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        "quote": "而 $v$ 称为 $S$ 对应于特征值 $\\lambda$ 的一个 <u id=\"concept-4-eigenvector\" class=\"concept-term\" data-concept-identity=\"urn:uuid:19c6e3d7-39e1-4c32-8812-3bd9d8a9958e\" data-concept-route=\"/concept-identities#concept-19c6e3d7-39e1-4c32-8812-3bd9d8a9958e\">特征向量(eigenvector)</u>。特征向量是那些在算符 $S$ 作用下被“拉伸”了 $\\lambda$ 的非零向量。必须规定 $v \\neq 0$，因为方程 <a href=\"#equation-4-6\" class=\"cross-reference\" id=\"xref-01-eigenspaces-and-characteristic-equations-3\" data-reference=\"equation-4-6\">Eq. (4.6)</a> 对零向量 ${ \\cal S } 0 = 0 = \\lambda 0 .$ 总是成立。",
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        "quote": "复数$\\alpha$称为其<u id=\"concept-13-eigenvalue\" class=\"concept-term\" data-concept-identity=\"urn:uuid:76dd2837-afb9-4545-a9b3-37d92e61b023\" data-concept-route=\"/concept-identities#concept-76dd2837-afb9-4545-a9b3-37d92e61b023\">特征值(eigenvalue)</u>，$u$ 称为 $A$ 对应于特征值$\\alpha$的<u id=\"concept-13-eigenvector\" class=\"concept-term\" data-concept-identity=\"urn:uuid:19c6e3d7-39e1-4c32-8812-3bd9d8a9958e\" data-concept-route=\"/concept-identities#concept-19c6e3d7-39e1-4c32-8812-3bd9d8a9958e\">特征向量(eigenvector)</u>，这在第四章我们都已经做过了。",
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        "quote": "$u$ is called the <u class=\"concept-term\" data-concept-identity=\"urn:uuid:19c6e3d7-39e1-4c32-8812-3bd9d8a9958e\" data-concept-route=\"/concept-identities#concept-19c6e3d7-39e1-4c32-8812-3bd9d8a9958e\" id=\"first-6e51c37775e465f0897f-en\">eigenvector</u> of $A$ corresponding to the eigenvalue $\\alpha$.",
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        "quote": "\n> [!theorem] 原理 · 可观测量\n> <span id=\"source-block-1041\" class=\"block-anchor\"></span>\n>\n> 若一个态对应的可观测量具有确定数值，则该态是相应自伴算符的<u class=\"concept-term\" data-concept-identity=\"urn:uuid:19c6e3d7-39e1-4c32-8812-3bd9d8a9958e\" data-concept-route=\"/concept-identities#concept-19c6e3d7-39e1-4c32-8812-3bd9d8a9958e\" id=\"first-84bccf053b8b34d182e5-zh\">特征向量(eigenvector)</u>。该可观测量在此态中的取值是实数，也就是该算符的<u class=\"concept-term\" data-concept-identity=\"urn:uuid:76dd2837-afb9-4545-a9b3-37d92e61b023\" data-concept-route=\"/concept-identities#concept-76dd2837-afb9-4545-a9b3-37d92e61b023\" id=\"first-db508acb9e0ea7c4aab8-zh\">特征值(eigenvalue)</u>。\n",
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        "quote": "回顾量子理论关于测量的基本原理：只有当[<u class=\"concept-term\" data-concept-reference=\"concepts/quantum-mechanics/quantum-state\" data-concept-identity=\"urn:uuid:8a86f3a5-8b52-4b89-ab39-b297da0ffe2d\" data-concept-route=\"/concept-identities#concept-8a86f3a5-8b52-4b89-ab39-b297da0ffe2d\">量子态(quantum state)</u>](../../../../../concepts/quantum-mechanics/quantum-state) 是相应矩阵的[<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/eigenvector\" data-concept-identity=\"urn:uuid:19c6e3d7-39e1-4c32-8812-3bd9d8a9958e\" data-concept-route=\"/concept-identities#concept-19c6e3d7-39e1-4c32-8812-3bd9d8a9958e\">特征向量(eigenvector)</u>](../../../../../concepts/linear-algebra/eigenvector) 时，相关可观测量才有确定值；测量值就是相应矩阵的[<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/eigenvalue\" data-concept-identity=\"urn:uuid:76dd2837-afb9-4545-a9b3-37d92e61b023\" data-concept-route=\"/concept-identities#concept-76dd2837-afb9-4545-a9b3-37d92e61b023\">特征值(eigenvalue)</u>](../../../../../concepts/linear-algebra/eigenvalue)。由于算符自伴，特征值为实数。第一个矩阵对应平凡的可观测量（恒等算符，对每个态都取相同值），而最后一个矩阵 $\\sigma_3$ 有两个特征向量：",
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      "chapter": "4",
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      "label": "特征向量(eigenvectors)",
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        "quote": "给定 $L$，其[<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/eigenvalue\" data-concept-identity=\"urn:uuid:76dd2837-afb9-4545-a9b3-37d92e61b023\" data-concept-route=\"/concept-identities#concept-76dd2837-afb9-4545-a9b3-37d92e61b023\" id=\"first-9396e5c51e389e4fcafd-zh\">特征值(eigenvalues)</u>](../../../../../concepts/linear-algebra/eigenvalue) $\\lambda_j$ 是[特征方程 (4.5)](07-eigenvalues-and-eigenvectors#equation-4-5) 的根；酉矩阵 $U$ 由相应的[<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/eigenvector\" data-concept-identity=\"urn:uuid:19c6e3d7-39e1-4c32-8812-3bd9d8a9958e\" data-concept-route=\"/concept-identities#concept-19c6e3d7-39e1-4c32-8812-3bd9d8a9958e\" id=\"first-b8509e0d59e70e8efc9b-zh\">特征向量(eigenvectors)</u>](../../../../../concepts/linear-algebra/eigenvector)确定。不同特征值对应的特征向量彼此正交。",
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        "quote": "在 $\\mathcal H=\\mathbf C^2$ 中，具有可观测量 $S_j$ 确定值的[<u id=\"concept-7-quantum-state\" class=\"concept-term\" data-concept-identity=\"urn:uuid:8a86f3a5-8b52-4b89-ab39-b297da0ffe2d\" data-concept-route=\"/concept-identities#concept-8a86f3a5-8b52-4b89-ab39-b297da0ffe2d\">量子态(quantum state)</u>](../../../../../concepts/quantum-mechanics/quantum-state)是 $S_j$ 的[<u id=\"concept-7-eigenvector\" class=\"concept-term\" data-concept-identity=\"urn:uuid:19c6e3d7-39e1-4c32-8812-3bd9d8a9958e\" data-concept-route=\"/concept-identities#concept-19c6e3d7-39e1-4c32-8812-3bd9d8a9958e\">特征向量(eigenvector)</u>](../../../../../concepts/linear-algebra/eigenvector)；相应的测量值就是其特征值，即 $+\\frac{1}{2}$ 或 $-\\frac{1}{2}$。测量理论假设，若我们对任意态$| \\psi \\rangle$进行对应于$S _ { j }$的测量，那么我们将会",
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      "label": "eigenvectors",
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        "quote": "[<u class=\"concept-term\" data-concept-identity=\"urn:uuid:6bb96444-d087-4f5c-8398-4f3dc18cac96\" data-concept-route=\"/concept-identities#concept-6bb96444-d087-4f5c-8398-4f3dc18cac96\" id=\"first-3608a5d0031de8861919-en\">States</u>](../../../../../concepts/quantum-mechanics/quantum-state) in $\\mathcal H=\\mathbf C^2$ with a definite value of the observable $S_j$ are [<u class=\"concept-term\" data-concept-identity=\"urn:uuid:19c6e3d7-39e1-4c32-8812-3bd9d8a9958e\" data-concept-route=\"/concept-identities#concept-19c6e3d7-39e1-4c32-8812-3bd9d8a9958e\" id=\"first-b8509e0d59e70e8efc9b-en\">eigenvectors</u>](../../../../../concepts/linear-algebra/eigenvector) of $S_j$. The corresponding measurement value is its eigenvalue, either $+\\frac{1}{2}$ or $-\\frac{1}{2}$. Measurement theory postulates that if we perform the measurement corresponding to $S _ { j }$ on an arbitrary state $| \\psi \\rangle$ , then we will",
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