[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki11\/archives\/188656#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki11\/archives\/188656","headline":"\u5909\u5206\u539f\u7406 – Wikipedia","name":"\u5909\u5206\u539f\u7406 – Wikipedia","description":"\u3053\u306e\u8a18\u4e8b\u306f\u691c\u8a3c\u53ef\u80fd\u306a\u53c2\u8003\u6587\u732e\u3084\u51fa\u5178\u304c\u5168\u304f\u793a\u3055\u308c\u3066\u3044\u306a\u3044\u304b\u3001\u4e0d\u5341\u5206\u3067\u3059\u3002\u51fa\u5178\u3092\u8ffd\u52a0\u3057\u3066\u8a18\u4e8b\u306e\u4fe1\u983c\u6027\u5411\u4e0a\u306b\u3054\u5354\u529b\u304f\u3060\u3055\u3044\u3002\u51fa\u5178\u691c\u7d22?:\u00a0“\u5909\u5206\u539f\u7406”\u00a0\u2013\u00a0\u30cb\u30e5\u30fc\u30b9\u00a0\u00b7 \u66f8\u7c4d\u00a0\u00b7 \u30b9\u30ab\u30e9\u30fc\u00a0\u00b7 CiNii\u00a0\u00b7 J-STAGE\u00a0\u00b7 NDL\u00a0\u00b7 dlib.jp\u00a0\u00b7 \u30b8\u30e3\u30d1\u30f3\u30b5\u30fc\u30c1\u00a0\u00b7 TWL\uff082011\u5e747\u6708\uff09 \u5909\u5206\u539f\u7406\uff08\u3078\u3093\u3076\u3093\u3052\u3093\u308a\u3001\u82f1\u8a9e: variational principle\uff09\u306f\u3001\u5909\u5206\u6cd5\u3092\u7528\u3044\u305f\u7269\u7406\u5b66\u306e\u539f\u7406\u3002 \u7279\u306b\u3001 \u5909\u5206\u539f\u7406\u306f\u7a4d\u5206\u306e\u5f62\u3067\u6271\u3046\u306e\u3067\u3001\u5ea7\u6a19\u7cfb\u306e\u53d6\u308a\u65b9\u306b\u4f9d\u5b58\u3057\u306a\u3044\u3002\u5f93\u3063\u3066\u62e1\u5f35\u6027\u306b\u512a\u308c\u3001\u3044\u308d\u3044\u308d\u306a\u5206\u91ce\u306b\u5fdc\u7528\u3001\u5229\u7528\u3055\u308c\u308b\u3002 Table of Contents \u53e4\u5178\u529b\u5b66[\u7de8\u96c6]\u96fb\u78c1\u6c17\u5b66[\u7de8\u96c6]\u91cf\u5b50\u529b\u5b66[\u7de8\u96c6]\u30ea\u30c3\u30c4\u306e\u5909\u5206\u539f\u7406[\u7de8\u96c6]\u30ae\u30d6\u30ba\u306e\u5909\u5206\u539f\u7406[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u95a2\u9023\u8a18\u4e8b[\u7de8\u96c6] \u53e4\u5178\u529b\u5b66[\u7de8\u96c6] \u4f5c\u7528\u7a4d\u5206S","datePublished":"2022-03-28","dateModified":"2022-03-28","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/jp\/wiki11\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/jp\/wiki11\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/6\/64\/Question_book-4.svg\/50px-Question_book-4.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/6\/64\/Question_book-4.svg\/50px-Question_book-4.svg.png","height":"39","width":"50"},"url":"https:\/\/wiki.edu.vn\/jp\/wiki11\/archives\/188656","about":["Wiki"],"wordCount":22659,"articleBody":"\u3053\u306e\u8a18\u4e8b\u306f\u691c\u8a3c\u53ef\u80fd\u306a\u53c2\u8003\u6587\u732e\u3084\u51fa\u5178\u304c\u5168\u304f\u793a\u3055\u308c\u3066\u3044\u306a\u3044\u304b\u3001\u4e0d\u5341\u5206\u3067\u3059\u3002\u51fa\u5178\u3092\u8ffd\u52a0\u3057\u3066\u8a18\u4e8b\u306e\u4fe1\u983c\u6027\u5411\u4e0a\u306b\u3054\u5354\u529b\u304f\u3060\u3055\u3044\u3002\u51fa\u5178\u691c\u7d22?:\u00a0“\u5909\u5206\u539f\u7406”\u00a0\u2013\u00a0\u30cb\u30e5\u30fc\u30b9\u00a0\u00b7 \u66f8\u7c4d\u00a0\u00b7 \u30b9\u30ab\u30e9\u30fc\u00a0\u00b7 CiNii\u00a0\u00b7 J-STAGE\u00a0\u00b7 NDL\u00a0\u00b7 dlib.jp\u00a0\u00b7 \u30b8\u30e3\u30d1\u30f3\u30b5\u30fc\u30c1\u00a0\u00b7 TWL\uff082011\u5e747\u6708\uff09\u5909\u5206\u539f\u7406\uff08\u3078\u3093\u3076\u3093\u3052\u3093\u308a\u3001\u82f1\u8a9e: variational principle\uff09\u306f\u3001\u5909\u5206\u6cd5\u3092\u7528\u3044\u305f\u7269\u7406\u5b66\u306e\u539f\u7406\u3002\u7279\u306b\u3001  \u5909\u5206\u539f\u7406\u306f\u7a4d\u5206\u306e\u5f62\u3067\u6271\u3046\u306e\u3067\u3001\u5ea7\u6a19\u7cfb\u306e\u53d6\u308a\u65b9\u306b\u4f9d\u5b58\u3057\u306a\u3044\u3002\u5f93\u3063\u3066\u62e1\u5f35\u6027\u306b\u512a\u308c\u3001\u3044\u308d\u3044\u308d\u306a\u5206\u91ce\u306b\u5fdc\u7528\u3001\u5229\u7528\u3055\u308c\u308b\u3002Table of Contents\u53e4\u5178\u529b\u5b66[\u7de8\u96c6]\u96fb\u78c1\u6c17\u5b66[\u7de8\u96c6]\u91cf\u5b50\u529b\u5b66[\u7de8\u96c6]\u30ea\u30c3\u30c4\u306e\u5909\u5206\u539f\u7406[\u7de8\u96c6]\u30ae\u30d6\u30ba\u306e\u5909\u5206\u539f\u7406[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u95a2\u9023\u8a18\u4e8b[\u7de8\u96c6]\u53e4\u5178\u529b\u5b66[\u7de8\u96c6]\u4f5c\u7528\u7a4d\u5206S \u3092\u3001  S[q(t)]:=\u222bt1t2L(q(t),q\u02d9(t),t)dt,{displaystyle Sleft[q(t)right]:=int _{t_{1}}^{t_{2}}L(q(t),{dot {q}}(t),t)dt,}\u3068\u3059\u308b\u3002L \u306f\u30e9\u30b0\u30e9\u30f3\u30b8\u30a2\u30f3\u3001q(t) \u306f\u4e00\u822c\u5316\u5ea7\u6a19\u3001q\u02d9(t):=dq(t)\/dt{displaystyle {dot {q}}(t):=dq(t)\/dt} \u306f\u305d\u306e\u6642\u9593\u5fae\u5206\u3001\u3059\u306a\u308f\u3061\u4e00\u822c\u5316\u901f\u5ea6\u3067\u3042\u308b\u3002\u3053\u3053\u3067\u3001\u3042\u308b\u6642\u523bt1\u3001t2 \u306b\u304a\u3044\u3066\u3001q(t1)\u3001q(t2) \u306f\u56fa\u5b9a\u3055\u308c\u3066\u3044\u308b\u3068\u3059\u308b\u3002\u3053\u306e\u4f5c\u7528\u7a4d\u5206 S \u306b\u5bfe\u3059\u308b\u5909\u5206\u539f\u7406\u306f\u3001\u4f5c\u7528\u7a4d\u5206\u306b\u5bfe\u3059\u308b\u505c\u7559\u5024\u554f\u984c\u3092\u8003\u3048\u308b\u3053\u3068\u3067\u3042\u308a\u3001  \u03b4S[q(t)]=\u03b4\u222bt1t2L(q(t),q\u02d9(t),t)dt=0{displaystyle delta Sleft[q(t)right]=delta int _{t_{1}}^{t_{2}}L(q(t),{dot {q}}(t),t)dt=0}\u3068\u3044\u3046\u3053\u3068\u306b\u76f8\u5f53\u3059\u308b\u3002\u5909\u5206\u306f\u3001\u4e00\u822c\u5316\u5ea7\u6a19 q \u3092\u3001q(t)\u2192q(t)+\u03b4q(t),{displaystyle q(t)to q(t)+delta q(t),}\u3068\u6642\u523b t \u4e0a\u3067 \u03b4q \u3060\u3051\u5fae\u5c0f\u5909\u5316\u3055\u305b\u308b\u3053\u3068\u306b\u76f8\u5f53\u3059\u308b\u3002\u5909\u5206\u306b\u304a\u3051\u308b\u3053\u306e\u5fae\u5c0f\u5909\u5316\u306f\u4eee\u60f3\u7684\u306a\u5909\u4f4d\u3092\u4e0e\u3048\u308b\u3053\u3068\u3067\u3042\u308a\u3001\u3053\u308c\u306f\u6642\u9593 t \u306b\u5bfe\u3059\u308b\u5fae\u5c0f\u5909\u4f4d dq \u3068\u306f\u7570\u306a\u3063\u305f\u6982\u5ff5\u3067\u3042\u308b\u3002\u03b4q \u306f\u5143\u306e\u7d4c\u8def q(t) \u8fd1\u508d\u306e\u5225\u306e\uff08\u4eee\u60f3\u7684\u306a\uff09\u7d4c\u8def\u3068\u306e\u5dee\u3067\u3042\u308a\u3001\u4ed6\u65b9\u3001\u6642\u9593\u5909\u5316 dq \u306f\u7d4c\u8def q \u306b\u6cbf\u3063\u305f\u5909\u5316\u306e\u5927\u304d\u3055\u3092\u8868\u3059\u3002\u4e00\u822c\u5316\u5ea7\u6a19 q \u306e\u5fae\u5c0f\u5909\u5316 \u03b4q \u306b\u3064\u3044\u3066\u3001\u59cb\u70b9 t =t1 \u3068\u7d42\u70b9 t =t2 \u306b\u304a\u3044\u3066\u306f\u7d4c\u8def\u304c\u56fa\u5b9a\u3055\u308c\u3066\u3044\u308b\u306e\u3067\u3001\u03b4q(t1)=\u03b4q(t2)=0{displaystyle delta q(t_{1})=delta q(t_{2})=0}\u306f\u5e38\u306b\u6e80\u305f\u3055\u308c\u308b\u3002\u4e00\u822c\u5316\u5ea7\u6a19 q \u306e\u8868\u3059\u7d4c\u8def\u306e\u5909\u5316\u306b\u4f34\u3044\u3001\u4e00\u822c\u5316\u901f\u5ea6 q\u02d9{displaystyle {dot {q}}} \u3082\u5fae\u5c0f\u5909\u5316\u3059\u308b\u3002q\u02d9(t)\u2192q\u02d9(t)+\u03b4q\u02d9(t).{displaystyle {dot {q}}(t)to {dot {q}}(t)+delta {dot {q}}(t).}\u3053\u3053\u3067\u3001\u4e00\u822c\u5316\u901f\u5ea6\u306e\u5fae\u5c0f\u5909\u5316 \u03b4q\u02d9(t){displaystyle delta {dot {q}}(t)} \u306f\u3001\u3042\u308b\u6642\u523bt \u306b\u304a\u3051\u308b\u3001\u4e8c\u3064\u306e\u7d4c\u8def\u3067\u306e\u4e00\u822c\u5316\u901f\u5ea6\u306e\u5dee\u3092\u8868\u3059\u3002\u03b4q\u02d9(t)=ddt\u03b4q(t).{displaystyle delta {dot {q}}(t)={frac {d}{dt}}delta q(t).}\u4f5c\u7528\u7a4d\u5206\u306e\u5909\u5206\u3092\u8a08\u7b97\u3059\u308b\u3068\u3001\u03b4S[q(t)]=S[q+\u03b4q]\u2212S[q]=\u222bt1t2L(q(t)+\u03b4q(t),q\u02d9(t)+\u03b4q\u02d9(t),t)dt\u2212\u222bt1t2L(q(t),q\u02d9(t),t)dt=\u222bt1t2[L(q+\u03b4q,q\u02d9+\u03b4q\u02d9,t)\u2212{L(q,q\u02d9+\u03b4q\u02d9,t)\u2212L(q,q\u02d9+\u03b4q\u02d9,t)}\u2212L(q,q\u02d9,t)]dt,{displaystyle {begin{aligned}delta Sleft[q(t)right]&=Sleft[q+delta qright]-Sleft[qright]\\&=int _{t_{1}}^{t_{2}}L(q(t)+delta q(t),{dot {q}}(t)+delta {dot {q}}(t),t)dt-int _{t_{1}}^{t_{2}}L(q(t),{dot {q}}(t),t)dt\\&=int _{t_{1}}^{t_{2}}left[L(q+delta q,{dot {q}}+delta {dot {q}},t)-left{L(q,{dot {q}}+delta {dot {q}},t)-L(q,{dot {q}}+delta {dot {q}},t)right}-L(q,{dot {q}},t)right]dt,end{aligned}}}\u3068\u5909\u5f62\u3067\u304d\u308b\u3002\u3053\u3053\u3067 \u03b4q{displaystyle delta q} \u304a\u3088\u3073 \u03b4q\u02d9{displaystyle delta {dot {q}}} \u306f\u5145\u5206\u5c0f\u3055\u3044\u306e\u3067\u3001\u7a4d\u5206\u4e2d\u306e\u7b2c\u4e00\u9805\u3068\u7b2c\u4e8c\u9805\u3001\u7b2c\u4e09\u9805\u3068\u7b2c\u56db\u9805\u306e\u7d44\u306f\u305d\u308c\u305e\u308c\u504f\u5fae\u5206\u306e\u5f62\u306b\u66f8\u304d\u63db\u3048\u3089\u308c\u3001\u03b4S[q(t)]=\u222bt1t2[\u2202L\u2202q\u03b4q+\u2202L\u2202q\u02d9\u03b4q\u02d9]dt=\u222bt1t2[\u2202L\u2202q\u03b4q+ddt(\u2202L\u2202q\u02d9\u03b4q)\u2212ddt(\u2202L\u2202q\u02d9)\u03b4q]dt=\u2202L\u2202q\u02d9\u03b4q|t1t2+\u222bt1t2[\u2202L\u2202q\u2212ddt(\u2202L\u2202q\u02d9)]\u03b4qdt,{displaystyle {begin{aligned}delta Sleft[q(t)right]&=int _{t_{1}}^{t_{2}}left[{frac {partial L}{partial q}}delta q+{frac {partial L}{partial {dot {q}}}}delta {dot {q}}right]dt\\&=int _{t_{1}}^{t_{2}}left[{frac {partial L}{partial q}}delta q+{frac {d}{dt}}left({frac {partial L}{partial {dot {q}}}}delta qright)-{frac {d}{dt}}left({frac {partial L}{partial {dot {q}}}}right)delta qright]dt\\&=left.{frac {partial L}{partial {dot {q}}}}delta {q}right|_{t_{1}}^{t_{2}}+int _{t_{1}}^{t_{2}}left[{frac {partial L}{partial q}}-{frac {d}{dt}}left({frac {partial L}{partial {dot {q}}}}right)right]delta qdt,end{aligned}}}\u3068\u306a\u308b\u3002\u03b4q (t1) = \u03b4q (t2) = 0 \u304b\u3089\u7b2c\u4e00\u9805\u306f 0 \u3068\u306a\u308b\u3002q(t) \u306e\u4efb\u610f\u306e\u5fae\u5c0f\u5909\u5316 \u03b4q(t) \u306b\u5bfe\u3057\u3066\u3001\u4f5c\u7528\u7a4d\u5206\u306e\u5909\u5206\u304c\u30bc\u30ed \u03b4S = 0 \u3067\u3042\u308b\u6761\u4ef6\u3068\u3057\u3066\u3001\u2202L\u2202q(q(t),q\u02d9(t),t)\u2212ddt(\u2202L\u2202q\u02d9(q(t),q\u02d9(t),t))=0,{displaystyle {frac {partial L}{partial q}}(q(t),{dot {q}}(t),t)-{frac {d}{dt}}left({frac {partial L}{partial {dot {q}}}}(q(t),{dot {q}}(t),t)right)=0,}\u3092\u5f97\u308b\u3002\u3053\u308c\u306f\u30aa\u30a4\u30e9\u30fc\uff1d\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u65b9\u7a0b\u5f0f\u306b\u306a\u3063\u3066\u3044\u308b\u3002\u540c\u69d8\u306b\u3057\u3066\u5909\u5206\u539f\u7406\u3092\u3001\u5e7e\u4f55\u5149\u5b66\uff08\u5149\u7dda\u5149\u5b66\uff09\u306b\u304a\u3051\u308b\u5149\u306e\u53cd\u5c04\u3084\u5c48\u6298\u306e\u554f\u984c\u306b\u3064\u3044\u3066\u9069\u7528\u3059\u308c\u3070\u3001\u30d5\u30a7\u30eb\u30de\u30fc\u306e\u539f\u7406\u304c\u5f97\u3089\u308c\u308b\u3002\u30d5\u30a7\u30eb\u30de\u30fc\u306e\u539f\u7406\u306b\u304a\u3044\u3066\u3001\u4f5c\u7528\u7a4d\u5206\u306b\u5bfe\u5fdc\u3059\u308b\u3082\u306e\u306f\u7a7a\u9593\u306e 2 \u70b9\u9593\u3092\u7d50\u3076\u7d4c\u8def\u306e\u5149\u8def\u9577\u3067\u3042\u308a\u3001\u30e9\u30b0\u30e9\u30f3\u30b8\u30a2\u30f3\u306b\u5bfe\u5fdc\u3059\u308b\u3082\u306e\u306f\u5c48\u6298\u7387\u3068\u306a\u308b\u3002\u96fb\u78c1\u6c17\u5b66[\u7de8\u96c6]\u5fae\u5206\u5f62\u306e\u30ac\u30a6\u30b9\u306e\u6cd5\u5247\u3001\u2207\u22c5E(r)=\u03c1(r)\u03b50{displaystyle nabla cdot {boldsymbol {E}}({boldsymbol {r}})={rho ({boldsymbol {r}}) over varepsilon _{0}}}\u304a\u3088\u3073\u9759\u78c1\u5834\u306b\u304a\u3051\u308b\u30d5\u30a1\u30e9\u30c7\u30fc\u306e\u96fb\u78c1\u8a98\u5c0e\u306e\u6cd5\u5247\u3001\u2207\u00d7E(r)=0{displaystyle nabla times {boldsymbol {E}}({boldsymbol {r}})={boldsymbol {0}}}\u304c\u6210\u308a\u7acb\u3064\u9759\u96fb\u5834\u306b\u3064\u3044\u3066\u3001\u96fb\u5834 E(r){displaystyle {boldsymbol {E}}({boldsymbol {r}})} \u3092\u9759\u96fb\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb \u03d5(r){displaystyle phi ({boldsymbol {r}})} \u3067\u66f8\u304d\u76f4\u305b\u3070[\u6ce8 1]\u3001E(r)=\u2212\u2207\u03d5(r){displaystyle {boldsymbol {E}}({boldsymbol {r}})=-nabla phi ({boldsymbol {r}})}\u6b21\u306e\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f\u304c\u5f97\u3089\u308c\u308b\u30020=\u22072\u03d5(r)+\u03c1(r)\u03b50.{displaystyle 0=nabla ^{2}phi ({boldsymbol {r}})+{rho ({boldsymbol {r}}) over varepsilon _{0}}.}\u3053\u3053\u3067\u3001\u03c1(r){displaystyle rho ({boldsymbol {r}})} \u306f\u4f4d\u7f6e r{displaystyle {boldsymbol {r}}} \u306b\u304a\u3051\u308b\u96fb\u8377\u5bc6\u5ea6\u3001\u03b50{displaystyle varepsilon _{0}} \u306f\u56fd\u969b\u5358\u4f4d\u7cfb\u306b\u304a\u3051\u308b\u771f\u7a7a\u306e\u8a98\u96fb\u7387\u3001\u22072{displaystyle nabla ^{2}} \u306f\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u3092\u8868\u3059\u3002\u3053\u306e\u65b9\u7a0b\u5f0f\u306f\u3001\u6b21\u306e \u03d5(r){displaystyle phi ({boldsymbol {r}})} \u306e\u6c4e\u95a2\u6570 F[\u03d5(r)]{displaystyle F[phi ({boldsymbol {r}})]} \u306b\u3064\u3044\u3066\u5909\u5206\u539f\u7406\u3092\u7528\u3044\u308b\u3053\u3068\u3067\u3082\u5f97\u3089\u308c\u308b\u3002F[\u03d5(r)]=\u222bV{12|\u2207\u03d5(r)|2\u2212\u03c1(r)\u03b50\u03d5(r)}dV.{displaystyle F[phi ({boldsymbol {r}})]=int _{V}left{{1 over 2}left|nabla phi ({boldsymbol {r}})right|^{2}-{rho ({boldsymbol {r}}) over varepsilon _{0}}phi ({boldsymbol {r}})right}dV.}\u7a4d\u5206\u4e2d\u306e\u9805\u3092 \u03b50{displaystyle varepsilon _{0}} \u500d\u3057\u305f\u3001\u03b502|\u2207\u03d5(r)|2{displaystyle {varepsilon _{0} over 2}left|nabla phi ({boldsymbol {r}})right|^{2}} \u306f\u9759\u96fb\u5834\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u5bc6\u5ea6\u3067\u3042\u308a\u3001\u03c1(r)\u03d5(r){displaystyle rho ({boldsymbol {r}})phi ({boldsymbol {r}})} \u306f\u96fb\u8377\u5bc6\u5ea6\u306e\u4f4d\u7f6e\u30a8\u30cd\u30eb\u30ae\u30fc\u3067\u3042\u308b\u3002\u5883\u754c\u4e0a \u2202V{displaystyle partial V} \u3067 \u03b4\u03d5(r)=0{displaystyle delta phi ({boldsymbol {r}})=0} \u3068\u3057\u3066\u3001 \u6c4e\u95a2\u6570 F[\u03d5(r)]{displaystyle F[phi ({boldsymbol {r}})]} \u306e\u5909\u5206\u3092\u8003\u3048\u308b\u3068\u3001\u03b4F[\u03d5(r)]=\u222bV{12|\u2207(\u03d5(r)+\u03b4\u03d5(r))|2\u2212\u03c1(r)\u03b50(\u03d5(r)+\u03b4\u03d5(r))}dV\u2212\u222bV{12|\u2207\u03d5(r)|2\u2212\u03c1(r)\u03b50\u03d5(r)}dV=\u222bV{12(\u2207\u03d5(r)+\u2207\u03b4\u03d5(r))\u22c5(\u2207\u03d5(r)+\u2207\u03b4\u03d5(r))\u2212\u03c1(r)\u03b50\u03b4\u03d5(r)\u221212|\u2207\u03d5(r)|2}dV=\u222bV{12(|\u2207\u03d5(r)|2+2\u2207\u03b4\u03d5(r)\u22c5\u2207\u03d5(r)+|\u2207\u03b4\u03d5(r)|2)\u2212\u03c1(r)\u03b50\u03b4\u03d5(r)\u221212|\u2207\u03d5(r)|2}dV=\u222bV{\u2207\u03b4\u03d5(r)\u22c5\u2207\u03d5(r)\u2212\u03c1(r)\u03b50\u03b4\u03d5(r)}dV{displaystyle {begin{aligned}delta F[phi ({boldsymbol {r}})]&=int _{V}left{{1 over 2}left|nabla left(phi ({boldsymbol {r}})+delta phi ({boldsymbol {r}})right)right|^{2}-{rho ({boldsymbol {r}}) over varepsilon _{0}}left(phi ({boldsymbol {r}})+delta phi ({boldsymbol {r}})right)right}dV-int _{V}left{{1 over 2}left|nabla phi ({boldsymbol {r}})right|^{2}-{rho ({boldsymbol {r}}) over varepsilon _{0}}phi ({boldsymbol {r}})right}dV\\&=int _{V}left{{1 over 2}left(nabla phi ({boldsymbol {r}})+nabla delta phi ({boldsymbol {r}})right)cdot left(nabla phi ({boldsymbol {r}})+nabla delta phi ({boldsymbol {r}})right)-{rho ({boldsymbol {r}}) over varepsilon _{0}}delta phi ({boldsymbol {r}})-{1 over 2}left|nabla phi ({boldsymbol {r}})right|^{2}right}dV\\&=int _{V}left{{1 over 2}left(left|nabla phi ({boldsymbol {r}})right|^{2}+2nabla delta phi ({boldsymbol {r}})cdot nabla phi ({boldsymbol {r}})+left|nabla delta phi ({boldsymbol {r}})right|^{2}right)-{rho ({boldsymbol {r}}) over varepsilon _{0}}delta phi ({boldsymbol {r}})-{1 over 2}left|nabla phi ({boldsymbol {r}})right|^{2}right}dV\\&=int _{V}left{nabla delta phi ({boldsymbol {r}})cdot nabla phi ({boldsymbol {r}})-{rho ({boldsymbol {r}}) over varepsilon _{0}}delta phi ({boldsymbol {r}})right}dVend{aligned}}}\u3068\u5909\u5f62\u3067\u304d\u308b\u3002\u3053\u3053\u3067\u3001\u03b4\u03d5(r){displaystyle delta phi ({boldsymbol {r}})} \u306e\u4e8c\u6b21\u306e\u9805\u306f\u7121\u8996\u3057\u305f\u3002\u30ca\u30d6\u30e9\u306e\u7a4d\u306e\u898f\u5247\u3088\u308a\u3001\u6b21\u306e\u5f0f\u304c\u6210\u308a\u7acb\u3064\u304b\u3089\u3001\u2207\u03b4\u03d5\u22c5\u2207\u03d5=\u2207\u22c5(\u03b4\u03d5\u2207\u03d5)\u2212\u03b4\u03d5\u22072\u03d5{displaystyle nabla delta phi cdot nabla phi =nabla cdot (delta phi nabla phi )-delta phi nabla ^{2}phi }\u5909\u5206\u306f\u3001\u03b4F[\u03d5(r)]=\u222bV{\u2207\u22c5(\u03b4\u03d5(r)\u2207\u03d5(r))\u2212\u03b4\u03d5(r)\u22072\u03d5(r)\u2212\u03c1(r)\u03b50\u03b4\u03d5(r)}dV=\u222b\u2202V\u03b4\u03d5(r)\u2207\u03d5(r)\u22c5dS\u2212\u222bV{\u22072\u03d5(r)+\u03c1(r)\u03b50}\u03b4\u03d5(r)dV=\u2212\u222bV{\u22072\u03d5(r)+\u03c1(r)\u03b50}\u03b4\u03d5(r)dV{displaystyle {begin{aligned}delta F[phi ({boldsymbol {r}})]&=int _{V}left{nabla cdot left(delta phi ({boldsymbol {r}})nabla phi ({boldsymbol {r}})right)-delta phi ({boldsymbol {r}})nabla ^{2}phi ({boldsymbol {r}})-{rho ({boldsymbol {r}}) over varepsilon _{0}}delta phi ({boldsymbol {r}})right}dV\\&=int _{partial V}delta phi ({boldsymbol {r}})nabla phi ({boldsymbol {r}})cdot d{boldsymbol {S}}-int _{V}left{nabla ^{2}phi ({boldsymbol {r}})+{rho ({boldsymbol {r}}) over varepsilon _{0}}right}delta phi ({boldsymbol {r}})dV\\&=-int _{V}left{nabla ^{2}phi ({boldsymbol {r}})+{rho ({boldsymbol {r}}) over varepsilon _{0}}right}delta phi ({boldsymbol {r}})dVend{aligned}}}\u3068\u306a\u308b\u3002\u3053\u3053\u3067\u3001\u30ac\u30a6\u30b9\u306e\u767a\u6563\u5b9a\u7406\u304a\u3088\u3073\u5883\u754c\u4e0a \u2202V{displaystyle partial V} \u3067\u9759\u96fb\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306e\u5909\u5206 \u03b4\u03d5(r){displaystyle delta phi ({boldsymbol {r}})} \u304c\u30bc\u30ed\u3067\u3042\u308b\u3053\u3068\u3092\u4f7f\u3063\u305f\u3002\u3053\u306e\u3053\u3068\u304b\u3089\u3001\u6c4e\u95a2\u6570 F[\u03d5(r)]{displaystyle F[phi ({boldsymbol {r}})]} \u306e\u5909\u5206\u304c\u4efb\u610f\u306e \u03b4\u03d5(r){displaystyle delta phi ({boldsymbol {r}})} \u306b\u5bfe\u3057\u30bc\u30ed\u306b\u306a\u308b\u6761\u4ef6\u306f\u3001\u03b4F[\u03d5(r)]=\u2212\u222bV{\u22072\u03d5(r)+\u03c1(r)\u03b50}\u03b4\u03d5(r)dV=0{displaystyle delta F[phi ({boldsymbol {r}})]=-int _{V}left{nabla ^{2}phi ({boldsymbol {r}})+{rho ({boldsymbol {r}}) over varepsilon _{0}}right}delta phi ({boldsymbol {r}})dV=0}\u95a2\u6570 \u03d5(r){displaystyle phi ({boldsymbol {r}})} \u304c\u9818\u57df V{displaystyle V} \u4e0a\u3067\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f\u3001\u22072\u03d5(r)+\u03c1(r)\u03b50=0{displaystyle nabla ^{2}phi ({boldsymbol {r}})+{rho ({boldsymbol {r}}) over varepsilon _{0}}=0}\u3092\u6e80\u305f\u3059\u3053\u3068\u3067\u3042\u308b\u3053\u3068\u304c\u78ba\u8a8d\u3067\u304d\u308b\u3002\u91cf\u5b50\u529b\u5b66[\u7de8\u96c6]\u30ea\u30c3\u30c4\u306e\u5909\u5206\u539f\u7406[\u7de8\u96c6]\u3053\u3053\u3067\u306f\u30ea\u30c3\u30c4\u306e\u5909\u5206\u539f\u7406 (Ritz variational principle) \u306e\u5fdc\u7528\u3068\u3057\u3066\u3001\u5909\u5206\u539f\u7406\u3092\u7528\u3044\u305f\u57fa\u5e95\u72b6\u614b\u306e\u6ce2\u52d5\u95a2\u6570\u306e\u8fd1\u4f3c\u306b\u3064\u3044\u3066\u8ff0\u3079\u308b\u3002\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3 H^{displaystyle {hat {H}}} \u306e\u56fa\u6709\u72b6\u614b\u3067\u3001\u56fa\u6709\u5024\u304c\u6700\u5c0f\u306e\u3082\u306e\u3092\u57fa\u5e95\u72b6\u614b\u3068\u547c\u3076\u3002\u3059\u306a\u308f\u3061\u57fa\u5e95\u72b6\u614b\u306f\u4ee5\u4e0b\u306e\u56fa\u6709\u5024\u65b9\u7a0b\u5f0f\u3092\u6e80\u305f\u3059\u3002H^|\u03c80\u27e9=E0|\u03c80\u27e9.{displaystyle {hat {H}}|psi _{0}rangle =E_{0}|psi _{0}rangle .}\u3053\u3053\u3067 E0{displaystyle E_{0}} \u306f\u57fa\u5e95\u72b6\u614b\u306e\u56fa\u6709\u5024\u3067\u3042\u308a\u3001\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u306e\u56fa\u6709\u5024\u306f\u7cfb\u306e\u56fa\u6709\u72b6\u614b\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u8868\u3059\u3002\u3053\u306e\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u306b\u3064\u3044\u3066\u6b21\u306e\u3053\u3068\u304c\u8a00\u3048\u308b\u3002\u300c\u9069\u5f53\u306a\u5883\u754c\u6761\u4ef6\u3092\u6301\u3064\u4efb\u610f\u306e\u72b6\u614b |\u03a8\u27e9{displaystyle |Psi rangle } \u306b\u5bfe\u3059\u308b\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3 H^{displaystyle {hat {H}}} \u306e\u671f\u5f85\u5024 E{displaystyle E} \u306f\u3001\u57fa\u5e95\u72b6\u614b\u306e\u30a8\u30cd\u30eb\u30ae\u30fc E0{displaystyle E_{0}} \u3088\u308a\u3082\u5e38\u306b\u5927\u304d\u3044\u304b\u7b49\u3057\u3044\u3002E[\u03a8]=\u27e8\u03a8|H^|\u03a8\u27e9\u27e8\u03a8|\u03a8\u27e9\u2265E0.{displaystyle E[Psi ]={frac {leftlangle Psi right|{hat {H}}left|Psi rightrangle }{leftlangle Psi |Psi rightrangle }}geq E_{0}.}\u7b49\u53f7\u306f |\u03a8\u27e9{displaystyle |Psi rangle } \u304c\u57fa\u5e95\u72b6\u614b |\u03c80\u27e9{displaystyle |psi _{0}rangle } \u3067\u3042\u308b\u5834\u5408\u306b\u6210\u308a\u7acb\u3064\u300d\u3002\u3053\u306e\u3053\u3068\u306f\u3001\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3 H^{displaystyle {hat {H}}} \u306e\u30a8\u30eb\u30df\u30fc\u30c8\u6027\u3088\u308a\u3001\u4efb\u610f\u306e\u72b6\u614b\u304c\u30a8\u30cd\u30eb\u30ae\u30fc\u56fa\u6709\u72b6\u614b\u306e\u7dda\u5f62\u7d50\u5408\u3067\u8868\u305b\u308b\u3053\u3068\u304b\u3089\u793a\u3055\u308c\u308b\u3002\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u306e\u56fa\u6709\u72b6\u614b |\u03c8\u03bb\u27e9{displaystyle |psi _{lambda }rangle } \u306f\u4ee5\u4e0b\u306e\u56fa\u6709\u5024\u65b9\u7a0b\u5f0f\u3092\u6e80\u305f\u3059\u3002H^|\u03c8\u03bb\u27e9=E\u03bb|\u03c8\u03bb\u27e9.{displaystyle {hat {H}}|psi _{lambda }rangle =E_{lambda }|psi _{lambda }rangle .}\u30a8\u30cd\u30eb\u30ae\u30fc\u56fa\u6709\u72b6\u614b\u3092\u57fa\u5e95\u3068\u3057\u3066\u72b6\u614b |\u03a8\u27e9{displaystyle |Psi rangle } \u3092\u5c55\u958b\u3059\u308c\u3070\u3001\u9069\u5f53\u306a\u8907\u7d20\u6570\u4fc2\u6570\u3092\u7528\u3044\u3066\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b\u3002|\u03a8\u27e9=\u2211\u03bbc\u03bb|\u03c8\u03bb\u27e9.{displaystyle |Psi rangle =sum _{lambda }c_{lambda }|psi _{lambda }rangle .}\u3053\u306e\u3068\u304d\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u306e\u671f\u5f85\u5024\u306f\u3001E[\u03a8]=\u27e8\u03a8|H^|\u03a8\u27e9\u27e8\u03a8|\u03a8\u27e9=\u2211\u03bb\u2211\u03bb\u2032\u27e8\u03c8\u03bb|c\u03bb\u2217H^c\u03bb\u2032|\u03c8\u03bb\u2032\u27e9\u2211\u03bb\u2211\u03bb\u2032\u27e8\u03c8\u03bb|c\u03bb\u2217c\u03bb\u2032|\u03c8\u03bb\u2032\u27e9=\u2211\u03bbE\u03bb|c\u03bb|2\u27e8\u03c8\u03bb|\u03c8\u03bb\u27e9\u2211\u03bb|c\u03bb|2\u27e8\u03c8\u03bb|\u03c8\u03bb\u27e9.{displaystyle {begin{aligned}E[Psi ]&={frac {leftlangle Psi right|{hat {H}}left|Psi rightrangle }{leftlangle Psi |Psi rightrangle }}\\&={frac {sum _{lambda }sum _{lambda ‘}leftlangle psi _{lambda }right|c_{lambda }^{*}{hat {H}}c_{lambda ‘}left|psi _{lambda ‘}rightrangle }{sum _{lambda }sum _{lambda ‘}leftlangle psi _{lambda }right|c_{lambda }^{*}c_{lambda ‘}left|psi _{lambda ‘}rightrangle }}\\&={frac {sum _{lambda }E_{lambda }left|c_{lambda }right|^{2}langle psi _{lambda }|psi _{lambda }rangle }{sum _{lambda }left|c_{lambda }right|^{2}langle psi _{lambda }|psi _{lambda }rangle }}.end{aligned}}}\u3068\u306a\u308b\u3002\u3053\u3053\u3067\u56fa\u6709\u72b6\u614b\u306e\u76f4\u4ea4\u6027\u3092\u7528\u3044\u305f\u3002\u27e8\u03c8\u03bb|\u03c8\u03bb\u2032\u27e9=0,\u03bb\u2260\u03bb\u2032.{displaystyle langle psi _{lambda }|psi _{lambda ‘}rangle =0,quad lambda neq lambda ‘.}\u30a8\u30cd\u30eb\u30ae\u30fc\u56fa\u6709\u5024\u306b\u3064\u3044\u3066\u3001\u4e0d\u7b49\u5f0f E\u03bb\u2265E0{displaystyle E_{lambda }geq E_{0}} \u304c\u6210\u308a\u7acb\u3064\u306e\u3067\u3001\u5206\u5b50\u306e\u56fa\u6709\u5024\u3092\u3059\u3079\u3066\u57fa\u5e95\u72b6\u614b\u306e\u56fa\u6709\u5024\u306b\u7f6e\u304d\u63db\u3048\u308c\u3070\u3001E[\u03a8]=\u2211\u03bbE\u03bb|c\u03bb|2\u27e8\u03c8\u03bb|\u03c8\u03bb\u27e9\u2211\u03bb|c\u03bb|2\u27e8\u03c8\u03bb|\u03c8\u03bb\u27e9\u2265\u2211\u03bbE0|c\u03bb|2\u27e8\u03c8\u03bb|\u03c8\u03bb\u27e9\u2211\u03bb|c\u03bb|2\u27e8\u03c8\u03bb|\u03c8\u03bb\u27e9=E0.{displaystyle E[Psi ]={frac {sum _{lambda }E_{lambda }left|c_{lambda }right|^{2}langle psi _{lambda }|psi _{lambda }rangle }{sum _{lambda }left|c_{lambda }right|^{2}langle psi _{lambda }|psi _{lambda }rangle }}geq {frac {sum _{lambda }E_{0}left|c_{lambda }right|^{2}langle psi _{lambda }|psi _{lambda }rangle }{sum _{lambda }left|c_{lambda }right|^{2}langle psi _{lambda }|psi _{lambda }rangle }}=E_{0}.}\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u306e\u671f\u5f85\u5024\u3068\u57fa\u5e95\u72b6\u614b\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u306b\u95a2\u3059\u308b\u4e0d\u7b49\u5f0f\u304c\u5f97\u3089\u308c\u308b\u3002\u3053\u306e\u539f\u7406\u306b\u3088\u3063\u3066\u3001\u4efb\u610f\u306e\u72b6\u614b |\u03a8\u27e9{displaystyle |Psi rangle } \u306b\u5bfe\u3059\u308b\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u306e\u671f\u5f85\u5024 E[\u03a8]{displaystyle E[Psi ]} \u306e\u6700\u5c0f\u5024\u304c\u57fa\u5e95\u72b6\u614b\u306e\u30a8\u30cd\u30eb\u30ae\u30fc E0{displaystyle E_{0}} \u3067\u3042\u308b\u4e8b\u304c\u4fdd\u8a3c\u3055\u308c\u3001\u305d\u306e\u3068\u304d\u306e\u72b6\u614b |\u03a8\u27e9{displaystyle |Psi rangle } \u304c\u57fa\u5e95\u72b6\u614b |\u03c80\u27e9{displaystyle |psi _{0}rangle } \u3067\u3042\u308b\u3068\u8a00\u3048\u308b\u3002\u305d\u306e\u305f\u3081\u3001\u3082\u3057\u3082\u57fa\u5e95\u72b6\u614b\u3068\u305d\u306e\u3068\u304d\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u5024\u3092\u6c42\u3081\u305f\u3044\u306e\u3067\u3042\u308c\u3070\u3001\u5909\u5206\u6cd5\u306b\u3088\u3063\u3066 |\u03a8\u27e9{displaystyle |Psi rangle } \u306e\u6c4e\u95a2\u6570 E[\u03a8]{displaystyle E[Psi ]} \u306e\u505c\u7559\u5024\u3092\u6c42\u3081\u308c\u3070\u3088\u3044\u4e8b\u306b\u306a\u308b\u3002\u5909\u5206\u539f\u7406\u3092\u5229\u7528\u3057\u305f\u3053\u306e\u624b\u6cd5\u3092\u6307\u3057\u3066\u300c\u5909\u5206\u539f\u7406\u300d\u3068\u8a00\u308f\u308c\u308b\u4e8b\u3082\u591a\u3044\u3002E[\u03a8]{displaystyle E[Psi ]} \u306e\u505c\u7559\u5024\u554f\u984c\u306f\u6b21\u306e\u3088\u3046\u306a\u3082\u306e\u306b\u306a\u308b\u3002\u03b4E[\u03a8]=\u03b4(\u27e8\u03a8|H^|\u03a8\u27e9\u27e8\u03a8|\u03a8\u27e9)=0.{displaystyle delta E[Psi ]=delta left({frac {langle Psi |{hat {H}}|Psi rangle }{langle Psi |Psi rangle }}right)=0.}|\u03a8\u27e9{displaystyle |Psi rangle } \u3092\u9069\u5f53\u306a\u8a66\u884c\u95a2\u6570 {|\u03d5\u03bb\u27e9}{displaystyle left{|phi _{lambda }rangle right}} \u3067\u8868\u305b\u3070\u3001|\u03a8\u27e9=\u2211\u03bbc\u03bb|\u03d5\u03bb\u27e9{displaystyle |Psi rangle =sum _{lambda }c_{lambda }|phi _{lambda }rangle }E[\u03a8]{displaystyle E[Psi ]} \u306e\u5909\u5206\u306f\u3001\u30d1\u30e9\u30e1\u30fc\u30bf\u30fc {c\u03bb}{displaystyle left{c_{lambda }right}} \u306e\u5909\u5206\u3067\u8868\u3055\u308c\u308b\u3002\u03b4E[\u03a8]=\u03b4(\u27e8\u03a8|H^|\u03a8\u27e9\u27e8\u03a8|\u03a8\u27e9)=\u03b4(\u2211\u03bb\u2211\u03bb\u2032c\u03bb\u2217c\u03bb\u2032\u27e8\u03d5\u03bb|H^|\u03d5\u03bb\u2032\u27e9\u2211\u03bb\u2211\u03bb\u2032c\u03bb\u2217c\u03bb\u2032\u27e8\u03d5\u03bb|\u03d5\u03bb\u2032\u27e9).{displaystyle {begin{aligned}delta E[Psi ]&=delta left({frac {langle Psi |{hat {H}}|Psi rangle }{langle Psi |Psi rangle }}right)\\&=delta left({frac {sum _{lambda }sum _{lambda ‘}c_{lambda }^{*}c_{lambda ‘}langle phi _{lambda }|{hat {H}}|phi _{lambda ‘}rangle }{sum _{lambda }sum _{lambda ‘}c_{lambda }^{*}c_{lambda ‘}langle phi _{lambda }|phi _{lambda ‘}rangle }}right).end{aligned}}}\u3053\u3053\u3067\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u306e |\u03d5\u27e9{displaystyle |phi rangle } \u8868\u793a\u306b\u304a\u3051\u308b\u884c\u5217\u6210\u5206\u3092 H\u03bb,\u03bb\u2032:=\u27e8\u03d5\u03bb|H^|\u03d5\u03bb\u2032\u27e9{displaystyle H_{lambda ,lambda ‘}:=langle phi _{lambda }|{hat {H}}|phi _{lambda ‘}rangle } \u3001\u8a66\u884c\u95a2\u6570\u306e\u5185\u7a4d\u3092 \u03a6\u03bb,\u03bb\u2032:=\u27e8\u03d5\u03bb|\u03d5\u03bb\u2032\u27e9{displaystyle Phi _{lambda ,lambda ‘}:=langle phi _{lambda }|phi _{lambda ‘}rangle } \u3068\u305d\u308c\u305e\u308c\u8868\u3059\u3053\u3068\u306b\u3059\u308b\u3068\u3001\u6b21\u306e\u3088\u3046\u306b\u306a\u308b\u3002\u03b4E[\u03a8]=\u03b4(\u2211\u03bb\u2211\u03bb\u2032c\u03bb\u2217c\u03bb\u2032H\u03bb,\u03bb\u2032\u2211\u03bb\u2211\u03bb\u2032c\u03bb\u2217c\u03bb\u2032\u03a6\u03bb,\u03bb\u2032).{displaystyle {begin{aligned}delta E[Psi ]&=delta left({frac {sum _{lambda }sum _{lambda ‘}c_{lambda }^{*}c_{lambda ‘}H_{lambda ,lambda ‘}}{sum _{lambda }sum _{lambda ‘}c_{lambda }^{*}c_{lambda ‘}Phi _{lambda ,lambda ‘}}}right).end{aligned}}}\u3053\u306e\u5909\u5206\u304c\u4efb\u610f\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u30fc\u306e\u5909\u5206 {\u03b4c\u03bb\u2217}{displaystyle left{delta c_{lambda }^{*}right}} \u306b\u5bfe\u3057\u3066\u30bc\u30ed\u306b\u306a\u308b\u3053\u3068\u306f\u3001\u5404\u30d1\u30e9\u30e1\u30fc\u30bf\u30fc {c\u03bb\u2217}{displaystyle left{c_{lambda }^{*}right}} \u306e\u504f\u5fae\u5206\u304c\u30bc\u30ed\u306b\u306a\u308b\u3053\u3068\u3068\u540c\u3058\u306a\u306e\u3067\u3001\u2202E\u2202c\u03bb\u2217=\u2202\u2202c\u03bb\u2217(\u2211\u03bb\u2032\u2211\u03bb\u2033c\u03bb\u2032\u2217c\u03bb\u2033H\u03bb\u2032,\u03bb\u2033\u2211\u03bb\u2032\u2211\u03bb\u2033c\u03bb\u2032\u2217c\u03bb\u2033\u03a6\u03bb\u2032,\u03bb\u2033)=\u2211\u03bb\u2033c\u03bb\u2033H\u03bb,\u03bb\u2033\u2211\u03bb\u2032\u2211\u03bb\u2033c\u03bb\u2032\u2217c\u03bb\u2033\u03a6\u03bb\u2032,\u03bb\u2033\u2212\u2211\u03bb\u2032\u2211\u03bb\u2033c\u03bb\u2032\u2217c\u03bb\u2033H\u03bb\u2032,\u03bb\u2033\u2211\u03bb\u2032\u2211\u03bb\u2033c\u03bb\u2032\u2217c\u03bb\u2033\u03a6\u03bb\u2032,\u03bb\u2033\u2211\u03bb\u2033c\u03bb\u2033\u03a6\u03bb,\u03bb\u2033\u2211\u03bb\u2032\u2211\u03bb\u2033c\u03bb\u2032\u2217c\u03bb\u2033\u03a6\u03bb\u2032,\u03bb\u2033=\u2211\u03bb\u2032c\u03bb\u2032(H\u03bb,\u03bb\u2032\u2212E\u03a6\u03bb,\u03bb\u2032)\u2211\u03bb\u2032\u2211\u03bb\u2033c\u03bb\u2032\u2217c\u03bb\u2033\u03a6\u03bb\u2032,\u03bb\u2033=0.{displaystyle {begin{aligned}{frac {partial E}{partial c_{lambda }^{*}}}&={frac {partial }{partial c_{lambda }^{*}}}left({frac {sum _{lambda ‘}sum _{lambda ”}c_{lambda ‘}^{*}c_{lambda ”}H_{lambda ‘,lambda ”}}{sum _{lambda ‘}sum _{lambda ”}c_{lambda ‘}^{*}c_{lambda ”}Phi _{lambda ‘,lambda ”}}}right)\\&={frac {sum _{lambda ”}c_{lambda ”}H_{lambda ,lambda ”}}{sum _{lambda ‘}sum _{lambda ”}c_{lambda ‘}^{*}c_{lambda ”}Phi _{lambda ‘,lambda ”}}}-{frac {sum _{lambda ‘}sum _{lambda ”}c_{lambda ‘}^{*}c_{lambda ”}H_{lambda ‘,lambda ”}}{sum _{lambda ‘}sum _{lambda ”}c_{lambda ‘}^{*}c_{lambda ”}Phi _{lambda ‘,lambda ”}}}{frac {sum _{lambda ”}c_{lambda ”}Phi _{lambda ,lambda ”}}{sum _{lambda ‘}sum _{lambda ”}c_{lambda ‘}^{*}c_{lambda ”}Phi _{lambda ‘,lambda ”}}}\\&={frac {sum _{lambda ‘}c_{lambda ‘}left(H_{lambda ,lambda ‘}-EPhi _{lambda ,lambda ‘}right)}{sum _{lambda ‘}sum _{lambda ”}c_{lambda ‘}^{*}c_{lambda ”}Phi _{lambda ‘,lambda ”}}}=0.end{aligned}}}\u3088\u308a\u3001\u6b21\u306e\u5f0f\u3092\u5f97\u308b\u3002\u2211\u03bb\u2032(H\u03bb,\u03bb\u2032\u2212E\u03a6\u03bb,\u03bb\u2032)c\u03bb\u2032\u2217=0.{displaystyle sum _{lambda ‘}left(H_{lambda ,lambda ‘}-EPhi _{lambda ,lambda ‘}right)c_{lambda ‘}^{*}=0.}\u3053\u306e\u6589\u6b21\u65b9\u7a0b\u5f0f\u304c\u975e\u81ea\u660e\u306a\u89e3\u3092\u6301\u3064\u305f\u3081\u306b\u306f\u3001\u30d9\u30af\u30c8\u30eb c{displaystyle {boldsymbol {c}}} \u306b\u304b\u304b\u308b\u884c\u5217 H\u2212E\u03a6{displaystyle mathrm {H} -Emathrm {Phi } } \u306e\u30c7\u30a3\u30bf\u30fc\u30df\u30ca\u30f3\u30c8\u304c\u30bc\u30ed\u3067\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044[\u6ce8 2]\u3002det[H\u2212E\u03a6]=0.{displaystyle det left[mathrm {H} -Emathrm {Phi } right]=0.}\u30ae\u30d6\u30ba\u306e\u5909\u5206\u539f\u7406[\u7de8\u96c6]\u5e73\u8861\u72b6\u614b\u306b\u304a\u3044\u3066\u5bc6\u5ea6\u884c\u5217\u306b\u3064\u3044\u3066\u5909\u5206\u3092\u8003\u3048\u308b\u30ae\u30d6\u30ba\u306e\u5909\u5206\u539f\u7406\u304c\u3042\u308b\u3002^ \u96fb\u5834 E(r){displaystyle scriptstyle {boldsymbol {E}}({boldsymbol {r}})} \u304c\u9759\u96fb\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306e\u52fe\u914d \u2212\u2207\u03d5(r){displaystyle scriptstyle -nabla phi ({boldsymbol {r}})} \u3067\u66f8\u304d\u76f4\u305b\u308b\u3053\u3068\u306f\u3001\u52fe\u914d\u306e\u56de\u8ee2 \u2207\u00d7\u2207\u03d5(r){displaystyle scriptstyle nabla times nabla phi ({boldsymbol {r}})} \u304c\u6052\u7b49\u7684\u306b\u30bc\u30ed\u306b\u306a\u308b\u3053\u3068\u304b\u3089\u5206\u304b\u308b\u3002^ \u884c\u5217\u306e\u5404\u5217\u3092\u5217\u30d9\u30af\u30c8\u30eb\u3067\u8868\u3057\u305f\u3068\u304d\u3001\u305d\u308c\u3089\u306e\u5217\u30d9\u30af\u30c8\u30eb\u304c\u7dda\u5f62\u5f93\u5c5e\u3067\u3042\u308c\u3070\u3001\u3059\u306a\u308f\u3061\u3044\u305a\u308c\u304b\u306e\u30d9\u30af\u30c8\u30eb\u304c\u4ed6\u306e\u30d9\u30af\u30c8\u30eb\u306e\u5b9a\u6570\u500d\u306e\u548c\u3068\u3057\u3066\u8868\u3055\u308c\u308b\u306a\u3089\u3001\u975e\u81ea\u660e\u306a\u89e3\u304c\u5b58\u5728\u3059\u308b\u3002\u307e\u305f\u3001\u30d9\u30af\u30c8\u30eb\u306e\u7d44\u304c\u7dda\u5f62\u5f93\u5c5e\u3067\u3042\u308c\u3070\u30c7\u30a3\u30bf\u30fc\u30df\u30ca\u30f3\u30c8\u306f\u30bc\u30ed\u306b\u306a\u308b\u3002\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u95a2\u9023\u8a18\u4e8b[\u7de8\u96c6]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki11\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki11\/archives\/188656#breadcrumbitem","name":"\u5909\u5206\u539f\u7406 – Wikipedia"}}]}]