{"id":1989,"date":"2023-03-15T08:59:41","date_gmt":"2023-03-15T00:59:41","guid":{"rendered":"http:\/\/ustb.cc\/zh\/?p=1989"},"modified":"2023-03-15T22:06:26","modified_gmt":"2023-03-15T14:06:26","slug":"ultra-fine","status":"publish","type":"post","link":"http:\/\/ustb.cc\/zh\/ultra-fine\/","title":{"rendered":"\u5b9a\u6001\u5fae\u6270\u8bba"},"content":{"rendered":"\n

1. \u5b9a\u6001\u5fae\u6270\u8bba<\/h1>\n\n\n\n

1.1 \u7b80\u4ecb<\/h2>\n\n\n\n

\u91cf\u5b50\u529b\u5b66\u4e2d\u7814\u7a76\u4fdd\u5b88\u7269\u7406\u4f53\u7cfb(\u5373)\u54c8\u5bc6\u987f\u7b97\u7b26\u4e0d\u660e\u663e\u5730\u4f9d\u8d56\u4e8e\u65f6\u95f4\u7684\u4f53\u7cfb)\u662f\u4ee5\u54c8\u5bc6\u987f\u7b97\u7b26\u7684\u672c\u5f81\u503c\u65b9\u7a0b\u4e3a\u57fa\u7840\u7684\u3002<\/p>\n\n\n\n

\u4e24\u4e2a\u91cd\u8981\u4f8b\u5b50\uff1a<\/p>\n\n\n\n

\u7b80\u8c10\u632f\u5b50 \u7279\u70b9\uff1a\u221a \u54c8\u5bc6\u987f\u7b97\u7b26\u975e\u5e38\u7b80\u5355<\/p>\n\n\n\n

\u6c22\u539f\u5b50 \u7279\u70b9\uff1a\u221a \u672c\u5f81\u65b9\u7a0b\u53ef\u7cbe\u786e\u6c42\u89e3<\/p>\n\n\n\n

\u7136\u800c\uff0c\u4ec5\u6709\u5c11\u6570\u4f53\u7cfb\u53ef\u7cbe\u786e\u6c42\u89e3\u3002<\/p>\n\n\n\n

\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c\u65b9\u7a0b\u8fc7\u4e8e\u590d\u6742\uff0c\u65e0\u6cd5\u6c42\u5f97\u89e3\u6790\u89e3\u3002<\/p>\n\n\n\n

\u4f8b\u5982\uff1a\u65e0\u6cd5\u7cbe\u786e\u6c42\u89e3\u591a\u7535\u5b50\u539f\u5b50\u4f53\u7cfb\uff0c\u54ea\u6015\u662f\u4ec5\u591a\u4e86\u4e00\u4e2a\u7535\u5b50\u7684\u6c26\u539f\u5b50\u3002<\/p>\n\n\n\n

\u8fd1\u4f3c\u65b9\u6cd5\uff1a\u89e3\u6790\u5730\u6c42\u5f97\u57fa\u672c\u7684\u672c\u5f81\u503c\u65b9\u7a0b\u7684\u8fd1\u4f3c\u89e3<\/p>\n\n\n\n

\u5b9a\u6001\u5fae\u6270\u8bba<\/strong>\uff1a\u5e94\u7528\u6781\u5176\u5e7f\u6cdb\uff0c\u5f88\u7b26\u5408\u7269\u7406\u5b66\u5bb6\u7684\u53e3\u5473<\/p>\n\n\n\n

\u2460\u9996\u5148\u7a81\u51fa\u4e3b\u8981\u7684\u6548\u5e94\uff0c\u4e5f\u5c31\u662f\u51b3\u5b9a\u8be5\u73b0\u8c61\u6216\u8be5\u4f53\u7cfb\u5168\u8c8c\u7684\u90a3\u4e9b\u6548\u5e94<\/p>\n\n\n\n

\u2461\u7406\u89e3\u4e86\u8fd9\u4e9b\u95ee\u9898\u4e4b\u540e\uff0c\u518d\u8003\u8651\u5728\u4e00\u7ea7\u8fd1\u4f3c\u4e2d\u5ffd\u7565\u4e86\u7684\u6b21\u8981\u6548\u5e94\uff0c\u5e76\u5c1d\u8bd5\u7740\u53bb\u89e3\u91ca\u66f4\u201c\u7cbe\u7ec6\u201d\u7684\u7ec6\u8282<\/p>\n\n\n\n

\u2462\u6b63\u662f\u5728\u5904\u7406\u8fd9\u4e9b\u6b21\u8981\u6548\u5e94\u65f6\uff0c\u9700\u8981\u5e94\u7528\u5fae\u6270\u7406\u8bba\u3002<\/p>\n\n\n\n

1.2 Nondegenerate perturbation theory<\/h2>\n\n\n\n

\u975e\u7b80\u5e76\u5fae\u6270\uff1a<\/strong><\/p>\n\n\n\n

\u5fae\u6270\u8bba\u9002\u7528\u7684\u524d\u63d0\u6761\u4ef6<\/p>\n\n\n\n

\u2460\u4f53\u7cfb\u7684\u54c8\u5bc6\u987f\u91cfH<\/em> \u53ef\u5199\u6210\u5982\u4e0b\u5f62\u5f0f\uff1a
$$H = {H_0} + W$$<\/p>\n\n\n\n

\u5176\u4e2dH<\/em>0<\/sub> \u7684\u672c\u5f81\u6001\u548c\u672c\u5f81\u503c\u5df2\u77e5\uff0cW<\/em>\u8fdc\u5c0f\u4e8eH<\/em>0<\/sub>\u7b97\u7b26\u600e\u4e48\u6bd4\u5927\u5c0f\uff1f<\/p>\n\n\n\n

\u2461\u5b9e\u9645\u4e0a\u6307\u7684\u662fW<\/em>\u7684\u77e9\u9635\u5143\u8981\u8fdc\u5c0f\u4e8eH<\/em>0<\/sub>\uff1b<\/p>\n\n\n\n

\u2462H<\/em>0<\/sub>\u4e0e\u65f6\u95f4\u65e0\u5173\uff0c\u79f0\u4e3a\u975e\u5fae\u6270\u54c8\u5bc6\u987f\u91cf(unperturbed Hamiltonian)\uff0c\u800cW<\/em>\u79f0\u4e3a\u5fae\u6270\uff1b<\/p>\n\n\n\n

\u2463\u5982\u679cW<\/em>\u4e5f\u4e0e\u65f6\u95f4\u65e0\u5173\uff0c\u5219\u79f0\u4e3a\u5b9a\u6001\u5fae\u6270\uff1b<\/p>\n\n\n\n

\u2464\u6c42\u89e3\u5fae\u6270\u9020\u6210\u7684\u80fd\u7ea7\u548c\u6ce2\u51fd\u6570\u53d8\u5316\u3002<\/p>\n\n\n\n

\u57fa\u672c\u5047\u8bbe\u4e0e\u65b9\u6cd5<\/h3>\n\n\n\n

\u5047\u8bbeW<\/em>\u6b63\u6bd4\u4e8e\u4e00\u4e2a\u5b9e\u6570\u03bb<\/em>\uff0c\u5176\u65e0\u91cf\u7eb2\u4e14\u8fdc\u5c0f\u4e8e1\u3002
$$W = \\lambda \\hat W$$
$$\\lambda \\ 1$$<\/p>\n\n\n\n

\u7b97\u7b26 W \u0302 \u7684\u77e9\u9635\u5143\u4e0eH<\/em>0<\/sub>\u53ef\u6bd4(comparable)<\/p>\n\n\n\n

\u4e00\u53e5\u8bdd\u5f04\u61c2\u5fae\u6270\u7406\u8bba\uff1a<\/p>\n\n\n\n

\u5c06H<\/em>\u7684\u672c\u5f81\u503c\u548c\u672c\u5f81\u6001\u6309\u03bb<\/em>\u5e42\u6b21\u5c55\u5f00\uff0c\u4fdd\u7559\u6709\u9650\u9879\u3002<\/p>\n\n\n\n

\u5177\u4f53\u6b65\u9aa4<\/h3>\n\n\n\n

\u2460\u5047\u8bbe\u672a\u5fae\u6270\u54c8\u5bc6\u987f\u91cfH<\/em>0<\/sub> \u7684\u672c\u5f81\u6001\u4e0e\u672c\u5f81\u503c\u5df2\u77e5\uff1b<\/p>\n\n\n\n

\u2461\u5047\u8bbe\u4e0a\u8ff0\u672c\u5f81\u503c\u6784\u6210\u5206\u7acb\u8c31(discrete spectrum<\/strong>)\uff0c\u5e76\u53ef\u4ee5\u7528\u4e00\u4e2a\u6574\u6570\u5e8f\u53f7p<\/em>\u4ee3\u8868\uff1aEp<\/sub><\/em>0<\/sup>\uff1b<\/p>\n\n\n\n

\u2462\u5bf9\u5e94\u7684\u672c\u5f81\u6001\u53ef\u8868\u793a\u4e3a |$\\varphi _p^i$\u3009\uff0c\u591a\u4f59\u7684\u5e8f\u53f7i<\/em>\u7528\u4e8e\u7b80\u5e76\u6001Ep<\/sub><\/em>0<\/sup>\uff0c\u533a\u5206\u672c\u5f81\u5b50\u7a7a\u95f4\u4e2d\u7684\u6b63\u4ea4\u5f52\u4e00\u57fa\u77e2\u3002
$${H_0}|\\varphi _p^i\\rangle = E_p^0|\\varphi _p^i\\rangle $$
$$\\langle \\varphi p^i|\\varphi {p’}^{i’}\\rangle = {\\delta {pp’}}{\\delta {ii’}},\\sum\\limits_p {\\sum\\limits_i {|\\varphi _p^{\\mathop i\\limits^. }\\rangle \\langle \\varphi _p^{\\mathop i\\limits^. }| = I} } $$<\/p>\n\n\n\n

\u5fae\u6270\u5f3a\u5ea6\u03bb<\/em><\/h3>\n\n\n\n

$$H(\\lambda ) = {H_0} + \\lambda \\hat W$$<\/p>\n\n\n\n

\u2460 \u4f53\u7cfb\u7684\u54c8\u5bc6\u987f\u91cf\u53ef\u770b\u4f5c\u03bb<\/em> \u7684\u51fd\u6570\uff0c\u800c\u03bb<\/em> \u5219\u8868\u5f81\u5fae\u6270\u7684\u5f3a\u5ea6\u5927\u5c0f\uff1b<\/p>\n\n\n\n

\u2461 \u03bb<\/em> \u7b49\u4e8e0\uff0cH<\/em>(\u03bb<\/em>) \u7b49\u4e8e\u672a\u5fae\u6270\u7684\u54c8\u5bc6\u987f\u91cfH<\/em>0<\/sub>\uff1b<\/p>\n\n\n\n

\u2462 H<\/em>(\u03bb<\/em>) \u7684\u672c\u5f81\u503cE<\/em>(\u03bb<\/em>) \u4e00\u822c\u60c5\u51b5\u4e0b\u4e0e\u03bb\u6709\u5173\uff1b<\/p>\n\n\n\n

\u2463 \u6bcf\u6761\u66f2\u7ebf\u90fd\u5bf9\u5e94\u7740H<\/em>(\u03bb<\/em>)\u7684\u4e00\u4e2a\u672c\u5f81\u77e2\uff1b<\/p>\n\n\n\n

\u2464 \u6bcf\u4e00\u4e2a\u03bb<\/em>\u5bf9\u5e94\u7684\u82e5\u5e72\u672c\u5f81\u77e2\u6784\u6210\u77e2\u91cf\u7a7a\u95f4\u7684\u4e00\u7ec4\u57fa\u77e2\uff1b<\/p>\n\n\n\n

\u2465 H<\/em>(\u03bb<\/em>) \u53ef\u4ee5\u5b58\u5728\u7b80\u5e76\uff1b<\/p>\n\n\n\n

\u2466 \u5f53\u03bb<\/em>\u21920\u65f6\uff0c\u4e0d\u540cE<\/em>(\u03bb<\/em>)\u53ef\u4ee5\u8d8b\u8fd1\u540c\u4e00\u4e2a\u672a\u5fae\u6270\u672c\u5f81\u503c Ep<\/sub><\/em>0<\/sup>\u3002<\/p>\n\n\n

\n
\"\"<\/figure>\n<\/div>\n\n\n

1.3 H<\/em>(\u03bb<\/em>) \u672c\u5f81\u65b9\u7a0b\u7684\u8fd1\u4f3c\u89e3<\/h2>\n\n\n\n

\u95ee\u9898\u63cf\u8ff0\uff1a<\/p>\n\n\n\n

\u2460 \u5bfb\u627eH<\/em>(\u03bb<\/em>)\u7684\u672c\u5f81\u6001 |\u03c8<\/em>(\u03bb<\/em>)\u3009\u548c\u672c\u5f81\u503cE<\/em>(\u03bb<\/em>)<\/p>\n\n\n\n

$$H(\\lambda )|\\psi (\\lambda )\\rangle = E(\\lambda )|\\psi (\\lambda )\\rangle $$<\/p>\n\n\n\n

\u2461\u5047\u8bbeE<\/em>(\u03bb<\/em>)\u548c|\u03c8<\/em>(\u03bb<\/em>)>\u53ef\u6309\u03bb<\/em>\u5e42\u6b21\u5c55\u5f00:<\/p>\n\n\n\n

$$E(\\lambda ) = {\\varepsilon _0} + \\lambda {\\varepsilon _1} + \u2026 + {\\lambda ^q}{\\varepsilon _q} + \u2026$$<\/p>\n\n\n\n

$$|\\psi (\\lambda )\\rangle = |0\\rangle + \\lambda |1\\rangle + \u2026 + {\\lambda ^q}|q\\rangle + \u2026$$<\/p>\n\n\n\n

\u6ce8\u610f\u8fd9\u4e2a\u5199\u6cd5\uff0c\u4e0e\u524d\u9762\u53f3\u77e2\u7684\u6807\u8bb0\u4e0d\u540c\u3002<\/p>\n\n\n\n

\u2462 \u66ff\u6362 $({H_0} + \\lambda \\hat W)\\left[ {\\sum\\limits_{q = 0}^\\infty {{\\lambda ^q}} |q\\rangle } \\right] = [\\sum\\limits_{q’ = 0}^\\infty {{\\lambda ^{q’}}} {\\varepsilon {q’}}]\\left[ {\\sum\\limits{q = 0}^\\infty {{\\lambda ^q}} |q\\rangle } \\right]$<\/p>\n\n\n\n

\u2463 \u03bb<\/em>\u53d6\u4efb\u610f\u503c(\u5c0f)\uff0c\u7b49\u5f0f\u4e24\u4fa7\u90fd\u5e94\u6210\u7acb\uff1b<\/p>\n\n\n\n

\u2464 \u4e3a\u6b64\u8981\u6c42\u4e24\u4fa7\u76f8\u540c\u03bb<\/em>\u5e42\u6b21\u7684\u7cfb\u6570\u76f8\u7b49<\/p>\n\n\n\n

$$({H_0} + \\lambda \\hat W)\\left[ {\\sum\\limits_{q = 0}^\\infty {{\\lambda ^q}} |q\\rangle } \\right] = [\\sum\\limits_{q’ = 0}^\\infty {{\\lambda ^{q’}}} {\\varepsilon {q’}}]\\left[ {\\sum\\limits{q = 0}^\\infty {{\\lambda ^q}} |q\\rangle } \\right]$$<\/p>\n\n\n\n

\u00b7 For 0th-order terms in \u03bb<\/em><\/p>\n\n\n\n

$${H_0}|0\\rangle = {\\varepsilon _0}|0\\rangle $$<\/p>\n\n\n\n

\u00b7 For 1st-order terms in \u03bb<\/em><\/p>\n\n\n\n

$$({H_0} – {\\varepsilon _0})|1\\rangle + (\\hat W – {\\varepsilon _1})|0\\rangle = 0$$<\/p>\n\n\n\n

\u00b7 For 2nd-order terms in \u03bb<\/em><\/p>\n\n\n\n

$$({H_0} – {\\varepsilon _0})|2\\rangle + (\\hat W – {\\varepsilon _1})|1\\rangle – {\\varepsilon _2}|0\\rangle = 0$$<\/p>\n\n\n\n

\u00b7 For q<\/em>th-order terms in \u03bb<\/em><\/p>\n\n\n\n

$$({H_0} – {\\varepsilon _0})|q\\rangle + (\\hat W – {\\varepsilon _1})|q – 1\\rangle – {\\varepsilon _2}|q – 2\\rangle \\; – {\\varepsilon _q}|0\\rangle = 0$$<\/p>\n\n\n\n

\u53ea\u5173\u6ce8\u524d\u4e09\u4e2a\u7b49\u5f0f\uff0c\u5373\u5ffd\u7565\u03bb<\/em>2<\/sup>\u4ee5\u4e0a\u7684\u9879<\/p>\n\n\n\n

\u524d\u9762\u5b66\u8fc7\uff0c\u672c\u5f81\u65b9\u7a0b<\/p>\n\n\n\n

$$H(\\lambda )|\\psi (\\lambda )\\rangle = E(\\lambda )|\\psi (\\lambda )\\rangle $$<\/p>\n\n\n\n

\u6240\u786e\u5b9a\u7684 |\u03c8<\/em>(\u03bb<\/em>)> \u53ef\u4e0e\u4efb\u610f\u7cfb\u6570\u76f8\u4e58\u3002<\/p>\n\n\n\n

\u56e0\u6b64\uff0c\u53ef\u6309\u4ee5\u4e0b\u89c4\u5219\u9009\u53d6|\u03c8(\u03bb<\/em>)> \u7684\u6a21\u548c\u76f8\u89d2(\u76f8\u4f4d\u56e0\u5b50)<\/p>\n\n\n\n

\u00b7 |\u03c8(\u03bb<\/em>)>\u5f52\u4e00\u5316\uff1b<\/p>\n\n\n\n

\u00b7 \u8c03\u6574\u76f8\u89d2\uff0c\u4f7f\u5f97\u5185\u79ef<0|\u03c8<\/em>(\u03bb<\/em>)> \u4e3a\u5b9e\u6570\u3002<\/p>\n\n\n\n

\u5bf9\u4e8e0\u7ea7\u5c0f\uff0c\u8fd9\u610f\u5473\u7740\u7528 |0> \u8868\u793a\u7684\u77e2\u91cf\u662f\u5f52\u4e00\u5316\u7684<\/p>\n\n\n\n

$$\\langle 0|0\\rangle = 1$$<\/p>\n\n\n\n

\u6b64\u65f6\u76f8\u89d2\u4efb\u610f\u9009\u62e9<\/p>\n\n\n\n

\u5bf9\u4e8e1\u7ea7\u5c0f\uff0c\u77e2\u91cf|\u03c8<\/em>(\u03bb<\/em>)>\u6a21\u7684\u5e73\u65b9\u53ef\u5199\u4e3a:<\/p>\n\n\n\n

$$\\langle \\psi (\\lambda )|\\psi (\\lambda )\\rangle = [\\langle 0| + \\lambda \\;\\langle \\;1|][|0\\rangle + \\lambda |1\\rangle ] + O({\\lambda ^2})$$<\/p>\n\n\n\n

$$ = \\;\\langle \\;0|0\\rangle + \\lambda [\\langle 1|0\\rangle + \\langle 0|1\\rangle ] + O({\\lambda ^2})$$<\/p>\n\n\n\n

\u8981\u4ee4\u4e0a\u5f0f\u7b49\u4e8e1\uff0c\u5219 \u03bb<\/em>\u7684\u4e00\u6b21\u9879\u5e94\u7b49\u4e8e0<\/p>\n\n\n\n

\u9009\u62e9\u76f8\u89d2\uff0c\u4f7f<0|1>\u4e3a\u5b9e\u6570(\u03bb<\/em>\u4e3a\u5b9e\u6570)\uff0c\u5219\u53ef\u4ee5\u5f97\u5230<\/p>\n\n\n\n

$$\\langle 0|1\\rangle = \\langle 1|0\\rangle = 0$$<\/p>\n\n\n\n

\u6ce8\u610f\u8fd9\u662f\u4e00\u4e2a\u4eba\u4e3a\u9009\u62e9\u7684\u7ed3\u679c<\/p>\n\n\n\n

\u7c7b\u4f3c\u5730\uff0c\u8ba8\u8bba\u03bb<\/em>\u7684\u4e8c\u7ea7\u5c0f\uff0c\u53ef\u4ee5\u5f97\u5230<\/p>\n\n\n\n

$$\\langle 0|2\\rangle = \\langle 2|0\\rangle = – \\frac{1}{2}\\langle 1|1\\rangle $$<\/p>\n\n\n\n

\u63a8\u5e7f\u5230q<\/em>\u7ea7\u5c0f\uff0c\u5219\u6709<\/p>\n\n\n\n

$$\\eqalign{
& \\langle 0|q\\rangle = \\langle q|0\\rangle = – \\frac{1}{2}[\\langle q – {\\text{1}}|1\\rangle + \\langle q – 2|2\\rangle + \u2026 \\cr
& {\\text{ }} + \\langle 2|q – 2\\rangle + \\langle 1|q – 1\\rangle ] \\cr} $$<\/p>\n\n\n\n

1.4 The perturbation subspace<\/h2>\n\n\n\n

\u5fae\u6270\u5b50\u7a7a\u95f4<\/strong><\/p>\n\n\n\n

$${H_0}|0\\rangle = {\\varepsilon _0}|0\\rangle $$<\/p>\n\n\n\n

\u4e0a\u5f0f\u8bf4\u660e\uff0c|0>\u662f H<\/em>0<\/sub> \u5177\u6709\u672c\u5f81\u503c \u03b5<\/em>0<\/sub>\u7684\u672c\u5f81\u77e2;<\/p>\n\n\n\n

\u03b5<\/em>0<\/sub>\u5c5e\u4e8eH<\/em>0<\/sub> \u7684\u80fd\u91cf\u8c31\uff0c\u56e0\u4e3aH<\/em>(\u03bb<\/em>)\u7684\u6bcf\u4e00\u4e2a\u672c\u5f81\u503c\uff0c\u5f53\u03bb<\/em>\u21920\u65f6\u90fd\u4f1a\u8d8b\u8fd1\u4e8e\u5176\u4e2d\u4e00\u4e2a\u672a\u5fae\u6270\u7684\u80fd\u7ea7;<\/p>\n\n\n\n

\u4e0d\u59a8\u4ee4\u03b5<\/em>0 <\/sub>= E<\/em>0<\/sub>\uff0c\u5219\u5f53\u03bb<\/em>\u21920\u65f6\uff0c\u4f1a\u6709\u4e00\u4e2a\u6216\u591a\u4e2a\u4e0d\u540c\u7684E<\/em>(\u03bb<\/em>) \u8d8b\u8fd1\u4e8eEn<\/sub><\/em>0<\/sup>;<\/p>\n\n\n\n

\u8003\u8651\u8fd9\u4e9b\u5c5e\u4e8e\u4e0a\u8ff0E<\/em>(\u03bb<\/em>)\u7684\u672c\u5f81\u6001;<\/p>\n\n\n\n

\u8fd9\u4e9b\u672c\u5f81\u77e2\u5f20\u5f00\u4e00\u4e2a\u5b50\u7a7a\u95f4\uff0c\u5f53\u03bb<\/em>\u57280\u9644\u8fd1\u53d8\u5316\u65f6\uff0c\u5176\u7ef4\u5ea6\u4e0d\u80fd\u7a81\u53d8<\/p>\n\n\n\n

\u6700\u7ec8\uff0c\u5b50\u7a7a\u95f4\u7684\u7ef4\u5ea6\u7b49\u4e8eE<\/em>n<\/sub><\/em>0<\/sup>\u7684\u7b80\u5e76\u5ea6 <\/em>g<\/em>n<\/sub><\/em><\/p>\n\n\n\n

\u7279\u522b\u5730\uff0c\u5982\u679cEn<\/sub><\/em>0<\/sup>\u662f\u975e\u7b80\u5e76\uff0c\u5219\u4ec5\u80fd\u5bf9\u5e94\u4e00\u4e2a\u672c\u5f81\u503cE<\/em>(\u03bb<\/em>), \u4e14\u8be5\u80fd\u7ea7\u662f\u975e\u7b80\u5e76\u7684<\/p>\n\n\n\n

\u4e00\u76f4\u5230\u8fd9\u91cc\uff0c\u5e76\u672a\u533a\u5206\u80fd\u7ea7\u7b80\u5e76\u4e0e\u5426<\/p>\n\n\n\n

1.5 Perturbation of a non-degenerate level<\/h2>\n\n\n\n

\u975e\u7b80\u5e76\u80fd\u7ea7\u7684\u5fae\u6270<\/strong>: \u6ce8\u610f\uff1a\u5fae\u6270\u7684\u5bf9\u8c61\u662f\u80fd\u7ea7\uff0c\u800c\u975e\u4f53\u7cfb<\/strong><\/p>\n\n\n\n

\u00b7 \u8003\u8651\u672a\u5fae\u6270\u54c8\u5bc6\u987f\u91cfH<\/em>0<\/sub>\u7684\u975e\u7b80\u5e76\u80fd\u7ea7En<\/sub><\/em>0<\/sup>\uff0c\u5176\u672c\u5f81\u6001\u4e3a|\u03c6n<\/sub><\/em>> \u6ce8\u610f\u8fd9\u4e2a\u7b26\u53f7\uff0c\u5c0f\u5199\u7684\u3002<\/p>\n\n\n\n

\u00b7 \u76ee\u6807\uff1a\u5fae\u6270W<\/em>\u4f5c\u7528\u540e\u80fd\u7ea7\u548c\u5b9a\u6001\u6ce2\u51fd\u6570\u7684\u53d8\u5316\uff1b<\/p>\n\n\n\n

\u00b7 \u5bf9\u4e8eH<\/em>(\u03bb<\/em>)\u5f53\u03bb<\/em>\u21920\u65f6\u8d8b\u8fd1\u4e8eEn<\/sub><\/em>0<\/sup> \u7684\u672c\u5f81\u503c\uff0c\u5b58\u5728\u5173\u7cfb\uff1a<\/p>\n\n\n\n

$${\\varepsilon _0} = E_n^0$$<\/p>\n\n\n\n

\u00b7 \u8fd9\u8868\u660e|0> \u6b63\u6bd4\u4e8e|\u03c6n<\/sub><\/em>>\u3002<\/p>\n\n\n\n

\u00b7 p|0>\u548c|\u03c6n<\/sub><\/em>>\u90fd\u662f\u5f52\u4e00\u5316\u7684\uff0c\u56e0\u6b64\u9009\u62e9<\/p>\n\n\n\n

$$|0\\rangle = |{\\varphi _n}\\rangle $$<\/p>\n\n\n\n

\u8fd9\u6837\uff0c\u5f53 \u03bb<\/em>\u21920\u65f6\uff0c\u5c31\u53c8\u5f97\u5230\u4e86\u5177\u6709\u76f8\u540c\u76f8\u89d2\u7684\u672a\u5fae\u6270\u6001 |\u03c6n<\/sub><\/em>><\/p>\n\n\n\n

H<\/em>(\u03bb<\/em>)\u7684\u672c\u5f81\u503cEn<\/sub><\/em>(\u03bb<\/em>) \uff0c\u5f53 \u03bb<\/em>\u21920\u65f6\u8d8b\u8fd1\u4e8eH<\/em>0<\/sub>\u7684\u672c\u5f81\u503cEn<\/sub><\/em>0<\/sup><\/p>\n\n\n\n

\u5047\u8bbe\u03bb<\/em> \u8db3\u591f\u5c0f\uff0c\u8fd9\u6837\u672c\u5f81\u503c\u4ecd\u7136\u662f\u975e\u7b80\u5e76\u7684\uff0c\u5373\u53ea\u5bf9\u5e94\u4e8e\u4e00\u4e2a\u7279\u5b9a\u7684 |\u03c8n<\/sub><\/em>(\u03bb<\/em>)>\u3002<\/p>\n\n\n\n

1.6 First-order corrections\u4e00\u7ea7\u4fee\u6b63<\/h2>\n\n\n\n

p\u5148\u786e\u5b9a\u03b5<\/em>1<\/sub> \u548c|1><\/p>\n\n\n\n

Energy correction<\/em>\u80fd\u91cf\u4fee\u6b63<\/strong><\/p>\n\n\n\n

$$({H_0} – {\\varepsilon _0})|1\\rangle + (\\hat W – {\\varepsilon _1})|0\\rangle = 0$$<\/p>\n\n\n\n

\u5c06\u4e0a\u5f0f\u6295\u5f71\u81f3|\u03c6n<\/sub><\/em>>, \u53ef\u5f97<\/p>\n\n\n\n

$$\\langle {\\varphi _n}|({H_0} – {\\varepsilon _0})|1\\rangle + \\langle {\\varphi _n}|(\\hat W – {\\varepsilon _1})|0\\rangle = 0$$<\/p>\n\n\n\n

\u7b2c\u4e00\u9879\u7b49\u4e8e0\uff0c\u56e0\u4e3a|\u03c6n<\/sub><\/em>>=|0> \u662fH<\/em>0<\/sub>\u7684\u672c\u5f81\u6001\uff0c\u5176\u672c\u5f81\u503c\u4e3a En<\/sub><\/em>0<\/sup>=\u03b5<\/em>0<\/sub><\/p>\n\n\n\n

$${\\varepsilon _1} = \\langle {\\varphi _n}|\\hat W|0\\rangle = \\langle {\\varphi _n}|\\hat W|{\\varphi _n}\\rangle $$<\/p>\n\n\n\n

$${\\varepsilon _1} = \\langle {\\varphi _n}|\\hat W|0\\rangle = \\langle {\\varphi _n}|\\hat W|{\\varphi _n}\\rangle $$<\/p>\n\n\n\n

\u5728\u975e\u7b80\u5e76\u6001\u7684\u60c5\u51b5\u4e0b\uff0c\u5bf9\u5e94\u4e8eEn<\/sub><\/em>0<\/sup> \u7684 H<\/em> \u7684\u672c\u5f81\u503cEn<\/sub><\/em>(\u03bb<\/em>) \u5728\u5fae\u6270\u7684\u4e00\u7ea7\u4fee\u6b63\u4e0b\u53ef\u5199\u4e3a<\/p>\n\n\n\n

$${E_n}(\\lambda ) = E_n^0 + \\langle {\\varphi _n}|W|{\\varphi _n}\\rangle + O({\\lambda ^2})$$<\/p>\n\n\n\n

$W = \\lambda \\hat W$ \u6ce8\u610f\u4e24\u4e2aW\u7684\u533a\u522b<\/p>\n\n\n\n

\u56e0\u6b64\uff0c\u80fd\u91cf\u7684\u4e00\u7ea7\u4fee\u6b63\uff0c\u7b49\u4e8e\u5fae\u6270\u9879\u5728\u672a\u5fae\u6270\u6001|\u03c6n<\/sub><\/em>>\u4e2d\u7684\u5e73\u5747\u503c<\/p>\n","protected":false},"excerpt":{"rendered":"

1. \u5b9a\u6001\u5fae\u6270\u8bba 1.1 \u7b80\u4ecb \u91cf\u5b50\u529b\u5b66\u4e2d\u7814\u7a76\u4fdd\u5b88\u7269\u7406\u4f53\u7cfb(\u5373)\u54c8\u5bc6\u987f\u7b97\u7b26\u4e0d\u660e\u663e\u5730\u4f9d\u8d56\u4e8e\u65f6\u95f4\u7684\u4f53\u7cfb)\u662f\u4ee5\u54c8\u5bc6\u987f […]<\/p>\n","protected":false},"author":11,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11],"tags":[],"class_list":["post-1989","post","type-post","status-publish","format-standard","hentry","category-2021qm"],"_links":{"self":[{"href":"http:\/\/ustb.cc\/zh\/wp-json\/wp\/v2\/posts\/1989","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/ustb.cc\/zh\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/ustb.cc\/zh\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/ustb.cc\/zh\/wp-json\/wp\/v2\/users\/11"}],"replies":[{"embeddable":true,"href":"http:\/\/ustb.cc\/zh\/wp-json\/wp\/v2\/comments?post=1989"}],"version-history":[{"count":53,"href":"http:\/\/ustb.cc\/zh\/wp-json\/wp\/v2\/posts\/1989\/revisions"}],"predecessor-version":[{"id":2255,"href":"http:\/\/ustb.cc\/zh\/wp-json\/wp\/v2\/posts\/1989\/revisions\/2255"}],"wp:attachment":[{"href":"http:\/\/ustb.cc\/zh\/wp-json\/wp\/v2\/media?parent=1989"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/ustb.cc\/zh\/wp-json\/wp\/v2\/categories?post=1989"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/ustb.cc\/zh\/wp-json\/wp\/v2\/tags?post=1989"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}