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        "quote": "> [!example] <a class=\"object-permalink\" href=\"#example-13-17\">例 13.17</a> \n>\n> 在平方可积函数等价类的希尔伯特空间 $L^{2}(\\mathbb{R})$ 中（见<a href=\"01-definitions-and-examples#example-13-4\" class=\"cross-reference\" id=\"xref-06-unbounded-operators-1\" data-reference=\"example-13-4\">例13.4</a>）， 设$D$为$\\mathbb{R}$上具紧支集的$C^{\\infty}$函数$\\varphi$的子空间，其元素记作$\\widetilde{\\varphi}$。这本质上是第 12 章中定义的测试函数空间 $\\mathcal{D}^{\\infty}(\\mathbb{R})$。参考<a href=\"02-expansion-theorems#example-13-5\" class=\"cross-reference\" id=\"xref-06-unbounded-operators-2\" data-reference=\"example-13-5\">例 13.5</a> 中的论述表明 $D$ 是 $L^{2}(\\mathbb{R})$ 的稠密子空间。我们定义<u id=\"concept-13-position-operator\" class=\"concept-term\" data-concept-identity=\"urn:uuid:05d69eba-f971-4eda-9264-d2a2b9cb71b1\" data-concept-route=\"/concept-identities#concept-05d69eba-f971-4eda-9264-d2a2b9cb71b1\">位置算符(position operator)</u> $Q: D \\rightarrow D \\subset L^{2}(\\mathbb{R})$ 由 $Q \\widetilde{\\varphi}=\\widetilde{x \\varphi}$。我们可以更非正式地写成\n>\n> $$\n> (Q \\varphi)(x)=x \\varphi(x)\n> $$\n>\n> 类似地，<u id=\"concept-13-momentum-operator\" class=\"concept-term\" data-concept-identity=\"urn:uuid:34c7e92b-ccec-4288-a000-265917ff0ffb\" data-concept-route=\"/concept-identities#concept-34c7e92b-ccec-4288-a000-265917ff0ffb\">动量算符(momentum operator)</u> $P: D \\rightarrow D$ 定义为\n>\n> $$\n> P \\varphi(x)=-i \\frac{\\mathrm{d}}{\\mathrm{d} x} \\varphi(x)\n> $$\n>\n> 这两个算符在其定义域上显然都是线性的。",
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