) \u306f\u6700\u7d42\u7684\u306b\u306f\u5f0f\u304b\u3089\u6d88\u3048\u308b\u304c, \u5f0f\u5c55\u958b\u306e\u4e2d\u30672\u3064\u306e\u5927\u304d\u306a\u5f79\u5272\u3092\u6f14\u3058, \u4e00\u3064\u306f\u7121\u9650\u7b49\u6bd4\u7d1a\u6570\u306e\u548c\u306e\u516c\u5f0f\u3092\u4f7f\u7528\u3059\u308b\u5834\u9762, \u3082\u3046\u4e00\u3064\u306f \\(\\lim_{t \\rightarrow 0}\\) \u3084 \\(n\\) \u968e\u306e\u5fae\u5206\u3067\u307b\u3068\u3093\u3069\u306e\u9805\u304c\u6d88\u3055\u308c\u308b\u5834\u9762\u3060\u304c, \u5c0e\u5165\u6642\u70b9\u3067\u306f\u3053\u306e\u76ee\u7684\u304c\u5206\u304b\u308a\u3065\u3089\u3044. \u8abf\u548c\u632f\u52d5\u5b50\u306b\u5bfe\u3059\u308b\u591a\u91cd\u5ea6\u95a2\u6570 \\(g\\left(N, n\\right)\\) \u306b\u3064\u3044\u3066, \u4ee5\u4e0b\u306e\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3068\u4e3b\u5f35\u304c\u3055\u308c\u308b.<\/p>\n

\\[ \\left(\\sum_{s = 0}^{\\infty} t^{s}\\right)^{N} = \\sum_{n = 0}^{\\infty} g\\left(N, n\\right) t^{n} \\tag{1} \\]<\/p>\n

(1)\u5f0f\u306e\u53f3\u8fba\u3092\u5c55\u958b\u3059\u308b.<\/p>\n

\\[ \\sum_{n = 0}^{\\infty} g\\left(N, n\\right) t^{n} = g\\left(N, 0\\right) t^{0} + g\\left(N, 1\\right) t^{1} + g\\left(N, 2\\right) t^{2} + g\\left(N, 3\\right) t^{3} + \\cdots \\tag{2} \\]<\/p>\n

(1)\u5f0f\u306e\u5de6\u8fba\u3092\u5c55\u958b\u3059\u308b.<\/p>\n

\\[ \\begin{eqnarray} \\left(\\sum_{s = 0}^{\\infty} t^{s}\\right)^{N} &=& \\underbrace{ \\left(1 + t + t^{2} + \\cdots\\right) \\cdot \\left(1 + t + t^{2} + \\cdots\\right) \\cdot \\cdots \\cdot \\left(1 + t + t^{2} + \\cdots\\right) }_{N} \\\\ &=& a t^{0} + b t^{1} + c t^{2} + \\cdots \\end{eqnarray} \\]<\/p>\n

\\(t^{0}\\) \u306e\u9805\u306e\u500b\u6570\u306f \\(1\\) \u306e\u307f\u306e\u7a4d\u3067\u4e00\u901a\u308a\u3057\u304b\u306a\u3044\u3057, \\(t^{1}\\) \u306e\u9805\u306e\u500b\u6570\u306f \\(N\\) \u500b\u306a\u306e\u3067, \u6697\u7b97\u3067\u3082 \\(a = 1, b = N\\) \u3068\u306a\u308b\u306e\u306f\u5206\u304b\u308b. \\(c\\) \u306f\u3082\u306f\u3084\u6697\u7b97\u3067\u306f\u53b3\u3057\u3044\u304c, \\(c\\) \u3068\u306f\u3064\u307e\u308a\u3069\u3046\u3044\u3046\u4fc2\u6570\u306e\u6570\u3048\u65b9\u3092\u3059\u308b\u304b\u3068\u3044\u3046\u3068, \u6b63\u306e\u6574\u6570\u304b\u30bc\u30ed\u3092 \\(N\\) \u500b\u96c6\u3081\u305f\u6642\u306b\u5408\u8a08\u304c \\(2\\) \u3068\u306a\u308b\u3088\u3046\u306a\u7d44\u307f\u5408\u308f\u305b\u306e\u6570\u3092\u6570\u3048\u3066\u3044\u308b. \u3053\u308c\u306f\u4eca\u8003\u3048\u3066\u3044\u308b \\(g\\left(N, 2\\right)\\) \u306e\u5b9a\u7fa9\u3068\u540c\u3058\u306a\u306e\u3067, \\(a = g\\left(N, 0\\right), b = g\\left(N, 1\\right), c = g\\left(N, 2\\right), \\ldots \\) \u3068\u66f8\u3051\u308b. \u5f93\u3063\u3066,<\/p>\n

\\[ \\left(\\sum_{s = 0}^{\\infty} t^{s}\\right)^{N} = g\\left(N, 0\\right) t^{0} + g\\left(N, 1\\right) t^{1} + g\\left(N, 2\\right) t^{2} + \\cdots \\tag{3} \\]<\/p>\n

\u3068\u306a\u308b. \u3053\u308c\u3067(1)\u5f0f\u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u304c\u78ba\u8a8d\u3067\u304d\u305f.<\/p>\n

\\[ \\left(\\sum_{s = 0}^{\\infty} t^{s}\\right)^{N} = \\left(\\frac{1}{1 – t}\\right)^{N} \\]<\/p>\n

\u306b\u3064\u3044\u3066\u306f, \u3053\u308c\u306f\u305f\u3060\u306e\u7121\u9650\u7b49\u6bd4\u7d1a\u6570\u306e\u548c\u3067\u3042\u308b.<\/p>\n

\\(n\\) \u968e\u306e\u5fae\u5206\u3068\u6975\u9650\u3092\u7528\u3044\u3066\u591a\u91cd\u5ea6\u95a2\u6570\u304b\u3089 \\(t\\) \u3092\u6d88\u53bb\u3059\u308b<\/h2>\n

\\(t\\) \u3092\u6d88\u53bb\u3057\u306b\u304b\u304b\u308b\u304c, \u672c\u6587\u4e2d\u3067\u306f\u300c\u6b21\u306e\u3088\u3046\u306b\u3059\u308c\u3070\u3088\u3044\u300d\u3068\u66f8\u304b\u308c\u3066\u4ee5\u4e0b\u306e\u8868\u5f0f\u304c\u3042\u308b.<\/p>\n

\\[ g\\left(N, n\\right) = \\lim_{t \\rightarrow 0} \\frac{1}{n!} \\left(\\frac{d}{dt}\\right)^{n} \\sum_{s = 0}^{\\infty} g\\left(N, s\\right) t^{s} \\tag{4} \\]<\/p>\n

\u53f3\u8fba\u306e\u4e00\u90e8\u3092\u5c55\u958b\u3059\u308b.<\/p>\n

\\[ \\sum_{s = 0}^{\\infty} g\\left(N, s\\right) t^{s} = \\underbrace{ g\\left(N, 0\\right)t^{0} + g\\left(N, 1\\right)t^{1} + g\\left(N, 2\\right)t^{2} + \\cdots }_{ \\left(\\frac{d}{dt}\\right)^{n} \\mbox{\u306b\u3088\u3063\u3066\u6d88\u3048\u308b\u9805}} + g\\left(N, n\\right)t^{n} + \\underbrace{ g\\left(N, n + 1\\right)t^{n + 1} + g\\left(N, n + 2\\right)t^{n + 2} + \\cdots}_{\\lim_{t \\rightarrow 0} \\mbox{\u306b\u3088\u3063\u3066\u6d88\u3048\u308b\u9805}} \\]<\/p>\n

\\(t^{n}\\) \u306e\u524d\u5f8c\u306e\u9805\u306f\u5fae\u5206\u3084\u5fae\u5206\u5f8c\u306e\u6975\u9650\u306e\u64cd\u4f5c\u306b\u3088\u3063\u3066\u6d88\u53bb\u3055\u308c, \u6b8b\u3063\u305f\u9805\u306e \\(t^{n}\\) \u306f \\(n\\) \u968e\u306e\u5fae\u5206\u306b\u3088\u308a \\(n!\\) \u306b\u306a\u308a\u3001\\(\\frac{1}{n!}\\) \u3067\u6253\u3061\u6d88\u3055\u308c\u3066, \u6700\u7d42\u7684\u306b \\(g\\left(N, n\\right)\\) \u3068\u306a\u308a, (4)\u5f0f\u304c\u6210\u7acb\u3059\u308b. \u3042\u3068\u306f\u5f0f\u5909\u5f62\u3060\u3051\u3067,<\/p>\n

\\[ \\frac{d}{dt} \\left(1 – t\\right)^{-N} = (-N) \\cdot \\left(1 – t\\right)^{-N-1} \\cdot (-1) = N \\cdot \\left(1 – t\\right)^{-(N+1)} \\]<\/p>\n

\u306e\u3088\u3046\u306b\u5730\u9053\u306b\u5fae\u5206\u3092\u3057\u3066\u3044\u3051\u3070,<\/p>\n

\\[ g\\left(N, n\\right) = \\frac{1}{n!} N(N+1)(N+2) \\cdots (N+n-1) \\]<\/p>\n

\u3068\u306a\u308a,<\/p>\n

\\[ g\\left(N, n\\right) = \\frac{\\left(N + n – 1\\right)!}{n!\\left(N – 1\\right)!} \\]<\/p>\n

\u3092\u5f97\u308b.<\/p>\n","protected":false},"excerpt":{"rendered":"

\u306a\u3093\u3068\u306a\u304f\u30ad\u30c3\u30c6\u30eb\u306e\u71b1\u7269\u7406\u5b66\u7b2c2\u7248\u3092\u518d\u8aad\u3057\u3066\u307f\u3066\u3044\u307e\u3059. \u305b\u3063\u304b\u304f\u306a\u306e\u3067\u5f0f\u3092\u8ffd\u3044\u306a\u304c\u3089\u8aad\u3093\u3067\u3044\u308b\u306e\u3067\u3059\u304c, 1\u7ae0\u306e\u8abf\u548c\u632f\u52d5\u5b50\u306e\u591a\u91cd\u5ea6\u95a2\u6570\u3092\u6c42\u3081\u308b\u4f8b\u984c(p.20)\u3067\u6c17\u6301\u3061\u304c\u5206\u304b\u308b\u307e\u3067\u5c11\u3057\u8003\u3048\u308b\u5fc5\u8981\u304c\u3042\u3063\u305f\u306e\u3067, \u88dc\u5b8c\u3057\u305f\u60c5 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[10],"tags":[],"_links":{"self":[{"href":"https:\/\/naru.jpn.com\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/461"}],"collection":[{"href":"https:\/\/naru.jpn.com\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/naru.jpn.com\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/naru.jpn.com\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/naru.jpn.com\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=461"}],"version-history":[{"count":38,"href":"https:\/\/naru.jpn.com\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/461\/revisions"}],"predecessor-version":[{"id":501,"href":"https:\/\/naru.jpn.com\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/461\/revisions\/501"}],"wp:attachment":[{"href":"https:\/\/naru.jpn.com\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=461"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/naru.jpn.com\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=461"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/naru.jpn.com\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=461"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}