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        "quote": "\n> [!theorem] 公理 · <u id=\"concept-1-quantum-observable\" class=\"concept-term\" data-concept-id=\"concept-1-quantum-observable\" data-concept-identity=\"urn:uuid:1f30c262-f238-4dfd-acec-55532f758890\" data-concept-route=\"/concept-identities#concept-1f30c262-f238-4dfd-acec-55532f758890\">量子可观测量(quantum observable)</u>\n> <span id=\"source-block-1023\" class=\"block-anchor\"></span>\n>\n> 量子力学系统的可观测量由 $\\mathcal{H}$ 上的自伴线性算符给出。\n",
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        "quote": "因此，至少在有限维态空间 $\\mathcal{H}$ 的情形下，由群 $G$ 在物理系统上的作用所诱导的 $G$ 在 $\\mathcal{H}$ 上的酉表示 $\\pi$，不仅给出酉矩阵 $\\pi(g)$，还给出相应的<u id=\"concept-1-self-adjoint-operator\" class=\"concept-term\" data-concept-id=\"concept-1-self-adjoint-operator\" data-concept-role=\"discussion\" data-concept-identity=\"urn:uuid:d5f9da17-e94f-479d-90f0-7732d12ee03a\" data-concept-route=\"/concept-identities#concept-d5f9da17-e94f-479d-90f0-7732d12ee03a\">自伴算符(self-adjoint operator)</u> $B$。李群作用因此为我们提供了一类[<u class=\"concept-term\" data-concept-reference=\"concepts/quantum-mechanics/quantum-observable\" data-concept-identity=\"urn:uuid:1f30c262-f238-4dfd-acec-55532f758890\" data-concept-route=\"/concept-identities#concept-1f30c262-f238-4dfd-acec-55532f758890\">量子可观测量(quantum observable)</u>](../../../../../concepts/quantum-mechanics/quantum-observable)；这些算符的自伴性恰好对应着态空间上表示的酉性。令人惊讶的是，在许多物理系统中，以这种方式得到的可观测量包括物理上最重要的那些。",
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        "quote": "对于态空间为二维的量子系统 $\\mathcal H=\\mathbf C^2$，[<u class=\"concept-term\" data-concept-reference=\"concepts/quantum-mechanics/quantum-observable\" data-concept-identity=\"urn:uuid:1f30c262-f238-4dfd-acec-55532f758890\" data-concept-route=\"/concept-identities#concept-1f30c262-f238-4dfd-acec-55532f758890\">可观测量(quantum observable)</u>](../../../../../concepts/quantum-mechanics/quantum-observable) 是 $\\mathbf C^2$ 上的 [<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/adjoint-operator\" data-concept-identity=\"urn:uuid:a7971b9e-8f7b-49a0-a412-0ae28a65c35b\" data-concept-route=\"/concept-identities#concept-a7971b9e-8f7b-49a0-a412-0ae28a65c35b\">自伴线性算符(self-adjoint linear operator)</u>](../../../../../concepts/linear-algebra/adjoint-operator)。选定一组基后，这些算符可表示为满足 $M=M^{\\dagger}$ 的 $2\\times2$ 复矩阵 $M$（其中 $M^{\\dagger}$ 是 $M$ 的共轭转置）。每个这样的矩阵都可写成以下四个矩阵的实线性组合：",
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        "quote": "在[第3](../chapter-03/index#section-3)章中我们看到，一个具有$\\mathcal { H } = \\mathbf { C } ^ { 2 }$的一般量子系统可以通过$U ( 2 )$在$\\mathbf { C } ^ { 2 }$上的作用来理解。自伴算符所表示的[<u id=\"concept-7-quantum-observable\" class=\"concept-term\" data-concept-identity=\"urn:uuid:1f30c262-f238-4dfd-acec-55532f758890\" data-concept-route=\"/concept-identities#concept-1f30c262-f238-4dfd-acec-55532f758890\">可观测量(quantum observable)</u>](../../../../../concepts/quantum-mechanics/quantum-observable)对应于相应的[<u id=\"concept-7-lie-algebra-representation\" class=\"concept-term\" data-concept-identity=\"urn:uuid:5bad1df2-281a-4c11-8362-128d54fb523a\" data-concept-route=\"/concept-identities#concept-5bad1df2-281a-4c11-8362-128d54fb523a\">李代数表示(Lie algebra representation)</u>](../../../../../concepts/lie-theory/lie-algebra-representation)（相差一个因子 $i$）。$U ( 1 ) \\subset U ( 2 )$子群与所有其他部分对易，可以单独分析，这里我们只考虑$S U ( 2 )$子群。对于任意这样的系统，群$S U ( 2 )$没有特别的几何意义。当它作为旋转群的双覆盖出现时，称该量子系统携带[<u id=\"concept-7-spin\" class=\"concept-term\" data-concept-identity=\"urn:uuid:3e11a671-c004-44f2-8d6e-bf7933e08bbb\" data-concept-route=\"/concept-identities#concept-3e11a671-c004-44f2-8d6e-bf7933e08bbb\">自旋(spin)</u>](../../../../../concepts/quantum-mechanics/spin)；特别是，二维[<u id=\"concept-7-irreducible-representation\" class=\"concept-term\" data-concept-identity=\"urn:uuid:4e9e2c89-0422-42cf-8762-6b5e59837ef0\" data-concept-route=\"/concept-identities#concept-4e9e2c89-0422-42cf-8762-6b5e59837ef0\">不可约表示(irreducible representation)</u>](../../../../../concepts/representation-theory/irreducible-representation)对应自旋 $1/2$（在[第8](../chapter-08/index#section-8)章中我们将讨论更高自旋值的态空间）。",
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