- \n
- \u7df4\u7fd2\uff1a\u95a2\u6570\u306e\u6975\u5927\u5024<\/a><\/li>\n<\/ul>\n<\/li>\n
- SciPy \u3067\u6570\u5024\u89e3\u6790<\/a>\n
- \n
- \u7df4\u7fd2\uff1a\u95a2\u6570\u306e\u6975\u5927\u5024<\/a><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n
\u5358\u306b\u6f14\u7fd2\u306e\u305f\u3081\u306e\u6f14\u7fd2\u554f\u984c\u3068\u3044\u3046\u306e\u3067\u306f\u306a\u304f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u610f\u7fa9\u304c\u3042\u308b\u3093\u3060\u3088\u3068\u3044\u3046\u8a71\u3002<\/p>\n
<\/p>\n
\u53c2\u8003\uff1a<\/p>\n
- \n
- \u30d7\u30e9\u30f3\u30af\u306e\u6cd5\u5247 – Wikipedia<\/a><\/li>\n
- \u30d7\u30e9\u30f3\u30af\u306e\u6cd5\u5247 | \u5929\u6587\u5b66\u8f9e\u5178<\/a><\/li>\n<\/ul>\n
\u6e29\u5ea6 $T$ \u306e\u9ed2\u4f53\u304b\u3089\u653e\u5c04\u3055\u308c\u308b\u653e\u5c04\u5f37\u5ea6\u306f\uff0c\u5468\u6ce2\u6570 $\\nu$ \u306e\u95a2\u6570\u3068\u3057\u3066\u5358\u4f4d\u5468\u6ce2\u6570\u3042\u305f\u308a\u3067\u3042\u3089\u308f\u3059\u3068<\/p>\n
\\begin{eqnarray}
\nI(\\nu) &=& \\frac{2 h \\nu^3}{c^2} \\frac{1}{\\exp\\left(\\frac{h\\nu}{k T} \\right) -1} \\\\
\n&=& \\frac{2h}{c^2} \\left( \\frac{kT}{h}\\right)^3 \\left( \\frac{h \\nu}{kT} \\right)^3 \\frac{1}{\\exp\\left(\\frac{h\\nu}{k T} \\right) -1} \\\\
\n&=& \\mbox{const.} \\times \\frac{x^3}{e^x-1}, \\quad x \\equiv \\frac{h\\nu}{kT}
\n\\end{eqnarray}<\/p>\n\u3068\u66f8\u3051\u308b\u3002\u3053\u3053\u3067 $h$ \u306f\u30d7\u30e9\u30f3\u30af\u5b9a\u6570\uff0c$k$ \u306f\u30dc\u30eb\u30c4\u30de\u30f3\u5b9a\u6570\uff0c$c$ \u306f\u5149\u901f\u3002\u3053\u308c\u304c\u30d7\u30e9\u30f3\u30af\u306e\u6cd5\u5247\u3067\u3042\u308a\uff0c\u30d7\u30e9\u30f3\u30af\u306e\u6cd5\u5247\u3067\u3042\u3089\u308f\u3055\u308c\u308b\u653e\u5c04\u5f37\u5ea6\u5206\u5e03\u306e\u3053\u3068\u3092\u30d7\u30e9\u30f3\u30af\u5206\u5e03\u3068\u3044\u3046\u3002<\/p>\n
\u3064\u307e\u308a\uff0c\u95a2\u6570 $f(x) \\equiv \\dfrac{x^3}{e^x-1}$ \u306e\u30b0\u30e9\u30d5\u3092\u63cf\u304d\uff0c\u6975\u5927\u5024\u3092\u8abf\u3079\u308b\u3068\u3044\u3046\u3053\u3068\u306f\u30d7\u30e9\u30f3\u30af\u5206\u5e03\u3092\u77e5\u308b\u3068\u3044\u3046\u610f\u7fa9\u304c\u3042\u308b\u3093\u3060\u3088\u3002<\/p>\n
\u307e\u305f\uff0c\u30d7\u30e9\u30f3\u30af\u5206\u5e03\u3092\u5358\u4f4d\u6ce2\u9577\u3042\u305f\u308a\u306b\u3057\u3066\u6ce2\u9577 $\\lambda$ \u306e\u95a2\u6570\u3068\u3057\u3066\u3042\u3089\u308f\u3059\u3068<\/p>\n
\\begin{eqnarray}
\nI^{\\prime}(\\lambda) &=& \\frac{2hc^2}{\\lambda^5} \\frac{1}{\\exp\\left(\\frac{hc}{kT\\lambda} \\right)-1}\\\\
\n&=& 2 h c^2 \\left( \\frac{kT}{hc}\\right)^5 \\left( \\frac{hc}{kT \\lambda}\\right)^5\\frac{1}{\\exp\\left(\\frac{hc}{kT\\lambda} \\right)-1}\\\\
\n&=& \\mbox{const.} \\times \\frac{x^5}{e^x-1}, \\quad x \\equiv \\frac{hc}{kT \\lambda}
\n\\end{eqnarray}<\/p>\n\u3068\u66f8\u3051\u308b\u3002<\/p>\n
\u30d7\u30e9\u30f3\u30af\u5206\u5e03 $I^{\\prime}(\\lambda)$ \u304c\u6700\u5927\u3068\u306a\u308b\u6ce2\u9577 $\\lambda_{\\rm max}$ \u306f\u6e29\u5ea6 $T$ \u306b\u53cd\u6bd4\u4f8b\u3059\u308b\u3053\u3068\u304c\u77e5\u3089\u308c\u3066\u3044\u3066\uff0c\u3053\u308c\u3092\u30a6\u30a3\u30fc\u30f3\u306e\u5909\u4f4d\u5247\u3068\u3044\u3046\u3002<\/p>\n
$I^{\\prime}(\\lambda)$ \u306e\u6700\u5927\u5024\u3092\u8abf\u3079\u308b\u3053\u3068\u306f $g(x) \\equiv \\dfrac{x^5}{e^x-1}$ \u306e\u6975\u5927\u5024\u3092\u8abf\u3079\u308b\u3053\u3068\u3060\u304b\u3089\uff0c\u30a6\u30a3\u30fc\u30f3\u306e\u5909\u4f4d\u5247\u3092\u5c0e\u304f\u305f\u3081\u306b\u306f\u3053\u306e\u95a2\u6570\u304c\u6975\u5927\u3068\u306a\u308b\u70b9\u3092\u6570\u5024\u7684\u306b\u6c42\u3081\u308b\u5fc5\u8981\u304c\u3042\u308b\u308f\u3051\u3060\u3002<\/p>\n
$g(x)$ \u304c\u6975\u5927\u3068\u306a\u308b $x$ \u3092 $x_{\\rm max}$ \u3068\u3059\u308b\u3068\uff0c$g^{\\prime}(x_{\\rm max}) = 0$ \u3092\u6570\u5024\u7684\u306b\u89e3\u304d\uff0c<\/p>\n
\\begin{eqnarray}
\nx_{\\rm max} &=& \\frac{hc}{kT \\lambda_{\\rm max}} \\\\
\n\\therefore\\ \\ \\lambda_{\\rm max} &=& \\frac{hc}{k\\, x_{\\rm max}} \\frac{1}{T} \\propto \\frac{1}{T}
\n\\end{eqnarray}<\/p>\n\u3068\u306a\u308b\u3002\u6bd4\u4f8b\u5b9a\u6570\u306e\u5177\u4f53\u7684\u306a\u6570\u5024\u3082\u7c21\u5358\u306b\u6c42\u3081\u3089\u308c\u308b\u3088\u306d\u3002<\/p>\n
\n\n<\/div>\n\n\n\u30a6\u30a3\u30fc\u30f3\u306e\u5909\u4f4d\u5247<\/h3>\n
$g(x) \\equiv \\dfrac{x^5}{e^x-1}$ \u304c\u6975\u5927\u5024\u3092\u3068\u308b $x$ \u306e\u5024\u3092 $x_{\\rm max}$ \u3068\u3059\u308b\u3068\uff0c\u30d7\u30e9\u30f3\u30af\u5206\u5e03\u304c\u6700\u5927\u5024\u3092\u3068\u308b\u6ce2\u9577 $\\lambda_{\\rm max}$ \u306f<\/p>\n
\\begin{eqnarray}
\nx_{\\rm max} &=& \\frac{hc}{kT \\lambda_{\\rm max}} \\\\
\n\\therefore\\ \\ \\lambda_{\\rm max} &=& \\frac{hc}{k\\, x_{\\rm max}} \\frac{1}{T} \\propto \\frac{1}{T}
\n\\end{eqnarray}
\n\u3064\u307e\u308a\uff0c\u6e29\u5ea6 $T$ \u306b\u53cd\u6bd4\u4f8b\u3059\u308b\u3002\u3053\u308c\u3092\u30a6\u30a3\u30fc\u30f3\u306e\u5909\u4f4d\u5247\u3068\u3044\u3046\u3002\u3053\u3053\u3067 $h$ \u306f\u30d7\u30e9\u30f3\u30af\u5b9a\u6570\uff0c$k$ \u306f\u30dc\u30eb\u30c4\u30de\u30f3\u5b9a\u6570\uff0c$c$ \u306f\u5149\u901f\u3002\u3053\u308c\u30893\u3064\u306e\u5b9a\u6570\u306f\u3044\u305a\u308c\u3082\u5b9a\u7fa9\u5b9a\u6570\u3067\u3042\u308a\uff0c\u8aa4\u5dee\u306f\u306a\u3044\u3002<\/p>\n- \n
- \u30d7\u30e9\u30f3\u30af\u5b9a\u6570 – Wikipedia<\/a><\/li>\n
- \u30dc\u30eb\u30c4\u30de\u30f3\u5b9a\u6570 – Wikipedia<\/a><\/li>\n
- \u5149\u901f – Wikipedia<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n
\n<\/div>\n\n\n$x_{\\rm max}$ \u3092\u6c42\u3081\u308b<\/h3>\n
\u30e9\u30a4\u30d6\u30e9\u30ea\u306e import<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
\n\nIn\u00a0[1]:<\/div>\n\n\n\n# SymPy \u3092\u4f7f\u3046\u3068\u304d\u306e\u304a\u307e\u3058\u306a\u3044<\/span>\r\nfrom<\/span> sympy.abc<\/span> import<\/span> *<\/span>\r\nfrom<\/span> sympy<\/span> import<\/span> *<\/span>\r\n\r\n# \u30b0\u30e9\u30d5\u306f SymPy Plotting Backends (SPB) \u3067\u63cf\u304f<\/span>\r\nfrom<\/span> spb<\/span> import<\/span> *<\/span>\r\n\r\n# \u4ee5\u4e0b\u306f\u30b0\u30e9\u30d5\u3092 SVG \u3067 Notebook \u306b\u30a4\u30f3\u30e9\u30a4\u30f3\u8868\u793a\u3055\u305b\u308b\u8a2d\u5b9a<\/span>\r\n%<\/span>config<\/span> InlineBackend.figure_formats = ['svg']\r\n\r\nimport<\/span> matplotlib.pyplot<\/span> as<\/span> plt<\/span>\r\n# mathtext font \u306e\u8a2d\u5b9a<\/span>\r\nplt<\/span>.<\/span>rcParams<\/span>[<\/span>'mathtext.fontset'<\/span>]<\/span> =<\/span> 'cm'<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n<\/div>\n\n\n$g(x)$ \u306e\u5b9a\u7fa9<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
\n\nIn\u00a0[2]:<\/div>\n\n\n\ndef<\/span> g<\/span>(<\/span>x<\/span>):<\/span>\r\n return<\/span> x<\/span>**<\/span>5<\/span>\/<\/span>(<\/span>exp<\/span>(<\/span>x<\/span>)<\/span> -<\/span> 1<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n<\/div>\n\n\n$g'(x)$<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
\n\nIn\u00a0[3]:<\/div>\n\n\n\n# g(x) \u306e\u5fae\u5206<\/span>\r\n\r\ndg<\/span> =<\/span> diff<\/span>(<\/span>g<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>\r\ndisplay<\/span>(<\/span>dg<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<\/div>\n$\\displaystyle -\\frac{x^{5} e^{x}}{\\left(e^{x} -1\\right)^{2}} + \\frac{5 x^{4}}{e^{x} -1}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n<\/div>\n\n\n$g'(x)$ \u306e\u30b0\u30e9\u30d5\u3092\u63cf\u304d\uff0c$g'(x) = 0$ \u3068\u306a\u308b $x$ \u306e\u3042\u305f\u308a\u3092\u3064\u3051\u308b<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
\n\nIn\u00a0[4]:<\/div>\n\n\n\nxaxis<\/span> =<\/span> hline<\/span>(<\/span>0<\/span>,<\/span> rendering_kw<\/span>=<\/span>{<\/span>'c'<\/span>:<\/span>'k'<\/span>})<\/span>\r\ngraphics<\/span>(<\/span>\r\n line<\/span>(<\/span>dg<\/span>,<\/span> (<\/span>x<\/span>,<\/span> 0.001<\/span>,<\/span> 10<\/span>),<\/span> r<\/span>'$g^{\\prime}(x)$'<\/span>),<\/span> \r\n xaxis<\/span>\r\n)<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n<\/div>\n\n\n\u4e0a\u306e\u30b0\u30e9\u30d5\u304b\u3089\u308f\u304b\u308b\u3088\u3046\u306b\uff0c$g^{\\prime}(x_{\\rm max}) = 0$ \u3068\u306a\u308b $x_{\\rm max}$ \u306f $4 < x_{\\rm max} < 6$ \u306e\u7bc4\u56f2\u306b\u3042\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n<\/div>\n\n\n$g'(x_{\\rm max}) = 0$ \u3068\u306a\u308b $x_{\\rm max}$ \u3092\u6570\u5024\u7684\u306b\u6c42\u3081\u308b<\/h4>\n
dg = 0<\/code> \u3068\u306a\u308b $x$ \u3092 $4 < x < 6$ \u306e\u7bc4\u56f2\u3067\u6570\u5024\u7684\u306b\u6c42\u3081\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\nIn\u00a0[5]:<\/div>\n\n\n\nx_max<\/span> =<\/span> nsolve<\/span>(<\/span>dg<\/span>,<\/span> x<\/span>,<\/span> [<\/span>4<\/span>,<\/span> 6<\/span>])<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n<\/div>\n\n\n\u30a6\u30a3\u30fc\u30f3\u306e\u5909\u4f4d\u5247\u306e\u6bd4\u4f8b\u5b9a\u6570\u3092\u6c42\u3081\u308b<\/h3>\n
$$ W_{\\rm const} \\equiv \\frac{h c}{k\\, x_{\\rm max}}$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n\nIn\u00a0[6]:<\/div>\n\n\n\nh<\/span> =<\/span> 6.62607015E-34<\/span>\r\nk<\/span> =<\/span> 1.380649E-23<\/span>\r\nc<\/span> =<\/span> 299792458<\/span>\r\n\r\nW_const<\/span> =<\/span> h<\/span> *<\/span> c<\/span>\/<\/span>(<\/span>k<\/span> *<\/span> x_max<\/span>)<\/span>\r\nprint<\/span>(<\/span>'<\/span>%e<\/span>'<\/span> %<\/span> W_const<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<\/div>\n - \u30dc\u30eb\u30c4\u30de\u30f3\u5b9a\u6570 – Wikipedia<\/a><\/li>\n
- \u30d7\u30e9\u30f3\u30af\u306e\u6cd5\u5247 | \u5929\u6587\u5b66\u8f9e\u5178<\/a><\/li>\n<\/ul>\n
- \u30d7\u30e9\u30f3\u30af\u306e\u6cd5\u5247 – Wikipedia<\/a><\/li>\n
- SciPy \u3067\u6570\u5024\u89e3\u6790<\/a>\n