Lesson 9 Hilbert 空间

这节课讲完要考期中考试了.

期中考试的主要内容都是我们布置的第二章作业.

Hilbert 空间

我们现在学的量子力学研究的都是纯态，可以通过一个波函数来描述这个系统的物理.

我们定义：

∣...剩余内容已隐藏 查看完整文章以阅读更多查看完整文章(self.__next_f=self.__next_f||[]).push([0])self.__next_f.push([1,"1:\"$Sreact.fragment\"\n2:I[39756,[\"/_next/static/chunks/ff1a16fafef87110.js\",\"/_next/static/chunks/7dd66bdf8a7e5707.js\"],\"default\"]\n3:I[37457,[\"/_next/static/chunks/ff1a16fafef87110.js\",\"/_next/static/chunks/7dd66bdf8a7e5707.js\"],\"default\"]\n4:I[79520,[\"/_next/static/chunks/05c08c33f311058e.js\"],\"\"]\n6:I[97367,[\"/_next/static/chunks/ff1a16fafef87110.js\",\"/_next/static/chunks/7dd66bdf8a7e5707.js\"],\"OutletBoundary\"]\n8:I[11533,[\"/_next/static/chunks/ff1a16fafef87110.js\",\"/_next/static/chunks/7dd66bdf8a7e5707.js\"],\"AsyncMetadataOutlet\"]\na:I[97367,[\"/_next/static/chunks/ff1a16fafef87110.js\",\"/_next/static/chunks/7dd66bdf8a7e5707.js\"],\"ViewportBoundary\"]\nc:I[97367,[\"/_next/static/chunks/ff1a16fafef87110.js\",\"/_next/static/chunks/7dd66bdf8a7e5707.js\"],\"MetadataBoundary\"]\nd:\"$Sreact.suspense\"\nf:I[68027,[\"/_next/static/chunks/05c08c33f311058e.js\"],\"default\"]\n:HL[\"/_next/static/chunks/aeb0d837d331be8d.css\",\"style\"]\n:HL[\"/_next/static/media/797e433ab948586e-s.p.dbea232f.woff2\",\"font\",{\"crossOrigin\":\"\",\"type\":\"font/woff2\"}]\n:HL[\"/_next/static/media/caa3a2e1cccd8315-s.p.853070df.woff2\",\"font\",{\"crossOrigin\":\"\",\"type\":\"font/woff2\"}]\n"])self.__next_f.push([1,"0:{\"P\":null,\"b\":\"1-eHBNv7Or-ljjbwnF-2i\",\"p\":\"\",\"c\":[\"\",\"article?id=064cdef4-9c0a-4f2d-ba0d-95119ed1f89d\u0026link=https%3A%2F%2Fphysnya.top%2Fquantum-mechanics%2Flesson-9%2F\"],\"i\":false,\"f\":[[[\"\",{\"children\":[\"article\",{\"children\":[\"__PAGE__?{\\\"id\\\":\\\"064cdef4-9c0a-4f2d-ba0d-95119ed1f89d\\\",\\\"link\\\":\\\"https://physnya.top/quantum-mechanics/lesson-9/\\\"}\",{}]}]},\"$undefined\",\"$undefined\",true],[\"\",[\"$\",\"$1\",\"c\",{\"children\":[[[\"$\",\"link\",\"0\",{\"rel\":\"stylesheet\",\"href\":\"/_next/static/chunks/aeb0d837d331be8d.css\",\"precedence\":\"next\",\"crossOrigin\":\"$undefined\",\"nonce\":\"$undefined\"}],[\"$\",\"script\",\"script-0\",{\"src\":\"/_next/static/chunks/05c08c33f311058e.js\",\"async\":true,\"nonce\":\"$undefined\"}]],[\"$\",\"html\",null,{\"lang\":\"zh-CN\",\"children\":[[\"$\",\"head\",null,{\"children\":[\"$\",\"link\",null,{\"rel\":\"stylesheet\",\"href\":\"/fontawesome.min.css\"}]}],[\"$\",\"body\",null,{\"className\":\"geist_a71539c9-module__T19VSG__variable geist_mono_8d43a2aa-module__8Li5zG__variable antialiased\",\"children\":[[\"$\",\"$L2\",null,{\"parallelRouterKey\":\"children\",\"error\":\"$undefined\",\"errorStyles\":\"$undefined\",\"errorScripts\":\"$undefined\",\"template\":[\"$\",\"$L3\",null,{}],\"templateStyles\":\"$undefined\",\"templateScripts\":\"$undefined\",\"notFound\":[[[\"$\",\"title\",null,{\"children\":\"404: This page could not be found.\"}],[\"$\",\"div\",null,{\"style\":{\"fontFamily\":\"system-ui,\\\"Segoe UI\\\",Roboto,Helvetica,Arial,sans-serif,\\\"Apple Color Emoji\\\",\\\"Segoe UI Emoji\\\"\",\"height\":\"100vh\",\"textAlign\":\"center\",\"display\":\"flex\",\"flexDirection\":\"column\",\"alignItems\":\"center\",\"justifyContent\":\"center\"},\"children\":[\"$\",\"div\",null,{\"children\":[[\"$\",\"style\",null,{\"dangerouslySetInnerHTML\":{\"__html\":\"body{color:#000;background:#fff;margin:0}.next-error-h1{border-right:1px solid rgba(0,0,0,.3)}@media (prefers-color-scheme:dark){body{color:#fff;background:#000}.next-error-h1{border-right:1px solid rgba(255,255,255,.3)}}\"}}],[\"$\",\"h1\",null,{\"className\":\"next-error-h1\",\"style\":{\"display\":\"inline-block\",\"margin\":\"0 20px 0 0\",\"padding\":\"0 23px 0 0\",\"fontSize\":24,\"fontWeight\":500,\"verticalAlign\":\"top\",\"lineHeight\":\"49px\"},\"children\":404}],[\"$\",\"div\",null,{\"style\":{\"display\":\"inline-block\"},\"children\":[\"$\",\"h2\",null,{\"style\":{\"fontSize\":14,\"fontWeight\":400,\"lineHeight\":\"49px\",\"margin\":0},\"children\":\"This page could not be found.\"}]}]]}]}]],[]],\"forbidden\":\"$undefined\",\"unauthorized\":\"$undefined\"}],[\"$\",\"$L4\",null,{\"id\":\"baidu-analytics\",\"strategy\":\"afterInteractive\",\"src\":\"https://hm.baidu.com/hm.js?4e30da8cba2361458b8c235bb1fd6d89\"}]]}]]}]]}],{\"children\":[\"article\",[\"$\",\"$1\",\"c\",{\"children\":[null,[\"$\",\"$L2\",null,{\"parallelRouterKey\":\"children\",\"error\":\"$undefined\",\"errorStyles\":\"$undefined\",\"errorScripts\":\"$undefined\",\"template\":[\"$\",\"$L3\",null,{}],\"templateStyles\":\"$undefined\",\"templateScripts\":\"$undefined\",\"notFound\":\"$undefined\",\"forbidden\":\"$undefined\",\"unauthorized\":\"$undefined\"}]]}],{\"children\":[\"__PAGE__\",[\"$\",\"$1\",\"c\",{\"children\":[\"$L5\",[[\"$\",\"script\",\"script-0\",{\"src\":\"/_next/static/chunks/f8e5d293c91c965f.js\",\"async\":true,\"nonce\":\"$undefined\"}],[\"$\",\"script\",\"script-1\",{\"src\":\"/_next/static/chunks/cff04410b4f29ca8.js\",\"async\":true,\"nonce\":\"$undefined\"}]],[\"$\",\"$L6\",null,{\"children\":[\"$L7\",[\"$\",\"$L8\",null,{\"promise\":\"$@9\"}]]}]]}],{},null,false]},null,false]},null,false],[\"$\",\"$1\",\"h\",{\"children\":[null,[[\"$\",\"$La\",null,{\"children\":\"$Lb\"}],[\"$\",\"meta\",null,{\"name\":\"next-size-adjust\",\"content\":\"\"}]],[\"$\",\"$Lc\",null,{\"children\":[\"$\",\"div\",null,{\"hidden\":true,\"children\":[\"$\",\"$d\",null,{\"fallback\":null,\"children\":\"$Le\"}]}]}]]}],false]],\"m\":\"$undefined\",\"G\":[\"$f\",[[\"$\",\"link\",\"0\",{\"rel\":\"stylesheet\",\"href\":\"/_next/static/chunks/aeb0d837d331be8d.css\",\"precedence\":\"next\",\"crossOrigin\":\"$undefined\",\"nonce\":\"$undefined\"}]]],\"s\":false,\"S\":false}\n"])self.__next_f.push([1,"b:[[\"$\",\"meta\",\"0\",{\"charSet\":\"utf-8\"}],[\"$\",\"meta\",\"1\",{\"name\":\"viewport\",\"content\":\"width=device-width, initial-scale=1\"}]]\n7:null\n"])self.__next_f.push([1,"10:I[76585,[\"/_next/static/chunks/05c08c33f311058e.js\",\"/_next/static/chunks/f8e5d293c91c965f.js\",\"/_next/static/chunks/cff04410b4f29ca8.js\"],\"I18nProvider\"]\n11:I[77646,[\"/_next/static/chunks/05c08c33f311058e.js\",\"/_next/static/chunks/f8e5d293c91c965f.js\",\"/_next/static/chunks/cff04410b4f29ca8.js\"],\"LanguageWrapper\"]\n12:I[22016,[\"/_next/static/chunks/05c08c33f311058e.js\",\"/_next/static/chunks/f8e5d293c91c965f.js\",\"/_next/static/chunks/cff04410b4f29ca8.js\"],\"default\"]\n13:I[60001,[\"/_next/static/chunks/05c08c33f311058e.js\",\"/_next/static/chunks/f8e5d293c91c965f.js\",\"/_next/static/chunks/cff04410b4f29ca8.js\"],\"LanguageSwitcher\"]\n14:I[34838,[\"/_next/static/chunks/05c08c33f311058e.js\",\"/_next/static/chunks/f8e5d293c91c965f.js\",\"/_next/static/chunks/cff04410b4f29ca8.js\"],\"NavigationClient\"]\n15:I[89581,[\"/_next/static/chunks/05c08c33f311058e.js\",\"/_next/static/chunks/f8e5d293c91c965f.js\",\"/_next/static/chunks/cff04410b4f29ca8.js\"],\"BackButton\"]\n"])self.__next_f.push([1,"5:[\"$\",\"$L10\",null,{\"initialLang\":\"zh-CN\",\"children\":[\"$\",\"$L11\",null,{\"lang\":\"zh-CN\",\"children\":[\"$\",\"main\",null,{\"className\":\"min-h-screen bg-gradient-to-br from-orange-50 to-amber-50\",\"children\":[[\"$\",\"nav\",null,{\"className\":\"citrus-navigation fixed top-0 left-0 right-0 z-50 bg-white/90 backdrop-blur-md border-b border-orange-100\",\"children\":[\"$\",\"div\",null,{\"className\":\"max-w-7xl mx-auto px-4 sm:px-6 lg:px-8\",\"children\":[\"$\",\"div\",null,{\"className\":\"flex justify-between items-center h-16\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex items-center space-x-4\",\"children\":[\"$\",\"$L12\",null,{\"href\":\"/\",\"className\":\"flex items-center space-x-2\",\"children\":[[\"$\",\"div\",null,{\"className\":\"w-8 h-8 gradient-citrus rounded-full flex items-center justify-center text-white font-bold\",\"children\":\"🍊\"}],[\"$\",\"span\",null,{\"className\":\"text-xl font-bold text-gray-800\",\"children\":\"柑橘 RSS\"}]]}]}],[\"$\",\"div\",null,{\"className\":\"hidden md:flex space-x-8\",\"children\":[[\"$\",\"$L12\",null,{\"href\":\"/feeds\",\"className\":\"nav-link text-gray-700 hover:text-orange-600 font-medium\",\"children\":\"订阅源\"}],[\"$\",\"$L12\",null,{\"href\":\"/browser\",\"className\":\"nav-link text-gray-700 hover:text-orange-600 font-medium\",\"children\":\"浏览器信息\"}]]}],[\"$\",\"div\",null,{\"className\":\"flex items-center space-x-4 relative\",\"children\":[[\"$\",\"$L13\",null,{\"className\":\"$undefined\"}],[\"$\",\"$L14\",null,{}]]}]]}]}]}],[\"$\",\"div\",null,{\"className\":\"max-w-4xl mx-auto px-6 py-8\",\"children\":[[\"$\",\"div\",null,{\"className\":\"mb-6\",\"children\":[\"$\",\"$L15\",null,{}]}],[\"$\",\"article\",null,{\"className\":\"bg-white rounded-lg shadow-lg p-8 mb-8\",\"children\":[[\"$\",\"div\",null,{\"className\":\"mb-6 pb-6 border-b border-gray-200\",\"children\":[\"$\",\"div\",null,{\"className\":\"flex items-center gap-3 mb-3\",\"children\":[\"\",[\"$\",\"div\",null,{\"children\":[[\"$\",\"p\",null,{\"className\":\"text-sm font-medium text-gray-900\",\"children\":\"菲兹克斯喵\"}],[\"$\",\"p\",null,{\"className\":\"text-xs text-gray-500\",\"children\":\"菲兹克斯喵\"}],[\"$\",\"p\",null,{\"className\":\"text-xs text-gray-500\",\"children\":[[\"$\",\"span\",null,{\"children\":[\"马上订阅 \",\"菲兹克斯喵\",\" RSS 更新: \"]}],[\"$\",\"a\",null,{\"href\":\"https://physnya.top/atom.xml\",\"target\":\"_blank\",\"rel\":\"noopener noreferrer\",\"className\":\"text-orange-600 hover:text-orange-800 underline break-all\",\"children\":\"https://physnya.top/atom.xml\"}]]}]]}]]}]}],[\"$\",\"h1\",null,{\"className\":\"text-3xl font-bold text-gray-900 mb-4 leading-tight\",\"children\":\"Lesson 9 Hilbert 空间\"}],[\"$\",\"div\",null,{\"className\":\"flex flex-wrap items-center gap-4 text-sm text-gray-600 mb-6\",\"children\":[\"\",[\"$\",\"div\",null,{\"className\":\"flex items-center gap-2\",\"children\":[[\"$\",\"svg\",null,{\"className\":\"w-4 h-4\",\"fill\":\"none\",\"stroke\":\"currentColor\",\"viewBox\":\"0 0 24 24\",\"children\":[\"$\",\"path\",null,{\"strokeLinecap\":\"round\",\"strokeLinejoin\":\"round\",\"strokeWidth\":2,\"d\":\"M8 7V3m8 4V3m-9 8h10M5 21h14a2 2 0 002-2V7a2 2 0 00-2-2H5a2 2 0 00-2 2v12a2 2 0 002 2z\"}]}],[\"$\",\"span\",null,{\"children\":\"2025年10月21日 02:39\"}]]}]]}],\"$undefined\",[\"$\",\"div\",null,{\"className\":\"relative mb-8\",\"children\":[[\"$\",\"div\",null,{\"className\":\"prose max-w-none text-gray-700 leading-relaxed\",\"dangerouslySetInnerHTML\":{\"__html\":\"\u003cp\u003e这节课讲完要考期中考试了.\u003c/p\u003e\\n\u003cp\u003e期中考试的主要内容都是我们布置的第二章作业.\u003c/p\u003e\\n\u003ch2\u003eHilbert 空间\u003c/h2\u003e\\n\u003cp\u003e我们现在学的量子力学研究的都是纯态，可以通过一个波函数来描述这个系统的物理.\u003c/p\u003e\\n\u003cp\u003e我们定义：\u003c/p\u003e\\n\u003cp v-pre=\\\"\\\" class=\\\"katex-block\\\"\u003e\u003cspan class=\\\"katex-display\\\"\u003e\u003cspan class=\\\"katex\\\"\u003e\u003cspan class=\\\"katex-mathml\\\"\u003e\u003cmath xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"block\\\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmpadded\u003e\u003cmi mathvariant=\\\"normal\\\"\u003e∣...\"}}],[\"$\",\"div\",null,{\"className\":\"relative -mt-24 h-24 bg-gradient-to-b from-transparent via-white/80 to-white flex items-end justify-center pb-4\",\"children\":[\"$\",\"div\",null,{\"className\":\"text-center\",\"children\":[[\"$\",\"p\",null,{\"className\":\"text-sm text-gray-500 font-medium mb-2\",\"children\":\"剩余内容已隐藏\"}],[\"$\",\"div\",null,{\"className\":\"flex items-center gap-1 text-xs text-gray-400\",\"children\":[[\"$\",\"svg\",null,{\"className\":\"w-3 h-3\",\"fill\":\"none\",\"stroke\":\"currentColor\",\"viewBox\":\"0 0 24 24\",\"children\":[\"$\",\"path\",null,{\"strokeLinecap\":\"round\",\"strokeLinejoin\":\"round\",\"strokeWidth\":2,\"d\":\"M19 9l-7 7-7-7\"}]}],[\"$\",\"span\",null,{\"children\":\"查看完整文章以阅读更多\"}]]}]]}]}]]}],[\"$\",\"div\",null,{\"className\":\"pt-6 border-t border-gray-200\",\"children\":[\"$\",\"a\",null,{\"href\":\"https://physnya.top/quantum-mechanics/lesson-9/\",\"target\":\"_blank\",\"rel\":\"noopener noreferrer\",\"className\":\"inline-flex items-center gap-2 bg-orange-500 hover:bg-orange-600 text-white font-medium px-6 py-3 rounded-lg transition-colors duration-200 shadow-md hover:shadow-lg\",\"children\":[\"$L16\",\"$L17\"]}]}]]}]]}]]}]}]}]\n"])self.__next_f.push([1,"18:T6af,"])self.__next_f.push([1,"订阅更新(RSS/Atom): 这节课讲完要考期中考试了.\n期中考试的主要内容都是我们布置的第二章作业.\nHilbert 空间\n我们现在学的量子力学研究的都是纯态，可以通过一个波函数来描述这个系统的物理.\n我们定义：\n∣α⟩=(α1α2⋮αn) ,∣β⟩=(β1β2⋮βn)\\ket{\\alpha}=\\begin{pmatrix}\n\\alpha_1\\\\\\alpha_2\\\\\\vdots\\\\\\alpha_n\n\\end{pmatrix}\\,,\\quad\\ket{\\beta}=\\begin{pmatrix}\n\\beta_1\\\\\\beta_2\\\\\\vdots\\\\\\beta_n\n\\end{pmatrix}\n∣α⟩=\n​α1​α2​⋮αn​​\n​,∣β⟩=\n​β1​β2​⋮βn​​\n​\n定义内积为 ⟨α∣β⟩=αiβi\\braket{\\alpha|\\beta} = \\alpha_i\\beta_i⟨α∣β⟩=αi​βi​.\n平方可积的函数 (Square-Integrate) 生活于 Hilbert 空间里.\n/Example/\n对于函数 f(x)=xνf(x)=x^\\nuf(x)=xν，在 0∼10\\sim10∼1 的区间上平方可积：\n∫01∣xν∣2dx=12ν+1\\int_0^1|x^\\nu|^2\\text{d}x=\\frac{1}{2\\nu+1}\n∫01​∣xν∣2dx=2ν+11​\n当然要求 ν\u003e1/2\\nu\u003e1/2ν\u003e1/2.\nObservables\n什么样的量被认为是可观测的？\nHermite 算符：能够作用在两边，也就是 ⟨f∣Q^f⟩=⟨Q^f∣f⟩\\braket{f|\\hat{Q}f}=\\braket{\\hat{Q}f|f}⟨f∣Q^​f⟩=⟨Q^​f∣f⟩. 我们说可观测量能够被 Hermite 算符所表示.\nDeterminant State：\nσ2=⟨(Q^−⟨Q^⟩)2⟩=⟨Q^2⟩+⟨Q^⟩2−2⟨Q^⟩2=⟨Q^2⟩−⟨Q^⟩2\\sigma^2 = \\braket{(\\hat{Q}-\\braket{\\hat{Q}})^2}=\\braket{\\hat{Q}^2}+\\braket{\\hat{Q}}^2-2\\braket{\\hat{Q}}^2=\\braket{\\hat{Q}^2}-\\braket{\\hat{Q}}^2\nσ2=⟨(Q^​−⟨Q^​⟩)2⟩=⟨Q^​2⟩+⟨Q^​⟩2−2⟨Q^​⟩2=⟨Q^​2⟩−⟨Q^​⟩2\n是被测量的可观测量的本征态，这个被测量量不存在量子涨落.\n简并：多个态有同一个本征值."])self.__next_f.push([1,"19:T6af,"])self.__next_f.push([1,"订阅更新(RSS/Atom): 这节课讲完要考期中考试了.\n期中考试的主要内容都是我们布置的第二章作业.\nHilbert 空间\n我们现在学的量子力学研究的都是纯态，可以通过一个波函数来描述这个系统的物理.\n我们定义：\n∣α⟩=(α1α2⋮αn) ,∣β⟩=(β1β2⋮βn)\\ket{\\alpha}=\\begin{pmatrix}\n\\alpha_1\\\\\\alpha_2\\\\\\vdots\\\\\\alpha_n\n\\end{pmatrix}\\,,\\quad\\ket{\\beta}=\\begin{pmatrix}\n\\beta_1\\\\\\beta_2\\\\\\vdots\\\\\\beta_n\n\\end{pmatrix}\n∣α⟩=\n​α1​α2​⋮αn​​\n​,∣β⟩=\n​β1​β2​⋮βn​​\n​\n定义内积为 ⟨α∣β⟩=αiβi\\braket{\\alpha|\\beta} = \\alpha_i\\beta_i⟨α∣β⟩=αi​βi​.\n平方可积的函数 (Square-Integrate) 生活于 Hilbert 空间里.\n/Example/\n对于函数 f(x)=xνf(x)=x^\\nuf(x)=xν，在 0∼10\\sim10∼1 的区间上平方可积：\n∫01∣xν∣2dx=12ν+1\\int_0^1|x^\\nu|^2\\text{d}x=\\frac{1}{2\\nu+1}\n∫01​∣xν∣2dx=2ν+11​\n当然要求 ν\u003e1/2\\nu\u003e1/2ν\u003e1/2.\nObservables\n什么样的量被认为是可观测的？\nHermite 算符：能够作用在两边，也就是 ⟨f∣Q^f⟩=⟨Q^f∣f⟩\\braket{f|\\hat{Q}f}=\\braket{\\hat{Q}f|f}⟨f∣Q^​f⟩=⟨Q^​f∣f⟩. 我们说可观测量能够被 Hermite 算符所表示.\nDeterminant State：\nσ2=⟨(Q^−⟨Q^⟩)2⟩=⟨Q^2⟩+⟨Q^⟩2−2⟨Q^⟩2=⟨Q^2⟩−⟨Q^⟩2\\sigma^2 = \\braket{(\\hat{Q}-\\braket{\\hat{Q}})^2}=\\braket{\\hat{Q}^2}+\\braket{\\hat{Q}}^2-2\\braket{\\hat{Q}}^2=\\braket{\\hat{Q}^2}-\\braket{\\hat{Q}}^2\nσ2=⟨(Q^​−⟨Q^​⟩)2⟩=⟨Q^​2⟩+⟨Q^​⟩2−2⟨Q^​⟩2=⟨Q^​2⟩−⟨Q^​⟩2\n是被测量的可观测量的本征态，这个被测量量不存在量子涨落.\n简并：多个态有同一个本征值."])self.__next_f.push([1,"1a:T6af,"])self.__next_f.push([1,"订阅更新(RSS/Atom): 这节课讲完要考期中考试了.\n期中考试的主要内容都是我们布置的第二章作业.\nHilbert 空间\n我们现在学的量子力学研究的都是纯态，可以通过一个波函数来描述这个系统的物理.\n我们定义：\n∣α⟩=(α1α2⋮αn) ,∣β⟩=(β1β2⋮βn)\\ket{\\alpha}=\\begin{pmatrix}\n\\alpha_1\\\\\\alpha_2\\\\\\vdots\\\\\\alpha_n\n\\end{pmatrix}\\,,\\quad\\ket{\\beta}=\\begin{pmatrix}\n\\beta_1\\\\\\beta_2\\\\\\vdots\\\\\\beta_n\n\\end{pmatrix}\n∣α⟩=\n​α1​α2​⋮αn​​\n​,∣β⟩=\n​β1​β2​⋮βn​​\n​\n定义内积为 ⟨α∣β⟩=αiβi\\braket{\\alpha|\\beta} = \\alpha_i\\beta_i⟨α∣β⟩=αi​βi​.\n平方可积的函数 (Square-Integrate) 生活于 Hilbert 空间里.\n/Example/\n对于函数 f(x)=xνf(x)=x^\\nuf(x)=xν，在 0∼10\\sim10∼1 的区间上平方可积：\n∫01∣xν∣2dx=12ν+1\\int_0^1|x^\\nu|^2\\text{d}x=\\frac{1}{2\\nu+1}\n∫01​∣xν∣2dx=2ν+11​\n当然要求 ν\u003e1/2\\nu\u003e1/2ν\u003e1/2.\nObservables\n什么样的量被认为是可观测的？\nHermite 算符：能够作用在两边，也就是 ⟨f∣Q^f⟩=⟨Q^f∣f⟩\\braket{f|\\hat{Q}f}=\\braket{\\hat{Q}f|f}⟨f∣Q^​f⟩=⟨Q^​f∣f⟩. 我们说可观测量能够被 Hermite 算符所表示.\nDeterminant State：\nσ2=⟨(Q^−⟨Q^⟩)2⟩=⟨Q^2⟩+⟨Q^⟩2−2⟨Q^⟩2=⟨Q^2⟩−⟨Q^⟩2\\sigma^2 = \\braket{(\\hat{Q}-\\braket{\\hat{Q}})^2}=\\braket{\\hat{Q}^2}+\\braket{\\hat{Q}}^2-2\\braket{\\hat{Q}}^2=\\braket{\\hat{Q}^2}-\\braket{\\hat{Q}}^2\nσ2=⟨(Q^​−⟨Q^​⟩)2⟩=⟨Q^​2⟩+⟨Q^​⟩2−2⟨Q^​⟩2=⟨Q^​2⟩−⟨Q^​⟩2\n是被测量的可观测量的本征态，这个被测量量不存在量子涨落.\n简并：多个态有同一个本征值."])self.__next_f.push([1,"9:{\"metadata\":[[\"$\",\"title\",\"0\",{\"children\":\"Lesson 9 Hilbert 空间 | 菲兹克斯喵 | 柑橘 RSS 阅读器\"}],[\"$\",\"meta\",\"1\",{\"name\":\"description\",\"content\":\"$18\"}],[\"$\",\"meta\",\"2\",{\"name\":\"keywords\",\"content\":\":blog\"}],[\"$\",\"meta\",\"3\",{\"name\":\"robots\",\"content\":\"index, follow\"}],[\"$\",\"meta\",\"4\",{\"property\":\"og:title\",\"content\":\"Lesson 9 Hilbert 空间\"}],[\"$\",\"meta\",\"5\",{\"property\":\"og:description\",\"content\":\"$19\"}],[\"$\",\"meta\",\"6\",{\"property\":\"og:site_name\",\"content\":\"Citrus\"}],[\"$\",\"meta\",\"7\",{\"property\":\"og:type\",\"content\":\"article\"}],[\"$\",\"meta\",\"8\",{\"property\":\"article:published_time\",\"content\":\"2025-10-20T18:39:36.000Z\"}],[\"$\",\"meta\",\"9\",{\"name\":\"twitter:card\",\"content\":\"summary_large_image\"}],[\"$\",\"meta\",\"10\",{\"name\":\"twitter:title\",\"content\":\"Lesson 9 Hilbert 空间\"}],[\"$\",\"meta\",\"11\",{\"name\":\"twitter:description\",\"content\":\"$1a\"}],\"$L1b\",\"$L1c\"],\"error\":null,\"digest\":\"$undefined\"}\n"])self.__next_f.push([1,"e:\"$9:metadata\"\n"])self.__next_f.push([1,"1d:I[27201,[\"/_next/static/chunks/ff1a16fafef87110.js\",\"/_next/static/chunks/7dd66bdf8a7e5707.js\"],\"IconMark\"]\n16:[\"$\",\"span\",null,{\"children\":\"查看完整文章\"}]\n17:[\"$\",\"svg\",null,{\"className\":\"w-4 h-4\",\"fill\":\"none\",\"stroke\":\"currentColor\",\"viewBox\":\"0 0 24 24\",\"children\":[\"$\",\"path\",null,{\"strokeLinecap\":\"round\",\"strokeLinejoin\":\"round\",\"strokeWidth\":2,\"d\":\"M10 6H6a2 2 0 00-2 2v10a2 2 0 002 2h10a2 2 0 002-2v-4M14 4h6m0 0v6m0-6L10 14\"}]}]\n1b:[\"$\",\"link\",\"12\",{\"rel\":\"icon\",\"href\":\"/favicon.ico?favicon.0b3bf435.ico\",\"sizes\":\"256x256\",\"type\":\"image/x-icon\"}]\n1c:[\"$\",\"$L1d\",\"13\",{}]\n"])