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            "51": {
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                "title": "Research Paper Notes on Particle Correlation in Wigner Function Approach",
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                        "*": "\ufeffResearch Paper Notes on Particle Correlation in Wigner Function Approach\n\n== nucl-th/9909018 Boson spectra and correlations for thermal locally equilibrium systems ==\n\nYuri M. Sinyukov Preprint ITP-93-8E, Heavy Ion Phys.10:113-136,1999\n\nP.4 (4) \nFirstly, <math>p^{\\mu}e^{ik\\cdot x}</math> can be seen as a conserved current, when <math>p^{\\mu}</math> is a given (constant) four-vector satisfying <math>p\\cdot k=0</math> (see below)\n\n:<math>\\begin{align}\n\\partial_{\\mu}(p^{\\mu}e^{ik\\cdot x})=ip^{\\mu}k_{\\mu}e^{ik\\cdot x}=i(p\\cdot k)e^{ik\\cdot x}=0\n\\end{align}</math>\n\nOne may apply Gauss's law to such conserved current, so that the surface integral of <math>p^{\\mu}e^{ik\\cdot x}</math> on any given surface can be replaced by an integral sharing the same boundary. In particular, we will replace the integral on surface <math>\\sigma_{\\mu}</math> by an integral on the surface $\\Sigma_{\\mu}$ with constant time. \n\n:<math>\\begin{align}\n\\int d\\Sigma_{\\mu}p^{\\mu}e^{ik\\cdot x}=\\int d\\sigma^t_{\\mu}p^{\\mu}e^{ik\\cdot x} \n\\end{align}</math>\n\nTherefore some terms can be simplified as follows\n\n:<math>\\begin{align}\n&d\\sigma^t=d^3x \\hat{e}_t\\\\\n&\\int d\\sigma^t_{\\mu}p^{\\mu}e^{ik\\cdot x}=\\int d\\sigma^t_{\\mu}p^{\\mu}e^{ik^0t}e^{-i\\mathbf{k\\cdot x}}=p^0e^{ik^0t}(\\int d^3x e^{-i\\mathbf{k\\cdot x}})=p^0e^{ik^0t}(2\\pi)^3\\delta(\\mathbf{k}) \n\\end{align}</math>\n\nWe note that the above argument maybe seems too clumsy and not necessary, one may simply argue that if a quantity is invariant, one can evaluate it in any frame of reference.\n\nP.4 (5) \nWe will first introduce some notations and some identities to be used later\n:<math>p=\\frac{p_1+p_2}{2},q=p_2-p_1</math>\ntherefore\n:<math>p_1=p-\\frac{q}{2},p_2=p+\\frac{q}{2}</math>\n\nNote that <math>p_1,p_2</math> are two 4-momenta of detected hadrons (on the mass shell), one immediate consequence is\n\n:<math>p\\cdot q = \\frac{1}{2}(p_2^2-p_1^2)=\\frac{1}{2}(m^2-m^2)=0 </math>\n\nAt a certain moment, we will want to write down <math>\\delta(p\\cdot u) </math> in terms of <math>\\delta^4</math> function. To this end, we define <math>u'</math> as <math>u=q+u'</math>. For any function <math>f(u)</math> one has,\n\n:<math>\\begin{align}\n&f(u) \\delta(p\\cdot u)\\delta(\\mathbf{q}-\\mathbf{u})  \\\\\n&=f(u) \\delta(p\\cdot (q+u'))\\delta(\\mathbf{q}-\\mathbf{u}) \\\\\n&=f(u) \\delta(p^0{u^0}')\\delta(\\mathbf{q}-\\mathbf{u}) \\\\\n&=f(u) \\delta({u^0}')\\delta(\\mathbf{q}-\\mathbf{u})/p^0 \\\\\n&=f(u) \\delta(q^0-u^0)\\delta(\\mathbf{q}-\\mathbf{u})/p^0 \\\\\n&=f(u) \\delta(q-u)/p^0\n\\end{align}</math>\n\nBoth <math>p,q</math> are seen as constant in the above expression, where <math>u</math> is the variable. We have made use of  <math>\\delta^3</math> function of 3-momentum <math>\\delta(\\mathbf{q}-\\mathbf{u}) </math>, where <math>\\mathbf{u}</math> is not necessarily on the mass shell, which implies <math>u'=({u^0}',0,0,0) </math>. We also note that\n\n:<math>\\begin{align}\nd^4u=du^0d\\mathbf{u}=du'^0d\\mathbf{u},d\\mathbf{u}\\equiv d^3u\n\\end{align}</math>\n\n:<math>\\begin{align}\n&d^4u p^0 e^{i(q^0-u^0)t}\\delta(p\\cdot u)\\delta(\\mathbf{q}-\\mathbf{u}) =d^4u p^0 e^{i(q^0-u^0)t} \\delta(q-u)/p^0 =d^4u e^{i(q^0-u^0)t}\\delta(q-u) =d^4u \\delta(q-u)\n\\end{align}</math>\n\nThe above expressions will be used below. Now we are in the position to derive (5)\n\n:<math>\\begin{align}\n&\\langle a^+(p_1)a(p_2)\\rangle\\\\\n&=\\langle a^+(p-\\frac{q}{2})a(p+\\frac{q}{2})\\rangle \\\\\n&=\\int d^4u \\delta(q-u)\\langle a^+(p-\\frac{u}{2})a(p+\\frac{u}{2})\\rangle \\\\\n&=\\int d^4u(2\\pi)^{-3}\\delta(p\\cdot u)\\langle a^+(p-\\frac{u}{2})a(p+\\frac{u}{2})\\rangle\\times[(2\\pi)^3p^0e^{i(q^0-u^0)t}\\delta(\\mathbf{q}-\\mathbf{u})] \\\\\n&= \\int d^4u\\int d\\Sigma_{\\mu} p^{\\mu}e^{i(q-u)\\cdot x}(2\\pi)^{-3}\\delta(p\\cdot u)\\langle a^+(p-\\frac{u}{2})a(p+\\frac{u}{2})\\rangle  \\\\\n&=\\int d\\Sigma_{\\mu} p^{\\mu}e^{iq\\cdot x} \\times\\int d^4u (2\\pi)^{-3}e^{-iu\\cdot x}\\delta(p\\cdot u)\\langle a^+(p-\\frac{u}{2})a(p+\\frac{u}{2})\\rangle \\\\\n&=\\int d\\Sigma_{\\mu} p^{\\mu}e^{iq\\cdot x} f_W(x,p)  =\\int d\\sigma_{\\mu} p^{\\mu}e^{iq\\cdot x} f_W(x,p) \n\\end{align}</math>\n\nWe would like to give a few comments on (5). The identity (5) involves an invariant form of <math>\\langle a^+(p_1)a(p_2)\\rangle</math>, it helps to express things in terms of hydrodynamic variables. However, by itself, (5) is irrelevant to physical quantities such as fluid velocity. In this respect, it merely transfers the problem into the calculation of Wigner function. In practice, for systems almost in equilibrium the Wigner function can be further approximated by the equilibrium distribution at local rest frame <math>f_W\\to f(p\\cdot U,T,\\mu)</math> (see [6] of the paper). This approximation is only valid when <math>\\vec{p}_1 \\sim \\vec{p}_2</math>, which is due to the fact that the density matrix is almost diagonal when the system is not far away from equilibrium. The later implies, one has to do the Bogoliubov transformation first, then apply (5). In particular, if the two 3-momenta are back to back (<math>\\vec{k}_1,\\vec{k}_2</math>), they will be parallel after the Bogoliubov transformation (consider, e.g., <math>\\langle  a^+(k_1)a^+(k_2)\\rangle \\to \\langle  b^+(k_1)b(-k_2)\\rangle</math>), this is when <math>f_W=f</math> is a good approximation. It is worth pointing out that the two corresponding 4-momenta in this case are <math>p_1=(k_1^0,\\vec{k}_1),p_2=(k_2^0,-\\vec{k}_2)</math> before substituting them into (5)."
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                "pageid": 116,
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                "title": "Research Paper Notes on Review of Superradiance",
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                        "*": "Research Paper Notes on Review of Superradiance\n\n\u672c\u6587\u6863\u9664\u4e86\u5305\u62ec\u63a8\u5bfc,<span style=\"color:red\">\u7591\u60d1</span>\u5916,\u505a<span style=\"color:#0000FF\">\u8bfb\u4e66\u91cd\u70b9</span>\u7684\u8bb0\u5f55 <math></math>\n\n== \u6587\u732e\u5217\u8868 ==\n\n* Superradiance, arXiv:1501.06570, by Richard Brito, Vitor Cardoso, and Paolo Pani\n\n== Superradiance, arXiv:1501.06570, by Richard Brito, Vitor Cardoso, and Paolo Pani ==\n\n\u672c\u7efc\u8ff0\u57282020\u5e743\u67083\u65e5\u7ed9\u51fa\u66f4\u65b0.\u8fd9\u662f\u5bf9\u66f4\u65b0\u7248\u7684\u9605\u8bfb\u7b14\u8bb0.\u672c\u7efc\u8ff0\u7ed9\u51fa\u7684\u91cd\u8981\u6587\u732e\u5217\u8868\u5f88\u6709\u610f\u4e49,\u5bf9\u521d\u5b66\u8005\u7684\u77e5\u8bc6\u4f53\u7cfb\u6784\u5efa\u5f88\u6709\u5e2e\u52a9.\n\n(3.4-5)\n\n\u8fd9\u91cc\u901a\u8fc7\u7fa4\u901f\u5ea6\u5bf9\u6ce2\u77e2\u4e0e\u9891\u7387\u7b26\u53f7\u7684\u5206\u6790\u662f\u5177\u6709\u4e00\u822c\u6027\u7684\u624b\u7eed.\n\n\u4e0e\u6587\u4e2d\u7684\u7ed3\u8bba\u4e0d\u540c,\u6ce8\u610f\u5230\u5176\u5b9e<math>\\omega</math>\u662f\u53ef\u4ee5\u5dee\u4e00\u4e2a\u8d1f\u53f7\u7684,\u8fd9\u6837\u5bf9\u5e94\u67d0\u79cd\u76f8\u4f4d\u6b63\u597d\u76f8\u53cd\u7684\u6ce2,\u800c\u91cd\u8981\u662f<math>\\omega</math>\u4e0e<math>k</math>\u7684\u76f8\u5bf9\u7b26\u53f7.\u5177\u4f53\u7684,\u5f53\u76f8\u5bf9\u7b26\u53f7\u4fdd\u6301\u4e00\u81f4\u662f,\u5bf9\u5e94\u7684\u8272\u6563\u5173\u7cfb\u5176\u5b9e\u662f\u4e00\u4e2a\u5947\u51fd\u6570<math>\\omega(k)=-\\omega(-k)</math>,\u5bf9\u4e0a\u8ff0\u51fd\u6570\u7fa4\u901f\u5ea6\u6ee1\u8db3<math>\\frac{\\partial\\omega}{\\partial k}(k)=\\frac{\\partial\\omega}{\\partial k}(-k)</math>,\u6362\u8a00\u4e4b,\u59cb\u7ec8\u6ee1\u8db3\u7269\u7406\u4e0a\u6b63\u786e\u7684\u5165\u5c04\u6ce2,\u53cd\u5c04\u6ce2\u4e0e\u900f\u5c04\u6ce2\u7684\u7fa4\u901f\u5ea6\u65b9\u5411.\n\n(3.6)\n\n\u6ce8\u610f\u5230(3.3)\u4ec5\u4ec5\u662f\u5173\u4e8e\u7a7a\u95f4\u5750\u6807\u7684\u51fd\u6570,\u6240\u4ee5\u8bc1\u660e\u6717\u65af\u57fa\u884c\u5217\u5f0f(3.6)\u5b88\u6052\u7684\u5173\u952e\u6b65\u9aa4\u662f\u6ce8\u610f\u5230(3.6)\u5bf9<math>x</math>\u7684\u5bfc\u6570\u4e3a\u96f6.\n\n\u5229\u7528(3.3),\u5e76\u6ce8\u610f\u5230<math>\\omega</math>\u662f\u7ed9\u5b9a\u7684,\u5373<math>\\omega,e,A_0</math>\u90fd\u662f\u5e38\u6570(\u4e0d\u662f<math>x</math>\u7684\u51fd\u6570),\u8fd9\u6837,\u5bf9\u65b9\u7a0b(3.3)\u4efb\u610f\u4e24\u4e2a\u89e3<math>(f_1,f_2)</math>,\u6613\u8bc1(3.6)\u5728\u4efb\u610f\u7a7a\u95f4\u70b9\u5bf9<math>x</math>\u7684\u5bfc\u6570\u90fd\u662f\u96f6.\u8fd9\u6837\u6717\u65af\u57fa\u53ea\u80fd\u662f\u5e38\u6570.\n\n(3.7)\n\n\u8fd9\u91cc,\u6211\u4eec\u53ea\u80fd\u7528\u6717\u65af\u57fa\u8fde\u63a5\u6ce2\u51fd\u6570\u5728\u6b63\u8d1f\u65e0\u7a77\u5927\u5904\u7684\u6e10\u8fdb\u884c\u4e3a\u800c\u5f97\u5230\u8fd9\u4e2a\u5173\u7cfb.\u8fd9\u4e2a\u7ed3\u679c\u5c31\u662f\u5206\u522b\u7528(3.4)\u4e2d\u7684\u4e24\u4e2a\u8868\u8fbe\u5f0f\u8ba1\u7b97(3.6),\u4ee4\u5b83\u4eec\u76f8\u7b49,\u5e76\u6ce8\u610f\u5230(3.5),\u5373\u5f97.\u5177\u4f53\u5df2\u9a8c\u8bc1.\u56e0\u4e3a\u6211\u4eec\u4ec5\u77e5\u9053\u52bf\u573a\u7684\u6e10\u8fdb\u5f62\u5f0f\u800c\u975e\u5177\u4f53\u5f62\u5f0f,\u6240\u4ee5\u65e0\u6cd5\u5728\u67d0\u7ed9\u5b9a\u7684\u7a7a\u95f4\u70b9\u505a\u6ce2\u51fd\u6570\u7684\u8fde\u63a5\u5173\u7cfb.\u4f46\u662f\u5047\u8bbe\u52bf\u573a\u5728<math>x=0</math>\u70b9\u8df3\u8dc3\u5728\u5176\u4ed6\u4efb\u4f55\u4f4d\u7f6e\u4e3a\u5e38\u6570\u53ef\u4ee5\u9a8c\u8bc1\u4e0a\u8ff0\u7ed3\u679c,\u5177\u4f53\u7684,\u4e0a\u8ff0\u5047\u8bbe\u5bfc\u81f4<math>\\mathcal{R}=\\frac{\\omega-k}{\\omega+k}\\mathcal{I}, \\mathcal{T}=\\frac{2\\omega}{\\omega+k}\\mathcal{I}</math>,\u6613\u8bc1,\u7531\u7ed3\u679c\u540c\u6837\u53ef\u4ee5\u5f97\u5230(3.7)\u7ed9\u51fa\u7684\u5173\u7cfb.\n\n\u6211\u4eec,\u5982\u679c\u6211\u4eec\u8003\u8651\u7ed9\u5b9a\u7684\u8fb9\u754c\u6761\u4ef6(3.4),\u90a3\u4e48\u5728<math>x\\to\\pm\\infty</math>\u7684\u4efb\u4f55\u4e00\u7aef,(3.6)\u90fd\u4e3a\u96f6.\u6545\u6717\u65af\u57fa\u4e0d\u4ec5\u4e3a\u5e38\u6570\u800c\u4e14\u4e3a\u96f6.\u5bf9\u4e8e\u4e00\u4e2a\u4e24\u9636\u5fae\u5206\u65b9\u7a0b(3.3),\u5b83\u7684\u901a\u89e3\u7531\u4e24\u4e2a\u7ebf\u6027\u72ec\u7acb\u7684\u89e3\u6784\u6210,\u6717\u65af\u57fa\u4e3a\u96f6\u610f\u5473\u7740\u4e24\u4e2a\u89e3\u662f\u7ebf\u6027\u72ec\u7acb\u7684,\u6362\u8a00\u4e4b,\u6ee1\u8db3\u8fb9\u754c\u6761\u4ef6(3.4)\u7684\u6240\u6709\u89e3\u4e4b\u95f4\u53ea\u53ef\u80fd\u76f8\u5dee\u4e00\u4e2a\u5e38\u6570,\u5b83\u4eec\u90fd\u662f\u7ebf\u6027\u76f8\u5173\u7684.\u8fb9\u754c\u6761\u4ef6(3.4)\u4ece(\u4e24\u4e2a\u7ebf\u6027\u72ec\u7acb\u7684)\u901a\u89e3\u4e2d\u6311\u51fa\u4e86\u4e00\u4e2a\u786e\u5b9a\u7684\u6ee1\u8db3\u7269\u7406\u4e0a\u6b63\u786e\u7684\u8fb9\u754c\u6761\u4ef6\u7684\u89e3.\n\n\u800c\u5bf9\u4e8e(3.7)\u4e2d\u8ba8\u8bba\u7684<math>f, f^*</math>,\u540e\u8005\u867d\u7136\u662f(3.3)\u7684\u89e3,\u4f46\u663e\u7136\u5b83\u5e76\u4e0d\u6ee1\u8db3\u8fb9\u754c\u6761\u4ef6(3.4).\u8fd9\u4e2a\u7ed3\u679c\u53ea\u662f\u5229\u7528(3.4)\u662f\u65b9\u7a0b(3.3)\u7684\u89e3\u7684\u6761\u4ef6\u5f97\u5230\u53cd\u5c04,\u6295\u5c04\u4e0e\u6298\u5c04\u7cfb\u6570\u95f4\u6ee1\u8db3\u7684\u5173\u7cfb\u800c\u5df2.\n\n(3.13)\n\n\u6ce8\u610f\u5230\u8fd9\u91cc\u8003\u8651\u7684\u662f(1+1)\u7ef4\u7684\u60c5\u51b5,\u800c\u8fd9\u4e2a\u8fb9\u754c\u6761\u4ef6\u65e2\u6ee1\u8db3(\u4e0e\u4e4b\u524d\u8ba8\u8bba\u5b8c\u5168\u7c7b\u4f3c\u7684)\u5bf9\u7fa4\u901f\u5ea6\u65b9\u5411\u7684\u8981\u6c42,\u53c8\u6ee1\u8db3\u8fd0\u52a8\u65b9\u7a0b(3.12).\n\n(3.14)\n\n\u53ef\u4ee5\u60f3\u8c61\u90e8\u5206\u6d89\u4e16\u672a\u6df1\u7684\u65e0\u77e5\u5c11\u5973\u4e0d\u7981\u8981\u95ee,\u4e3a\u4ec0\u4e48\u4e4b\u524d\u5bfc\u51fa(3.7)\u6211\u4eec\u8981\u501f\u52a9\u6717\u65af\u57fa,\u800c\u8fd9\u91cc(3.14)\u6211\u4eec\u5374\u8981\u4f7f\u7528\u5b88\u6052\u6d41.\u5982\u679c\u53cd\u8fc7\u6765,\u662f\u5426\u4f1a\u5bfc\u81f4\u5176\u4ed6\u4e0d\u5e73\u5eb8\u7684\u7ed3\u679c?\n\n\u5b9e\u9645\u4e0a,\u4e0d\u96be\u8bc1\u660e,\u4e4b\u524d\u7684\u6717\u65af\u57fa(3.6)\u5c31\u662f\u5b88\u6052\u6d41\u7684\u4e00\u822c\u60c5\u51b5.\u5177\u4f53\u7684,\u56e0\u4e3a\u7531(3.3)\u51fa\u53d1,\u6309\u91cf\u5b50\u529b\u5b66\u7684\u6807\u51c6\u65b9\u6cd5\u6784\u9020\u5b88\u6052\u6d41(\u5982\u53c2\u89c1\u82cf\u6c5d\u94ff\u91cf\u5b50\u529b\u5b66),\u6211\u4eec\u5f97\u5230\u7684\u6b63\u662f<math>j^1=f^*\\frac{df}{dx}-f\\frac{df^*}{dx}</math>.\u5982\u679c\u6211\u4eec\u8003\u8651\u7684\u662f\u975e\u76f8\u5bf9\u8bba\u7684\u859b\u5b9a\u8c14\u65b9\u7a0b\u800c\u975e\u514b\u83b1\u56e0\u9ad8\u767b\u65b9\u7a0b,\u90a3\u4e48\u6211\u4eec\u540c\u6837\u5c06\u5f97\u5230(3.14).\u6240\u4ee5\u5bf9\u6ce2\u8272\u573a\u800c\u8a00,\u8fd9\u662f\u4e00\u79cd\u76f8\u5bf9\u8bba\u6548\u5e94.\u800c\u5bf9\u6ee1\u8db3\u76f8\u5bf9\u8bba\u534f\u53d8\u7684\u72c4\u62c9\u514b\u65b9\u7a0b\u800c\u8a00,\u8d39\u7c73\u5b50\u5374\u6ca1\u6709\u8d85\u8f90\u5c04\u53d1\u6563.\n\n\u5982\u6587\u4e2d\u6240\u8ba8\u8bba\u7684,\u8fd9\u4e24\u4e2a\u4e0d\u540c\u7684\u6027\u8d28,\u672c\u8d28\u4e0a\u6765\u81ea\u4e8e\u5bf9\u8d39\u7c73\u5b50\u7684\u6ce1\u5229\u4e0d\u76f8\u5bb9\u539f\u7406.\n\n(3.15)\n\n\u8fd9\u4e2a\u7ed3\u679c\u7684\u63a8\u5bfc\u5b9e\u9645\u4e0a\u5c31\u662f\u5229\u7528\u6ce2\u8272\u5b50\u4e0e\u8d39\u7c73\u5168\u540c\u7c92\u5b50\u7edf\u8ba1\u5c5e\u6027,\u4ee5\u53ca\u6982\u7387\u7684\u5f52\u4e00\u6027\u4ee5\u8ba1\u7b97\u7c92\u5b50\u5e73\u5747\u6570,\u5177\u4f53\u53c2\u89c1Hansen\u6587\u7ae0\u7684(25)\u4e0e(28).\n\n\u5b9e\u9645\u4e0a,\u6211\u4eec\u987a\u5e26\u6307\u51fa,\u8be5\u6587\u7ed9\u51fa\u4e86\u514b\u83b1\u56e0\u4f6f\u8c2c\u975e\u5e38\u5168\u9762\u7684\u63cf\u8ff0,\u4e00\u4e9b\u5177\u4f53\u63a8\u5bfc\u548c\u8ba8\u8bba\u53c2\u89c1\u76f8\u5173\u7b14\u8bb0.\n\n(3.16)\n\n\u8fd9\u4e2a\u7ed3\u679c\u5728\u5f15\u6587\u4e2d\u5e76\u6ca1\u6709\u76f4\u63a5\u7ed9\u51fa,\u8fd9\u91cc\u6211\u4eec\u7ed3\u5408Hansen\u4e0eManogue\u7684\u6587\u732e\u6765\u63a8\u5bfc\u8fd9\u4e2a\u7ed3\u8bba.\n\n\u5bf9\u6807\u91cf\u573a,\u53c2\u8003Manogue\u4e00\u6587\u7684(22),\u8003\u8651\u5230\u7ed9\u5b9a\u9891\u7387\u4e0e\u52a8\u91cf\u7684\u60c5\u51b5,\u6ce8\u610f\u5230\u672c\u7efc\u8ff0\u4f7f\u7528\u7684\u52bf\u573a\u5f62\u5f0f\u4e0e\u4e0a\u8ff0\u6587\u732e\u4e0d\u540c,\u4ece\u6587\u732e\u7684\u8868\u8fbe\u5f0f\u51fa\u53d1\u9700\u8981\u505a\u53d8\u6362<math>+eV/2\\to 0,-eV/2\\to eV</math>,\u4e0d\u5b58\u5728\u6a2a\u5411\u7a7a\u95f4<math>k\\to 0</math>,\u540c\u65f6\u6807\u91cf\u573a\u8d28\u91cf\u4e3a\u96f6<math>m\\to 0</math>,\u7efc\u4e0a\u6211\u4eec\u5f97\u5230<math>q\\to\\omega,r\\to \\omega-eV</math>,\u8003\u8651\u5230\u900f\u5c04\u632f\u5e45\u5b9a\u4e49\u4e00\u81f4<math>T\\to\\mathcal{T}</math>,\u56e0\u6b64\u6587\u4e2d\u7684(22)\u610f\u5473\u7740\u7c92\u5b50\u5bf9\u7b97\u7b26\u7684\u671f\u5f85\u503c\u4e3a<math>\\bar{n}_B=\\left|\\frac{r}{q}\\right||T|^2\\to \\frac{\\omega-eV}{\\omega}|\\mathcal{T}|^2</math>,\u4e0e\u672c\u6587(3.16)\u7b2c\u4e00\u5f0f\u7684\u7ed3\u679c\u4e00\u81f4.\n\n\u53c2\u8003Hansen\u4e00\u6587\u7684\u7ed3\u679c(22),\u540c\u65f6\u53c2\u89c1(9),(22),(61)\u4ee5\u53ca\u76f8\u5173\u8ba8\u8bba,\u5e76\u6ce8\u610f\u5230\u6b64\u65f6<math>q</math>\u4e3a\u8d1f\u503c.\u6211\u4eec\u5229\u7528\u6587\u7ae0\u4e2d<math>T</math>\u7684\u8868\u8fbe\u5f0f,\u53c2\u89c1\u6587\u7ae0(3-4)\u7684\u7b14\u8bb0\u5e76\u6ce8\u610f\u5230\u4e0e<math>\\mathcal{T}</math>\u7684\u533a\u522b,\u8fd9\u65f6\u5bf9\u5e94\u66ff\u6362<math>q\\to \\omega-eV,p\\to \\omega</math>,\u6545<math>\\bar{n}_B=|T(-q)|^2=\\frac{|q|}{|p|}|\\mathcal{T}|^2\\to\\frac{\\omega-eV}{\\omega}|\\mathcal{T}|^2</math>.\u4e0e\u4e4b\u524d\u7684\u7ed3\u679c\u5b8c\u5168\u4e00\u81f4.\n\n\u5bf9\u8d39\u7c73\u573a,\u60c5\u51b5\u5176\u5b9e\u6bd4\u8f83\u590d\u6742.\u6211\u4eec\u76f2\u76ee\u7684\u540c\u6837\u53c2\u8003Hansen\u4e00\u6587\u7684\u7ed3\u679c(22),\u6ce8\u610f\u5230\u8fd9\u65f6<math>\\kappa</math>\u7684\u5b9a\u4e49\u4e0d\u540c,\u4f46\u662f\u66ff\u6362\u8fc7\u7a0b\u5b8c\u5168\u7c7b\u4f3c,\u5e76\u4e14\u6709<math>|\\kappa|\\to 1</math>,\u6211\u4eec\u5f97\u5230<math>\\bar{n}_F=|\\kappa||\\mathcal{T}|^2\\to|\\mathcal{T}|^2</math>,\u8fd9\u5c31\u662f\u672c\u6587(3.16)\u7b2c\u4e8c\u5f0f\u7684\u7ed3\u679c.\u6309\u4e4b\u540e\u7684\u8ba8\u8bba,\u5176\u5b9e\u6211\u4eec\u6ce8\u610f\u5230,\u6bcf\u4e00\u4e2a\u65cb\u91cf\u5206\u91cf\u7684\u6bd4\u503c\u90fd\u6ee1\u8db3<math>\\mathcal{T}=T</math>.\n\n\u53c2\u8003Manogue\u4e00\u6587\u7684\u6700\u540e\u4e00\u5f0f(\u6ca1\u6709\u7f16\u53f7),\u7ed3\u8bba\u4f3c\u4e4e\u4e0d\u663e\u7136.\u8fd9\u65f6\u6211\u4eec\u6ce8\u610f\u5230,\u6bd4\u8f83Hansen\u4e00\u6587\u7684\u9644\u5f55,\u53cd\u5c04\u6ce2\u5404\u81ea\u65cb\u5206\u91cf\u7684\u6bd4\u503c<math>R_i/I_i</math>\u5176\u5b9e\u662f\u4e0d\u540c,\u4e24\u7bc7\u6587\u7ae0\u90fd\u662f\u8003\u8651\u4e86(3+1)\u7ef4\u7a7a\u95f4,\u5bf9\u5165\u5c04\u6ce2\u5f52\u4e00\u5316\u7684\u7ea6\u5b9a\u4e0d\u540c,\u6545Hansen\u4e00\u6587\u6709\u7edf\u4e00\u7684\u53cd\u5c04\u632f\u5e45\u800cManogue\u4e00\u6587\u5bf9\u6bcf\u4e2a\u65cb\u91cf\u5206\u91cf\u90fd\u6d89\u53ca\u4e0d\u540c\u7684\u53cd\u5c04\u632f\u5e45,\u800c\u4e0e\u4e4b\u76f8\u6bd4,Cardoso\u4e00\u6587\u8003\u8651\u7684\u662f(1+1)\u7ef4,\u901a\u8fc7\u5b9a\u4e49\u4e5f\u4ec5\u4ec5\u4f7f\u7528\u4e86\u4e00\u4e2a\u53cd\u5c04\u632f\u5e45.\u65e2\u7136\u6211\u4eec\u5bf9\u8ba1\u7b97\u65b9\u6cd5\u5df2\u7ecf\u7ed9\u51fa\u4e86\u660e\u786e\u7684\u8ba8\u8bba,\u8fd9\u91cc\u5e76\u6ca1\u6709\u8fdb\u884c\u66f4\u4e3a\u8be6\u7ec6\u7684\u8ba1\u7b97.\n\n(3.17)\n\n\u6587\u4e2d\u8fd9\u90e8\u5206\u5173\u4e8e\u80fd\u91cf\u5b88\u6052\u7684\u8ba8\u8bba\u4f3c\u4e4e\u6bd4\u8f83\u6c11\u79d1.\u7528\u8d1f\u80fd\u91cf\u8054\u7cfb\u53cd\u7c92\u5b50\u7684\u8bf4\u6cd5\u5728\u73b0\u4ee3\u7269\u7406\u4e2d\u662f\u5e94\u8be5\u7aed\u529b\u907f\u514d\u7684,\u7ed9\u51fa\u7684\u5f15\u6587\u4e5f\u662f\u5728\u73b0\u4ee3\u91cf\u5b50\u573a\u8bba\u6ca1\u6709\u6210\u719f\u7684\u5e74\u4ee3,\u6587\u7ae0\u4e5f\u5b8c\u5168\u6ca1\u6709\u6d89\u53ca\u5230\u4ece\u573a\u8bba\u7684\u89d2\u5ea6\u5bf9\u95ee\u9898\u7684\u8ba8\u8bba.\n\n\u5bf9\u6ce2\u8272\u5b50,\u8003\u8651\u5e26\u6b63\u7535<math>+e</math>\u7684\u7c92\u5b50\u4ece\u771f\u7a7a\u5165\u5c04\u6253\u5728\u52bf\u5792<math>V</math>\u4e0a,\u5f53\u6709\u6b63\u53cd\u7c92\u5b50\u5bf9\u4ea7\u751f\u65f6,\u4ee5\u53cd\u5c04\u65b9\u5411\u51fa\u5c04\u7684\u6b63\u7c92\u5b50\u52a8\u80fd\u4e3a<math>\\omega</math>,\u6cbf\u5165\u5c04\u65b9\u5411\u8fdb\u5165\u52bf\u5792\u7684\u53cd\u7c92\u5b50\u52a8\u80fd\u4e3a<math>eV-\\omega</math>,\u4e24\u8005\u4e4b\u548c\u4e3a<math>eV</math>,\u800c\u53cd\u7c92\u5b50\u5e26\u8d1f\u7535,\u5176\u52bf\u80fd\u4e3a<math>-eV</math>,\u6545\u603b\u80fd\u91cf\u4e3a0.\u6574\u4e2a\u8fc7\u7a0b\u8fd8\u9700\u8003\u8651\u4ee5\u52a8\u80fd<math>\\omega</math>\u6253\u5728\u52bf\u5792\u4e0a\u76f4\u63a5\u88ab\u5f39\u6027\u53cd\u5f39\u7684\u7c92\u5b50,\u8fd9\u4e9b\u7c92\u5b50\u7684\u80fd\u91cf\u4e0d\u8db3\u4ee5\u8fdb\u5165\u52bf\u5792,\u5728\u88ab\u540e\u8005\u5f39\u6027\u53cd\u5f39\u540e\u52a8\u80fd\u540c\u6837\u4e5f\u662f<math>\\omega</math>,\u8fd0\u52a8\u65b9\u5411\u76f8\u53cd.\u6b63\u8d1f\u7c92\u5b50\u5bf9\u4ea7\u751f\u6982\u7387\u7531\u52a8\u529b\u5b66\u51b3\u5b9a,\u5c31\u662f(3.16)\u7b2c\u4e00\u5f0f.\n\n\u5982\u679c\u6ca1\u6709\u6ee1\u8db3\u8d85\u8f90\u5c04\u6761\u4ef6,\u90a3\u4e48\u900f\u5c04(\u6b63)\u7c92\u5b50\u7684\u52a8\u80fd\u4e3a<math>\\omega-eV</math>,\u800c\u52bf\u80fd\u4e3a<math>eV</math>,\u4e24\u8005\u4e4b\u548c\u4e3a<math>\\omega</math>.\u8fd9\u4e2a\u80fd\u91cf\u7531\u5165\u5c04\u7c92\u5b50\u7684\u52a8\u80fd<math>\\omega</math>\u63d0\u4f9b.\u548c\u4e0a\u9762\u4e00\u6837,\u5728\u4e0a\u8ff0\u900f\u5c04\u8fc7\u7a0b\u4e2d\u8fd8\u9700\u53e0\u52a0\u5f39\u6027\u53cd\u5f39\u7684\u7c92\u5b50,\u6574\u4e2a\u7269\u7406\u8fc7\u7a0b\u6ca1\u6709\u51fa\u73b0\u53cd\u7c92\u5b50.\u540c\u6837,\u53cd\u5c04\u4e0e\u900f\u5c04\u632f\u5e45\u7531\u52a8\u529b\u5b66\u51b3\u5b9a.\n\n\u4e0a\u8ff0\u5206\u6790,\u4e0eHansen\u4e00\u6587(14)\u4e0b\u8ba8\u8bba\u4e2d\u5f97\u5230\u7684,\u5bf9\u8d28\u91cf\u4e0d\u4e3a\u96f6\u7684\u60c5\u51b5\u4e0b\u5728\u6ee1\u8db3\u6761\u4ef6<math>eV>2m</math>\u65f6\u51fa\u73b0\u7c92\u5b50\u5bf9\u4ea7\u751f\u4e00\u81f4.\u8fd9\u65f6\u52bf\u573a\u63d0\u4f9b\u7ed9\u53cd\u7c92\u5b50\u7684\u80fd\u91cf\u7b49\u4e8e\u6b63\u53cd\u7c92\u5b50\u5bf9\u7684\u52a8\u80fd\u4e0e\u5176\u9759\u8d28\u91cf\u7684\u603b\u548c.\u5177\u4f53\u7684\u6ce8\u610f\u5230\u6761\u4ef6<math>E<eV</math>\u4ee5\u53ca\u8be5\u6587(1-2),\u6211\u4eec\u6709<math>\\sqrt{p^2+m^2}+\\sqrt{q^2+m^2}=eV</math>.\n\n(3.20)\n\n\u8fd9\u4e2a\u5173\u7cfb\u7684\u63a8\u5bfc\u662f\u4ece\u8fd0\u52a8\u5b66(\u80fd\u52a8\u5b88\u6052)\u7684\u89d2\u5ea6\u6765\u8003\u8651.\u8003\u8651Ginzburg\u8bb2\u5ea7\u4e2d\u65b9\u7a0b(10)\u7684\u89e3(11-12),\u5176\u4e2d(12)\u7b49\u5f0f\u5de6\u8fb9\u5fc5\u987b\u5927\u4e8e\u96f6,\u6545\u7531\u4e8e<math>n\\ge 1</math>\u7b49\u5f0f\u53f3\u8fb9\u7684\u5206\u5b50\u5fc5\u987b\u5927\u4e8e\u96f6,\u5373<math>v_0\\cos\\theta_0-c/n >0</math>.\u6ce8\u610f\u5230<math>v_0\\cdot k=v_0 \\cos\\theta_0 k</math>\u53ca<math>\\frac{ck}{n}=\\frac{2\\pi c}{n\\lambda}=\\frac{2\\pi }{T}=\\omega</math>,\u6211\u4eec\u6709<math>\\omega < v_0\\cdot k</math>,\u6b64\u5373(3.20).\n\n\u63a5\u7740,\u6587\u4e2d\u5bf9(3.20-21)\u7269\u7406\u4e0a\u7684\u8ba8\u8bba\u662f\u5f88\u6709\u610f\u4e49\u7684,\u6211\u4eec\u6307\u51fa,\u8fd9\u91cc\u5e76\u4e0d\u6d89\u53ca\u5916\u52bf\u573a\u4ee5\u53ca\u6b63\u53cd\u7c92\u5b50\u5bf9\u4ea7\u751f.\n\n(3.23)\n\n\u5b83\u63a8\u5bfc\u4e2d\u7684\u4e3b\u8981\u7269\u7406\u52a8\u673a\u662f\u70ed\u529b\u5b66\u7b2c\u4e8c\u539f\u7406,\u5177\u4f53\u53c2\u89c1\u8d1d\u6839\u65af\u5766\u4e00\u6587arXiv:gr-qc/9803033\u7684\u516c\u5f0f(12)\u4ee5\u53ca\u76f8\u5173\u7b14\u8bb0.\u4e0e\u91d1\u5179\u4f2f\u683c\u7684\u53cd\u5e38\u591a\u666e\u52d2\u6761\u4ef6\u76f8\u6bd4\u8f83,\u5438\u6536\u7cfb\u6570\u5fc5\u987b\u4e3a\u8d1f,\u53cd\u5c04\u7cfb\u6570\u5927\u4e8e1,\u8fd9\u65f6\u4f53\u7cfb\u53d1\u751f\u8d85\u8f90\u5c04\u73b0\u8c61.\n\n\u6587\u7ae0\u5728\u8fd9\u91cc\u6307\u51fa\u7684,\u8d85\u8f90\u5c04\u4e0e\u8017\u6563\u7684\u5173\u7cfb\u7684\u5177\u4f53\u8ba8\u8bba,\u53c2\u89c1\u4e0a\u8ff0\u6587\u732e(48ab)\u9644\u8fd1\u7684\u8ba8\u8bba.\u6587\u732e\u4e2d\u901a\u8fc7\u5bf9\u65cb\u8f6c\u5a92\u8d28\u8d85\u8f90\u5c04\u7684\u5177\u4f53\u8ba1\u7b97\u6307\u51fa,\u5982\u679c\u4e0d\u5b58\u5728\u8017\u6563(\u7535\u5bfc\u7cfb\u6570\u4e3a\u96f6),\u5219\u8d85\u8f90\u5c04\u4e5f\u4e0d\u80fd\u51fa\u73b0.\u6211\u4eec\u6ce8\u610f\u5230,\u8017\u6563\u7684\u7269\u7406\u672c\u8d28\u662f\u8fc7\u7a0b\u7684\u4e0d\u53ef\u9006\u6027,\u8fd9\u4e0e\u8d1d\u6839\u65af\u5766\u6587\u7ae0\u4e2d\u57fa\u4e8e\u70ed\u529b\u5b66\u7b2c\u4e8c\u5b9a\u5f8b\u5bf9\u8d85\u8f90\u5c04\u6761\u4ef6\u7684\u63a8\u5bfc\u662f\u5b8c\u5168\u81ea\u6d3d\u7684.\n\n(3.25)\n\n\u6b64\u5f0f\u7684\u7269\u7406\u610f\u4e49\u662f,\u538b\u5f3a\u68af\u5ea6\u4e0e\u5916\u529b\u573a\u6709\u5173.\u6bd4\u5982,\u5728\u91cd\u529b\u573a\u4e2d\u7a7a\u6c14\u7684\u4e0d\u540c\u9ad8\u5ea6\u7684\u538b\u5f3a\u4e0d\u540c.\n\n\u53c2\u8003\u539f\u6587,\u5f20\u5c11\u541b\u6307\u51fa,\u9759\u6001\u5e73\u8861\u65f6\u538b\u5f3a\u5dee\u5bfc\u81f4\u51c0\u5916\u529b,\u5373<math>\\frac1n\\frac{\\partial p}{\\partial r}=-(mg+eE)</math>,\u4ee3\u5165\u7406\u60f3\u6c14\u4f53\u72b6\u6001\u65b9\u7a0b<math>p=nkT</math>,\u5373\u5f97(3.25).\n\n\u6211\u4eec\u53ef\u4ee5\u8fdb\u4e00\u6b65\u5bfc\u51fa(3.26),\u6216\u539f\u6587\u4e2d\u7684(3).\u8fd9\u662f\u56e0\u4e3a\u5982\u679c\u4e24\u4e2a\u4e0d\u540c\u7ec4\u5206\u7684\u5206\u538b\u968f\u7740\u5750\u6807\u7684\u53d8\u5316\u4e0d\u540c,\u610f\u5473\u7740\u8fd9\u4e24\u4e2a\u7ec4\u5206\u5728\u7a7a\u95f4\u7684\u5206\u5e03\u6709\u5dee\u5f02,\u6362\u8a00\u4e4b,\u4e24\u4e2a\u7ec4\u5206\u5206\u79bb\u4e86.\n\n\u4e00\u4e2a\u88ab\u81ea\u5df1\u5e26\u5230\u6c9f\u91cc\u53bb\u7684\u7406\u89e3\u662f\u628a\u538b\u5f3a\u4e0e\u5de8\u52bf\u8054\u7cfb\u8d77\u6765.\u5229\u7528\u7cfb\u7efc\u7406\u8bba\u91cc\u5e7f\u4e49\u529b\u7684\u5f62\u5f0f,\u53c2\u8003\u82cf\u8001\u5e08\u7edf\u8ba1\u529b\u5b66\u5bf9\u6b63\u5219\u7cfb\u7efc\u5e7f\u4e49\u529b(\u538b\u5f3a)\u7684\u8ba1\u7b97\u516c\u5f0f(3.6.2),\u5373<math>\\frac{ X}{kT}=\\frac{\\partial \\ln P}{\\partial x}</math>\u5176\u4e2d<math>P=\\int e^{-\\beta E}d\\Omega</math>,\u63a8\u5e7f\u5230\u5de8\u6b63\u5219\u7cfb\u7efc\u7684\u60c5\u51b5(3.11.21),\u5373<math>\\frac{ X}{kT}=\\frac{\\partial \\ln \\Xi}{\\partial x}</math>\u5176\u4e2d<math>\\Xi=\\int e^{-\\beta (E-\\mu N)}d\\Omega</math>.\u4f46\u5bf9\u5de8\u6b63\u5219\u7cfb\u7efc,\u6709<math>-pV=\\tilde{\\Omega}=-kT\\ln \\Xi</math>.\u53e6\u4e00\u65b9\u9762,\u5982\u679c\u6d41\u4f53\u9759\u529b\u5b66\u7684\u53d7\u529b\u5e73\u8861,\u90a3\u4e48\u5e94\u8be5\u662f<math>{ X}=\\frac{\\partial p}{\\partial x}</math>,\u5176\u4e2d<math>X</math>\u662f\u5355\u4e2a\u7c92\u5b50\u53d7\u5230\u7684\u51c0\u5916\u529b.\u5982\u679c\u76f4\u63a5\u628a<math>X</math>\u66ff\u6362\u6210\u538b\u5f3a,\u4e0d\u8003\u8651\u5230\u7c92\u5b50\u6570\u5bc6\u5ea6\u662f\u53d8\u91cf.\u5219\u4e0e\u4e0a\u8ff0\u7ed3\u679c\u4e00\u81f4.\n\n(3.33)\n\n\u8fd9\u91cc<span style=\"color:red\">\u4e0d\u6e05\u695a</span>\u8d85\u8f90\u5c04\u51c6\u7c92\u5b50\u7684\u52a8\u91cf\u4e0e\u80fd\u91cf\u662f\u5982\u4f55\u88ab\u8868\u8fbe\u4e3a\u8d39\u7c73\u9762\u52a8\u91cf<math>k_{\\mathrm{F}}</math>\u548c\u80fd\u9699<math>\\Delta_0</math>\u7684.\u7efc\u8ff0\u4e2d\u7ed9\u51fa\u7684\u5f15\u6587[157]\u4ec5\u4ec5\u6d89\u53ca\u4e34\u754c\u78c1\u573a\u4e0e\u4e34\u754c\u7535\u6d41\u5bc6\u5ea6\u7684\u5173\u7cfb\u7684\u5386\u53f2\u56de\u987e.\n\n(3.36-37)\n\n\u5177\u4f53\u53c2\u89c1\u58f0\u5b66\u9ed1\u6d1e(arXiv:gr-qc/9712010v2)\u7684\u7efc\u8ff0\u58f0\u5b66\u9ed1\u6d1e\u7684(2-3)\u4ee5\u53ca\u63a8\u5bfc.\u6570\u5b66\u4e0a,\u8fd9\u4e2a\u5ea6\u89c4\u662f\u76f4\u63a5\u4ece\u6ce2\u52a8\u65b9\u7a0b\u5bf9\u5e94\u7684\u5fae\u6270\u65b9\u7a0b\u5f97\u5230\u7684.\n\n(3.41)\n\n\u8fd9\u90e8\u5206\u8ba8\u8bba\u57fa\u672c\u4e0a\u662f\u5bf9Landau\u843d\u4f53\u529b\u5b66\u4e00\u4e66\u7b2c\u4e8c\u7248P.322\u6fc0\u6ce2\u4e00\u7ae0\u4e60\u9898\u7684\u5177\u4f53\u9610\u8ff0,\u5185\u5bb9\u6bd4\u539f\u4e66\u66f4\u4e3a\u8be6\u5c3d.\u4e00\u4e9b\u8ba8\u8bba\u53ef\u4ee5\u53c2\u8003\u8be5\u4e66\u7684\u8bfb\u4e66\u7b14\u8bb0.\u6211\u4eec\u6ce8\u610f\u5230\u7531\u4e8e\u8fb9\u754c\u6761\u4ef6,\u6ce2\u77e2\u5728\u8fb9\u754c\u5207\u5411\u7684\u6295\u5f71\u5bf9\u6240\u6709\u7684\u6ce2\u4e00\u81f4,\u5728\u901a\u89e3\u4e2d\u5bf9\u5e94\u56e0\u5b50<math>e^{ik_xx+ik_yy}</math>.\u7c7b\u4f3c\u7684,\u5b58\u5728\u5171\u540c\u7684\u65f6\u95f4\u9707\u8361\u56e0\u5b50<math>e^{-i\\omega t}</math>.\n\n\u8fd9\u91cc,(3.41)\u53ef\u4ee5\u901a\u8fc7\u5c06\u5f62\u5f0f\u89e3(3.40)\u4ee3\u5165\u8fd0\u52a8\u65b9\u7a0b(3.36),\u6ce8\u610f\u5230\u5ea6\u89c4(3.37)\u5f97\u5230.\u5176\u4e2d,\u5ea6\u89c4\u7684\u4ea4\u53c9\u9879\u5bf9\u5e94(3.41)\u7b49\u5f0f\u5de6\u8fb9\u7684\u4ea4\u53c9\u9879,\u5ea6\u89c4\u7a7a\u95f4\u90e8\u5206\u4e0e\u5a92\u8d28\u901f\u5ea6\u6709\u5173\u7684\u90e8\u5206\u5bf9\u5e94(3.41)\u7b49\u5f0f\u53f3\u8fb9\u4e0e\u5a92\u8d28\u901f\u5ea6\u7684\u5e73\u65b9\u9879\u8d21\u732e.\n\n(3.44)\n\n\u8fd9\u91cc,\u7b49\u5f0f\u53f3\u8fb9\u7684\u8d21\u732e\u5206\u4e3a\u4e24\u90e8\u5206,\u7b2c\u4e00\u90e8\u5206\u662f\u7531\u4e8e\u673a\u68b0\u6ce2\u5728\u7ed9\u5b9a\u7a7a\u95f4\u70b9\u7684\u632f\u52a8\u5728\u4ea4\u754c\u9762\u6cd5\u5411\u7684\u6295\u5f71<math>v_z</math>,\u7b2c\u4e8c\u90e8\u5206\u662f\u6765\u6e90\u4e8e\u5a92\u8d28\u6cbf\u7740x\u65b9\u5411\u7684\u4f20\u64ad\u4ee5\u53ca\u6ce2\u5f62(\u5728\u7ed9\u5b9a\u65f6\u523b)\u6cbf\u7740x\u65b9\u5411\u7684\u53d8\u5316,\u5982\u679c\u5a92\u8d28\u9759\u6b62,\u7b2c\u4e8c\u90e8\u5206\u8d21\u732e\u4e3a\u96f6.\n\n(3.45)\n\n\u8fd9\u91cc<math>k</math>\u7684\u7b26\u53f7\u7531\u6298\u5c04\u6ce2\u7684\u7fa4\u901f\u5ea6(\u800c\u975e\u76f8\u901f\u5ea6)\u51b3\u5b9a.\u901a\u8fc7\u8fd9\u4e2a\u5173\u7cfb\u53ef\u4ee5\u5f97\u5230\u8d85\u8f90\u5c04\u7684\u6761\u4ef6.\n\n(3.50)\n\n\u8fd9\u662f\u4e00\u4e2a\"\u6709\u6548\"\u8fd0\u52a8\u65b9\u7a0b,\u5176\u4e2d\u4e0e<math>\\alpha</math>\u6b63\u6bd4\u7684\u9879\u4e3a\u963b\u5c3c\u9879.\u5728\u7269\u7406\u4e0a,\u8fd9\u4e0e\u5728\u7b80\u8c10\u632f\u5b50\u4e2d\u5f15\u5165\u963b\u5c3c,\u6216\u8005\u5728\u6ce2\u4f20\u64ad\u7684\u5a92\u8d28\u4e2d\u5f15\u5165\u963b\u5c3c\u7c7b\u4f3c.\u5728\u6570\u5b66\u4e0a,\u5b83\u4f7f\u5f97\u9891\u7387\u4ea7\u751f\u4e00\u4e2a\u865a\u90e8,\u4f7f\u5f97\u6ce2\u51fd\u6570\u7684\u6f14\u5316\u968f\u7740\u65f6\u95f4\u9010\u6e10\u8870\u51cf.\u5728\u6b64\u610f\u4e49\u4e0a,\u5982\u6587\u4e2d\u7684\u8ba8\u8bba,\u8f6c\u52a8\u4f7f\u5f97\u9891\u7387\u7684\u5b9e\u90e8\u6539\u53d8\u7b26\u53f7.(\u8fd9\u672c\u4e0d\u5e26\u6765\u4efb\u4f55\u5f71\u54cd,\u4f46)\u53c2\u8003(3.54),\u8fd9\u8fdb\u4e00\u6b65\u4f7f\u5f97\u963b\u5c3c<math>\\alpha</math>\u7b49\u6548\u7684\u6539\u53d8\u7b26\u53f7,\u6700\u7ec8\u5bfc\u81f4\u9891\u7387\u7684\u865a\u90e8\u6539\u53d8\u7b26\u53f7,\u4ece\u800c\u5bfc\u81f4\u8d85\u8f90\u5c04.\n\n\u6587\u4e2d\u540c\u6837\u63d0\u53ca,\u4ece\u8fd0\u52a8\u5b66\u89d2\u5ea6\u800c\u8a00,\u5728\u7ebf\u6027\u8fd0\u52a8\u60c5\u51b5\u4e0b,\u5f53\u7fa4\u901f\u5ea6\u8d85\u5149\u901f\u5bfc\u81f4\u8d85\u8f90\u5c04,\u5728\u8f6c\u52a8\u60c5\u51b5\u4e0b\u89d2(\u7fa4)\u901f\u5ea6\u8d85\u8fc7\u76f8\u5e94(\u89d2\u5411)\u76f8\u901f\u5ea6\u4e5f\u4f1a\u5bfc\u81f4\u8d85\u8f90\u5c04.\n\n()\n\n\n\n\u672c\u6587\u6863\u9664\u4e86\u5305\u62ec\u63a8\u5bfc,<span style=\"color:red\">\u7591\u60d1</span>\u5916,\u505a<span style=\"color:#0000FF\">\u8bfb\u4e66\u91cd\u70b9</span>\u7684\u8bb0\u5f55 <math></math>"
                    }
                ]
            }
        }
    }
}