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"quote": "回顾量子理论关于测量的基本原理：只有当[量子态(quantum state)](../../../../../concepts/quantum-mechanics/quantum-state) 是相应矩阵的[特征向量(eigenvector)](../../../../../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[特征值(eigenvalue)](../../../../../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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"quote": "给定 $L$，其[特征值(eigenvalues)](../../../../../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[特征向量(eigenvectors)](../../../../../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": "量子力学教材有时会指出，希尔伯特空间 $\\mathcal H$ 中物理态的表示方式存在一种歧义，以此解释这一现象：将 $\\mathcal H$ 中的向量乘以一个标量，不会改变算符的[特征值(eigenvalue)](../../../../../concepts/linear-algebra/eigenvalue)，也不会改变测得这些特征值的相对概率。因此，上述符号歧义没有物理后果；可以说，物理态应视为模去这一标量歧义的状态。不过，相较于使用射影表示，直接采用更大的群 $\\mathrm{Spin}(3)$ 似乎更清楚；作者接受它正是描述旋转对三维量子系统作用的恰当群。",
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"quote": "Quantum mechanics texts sometimes address this phenomenon by noting that a physical state in $\\mathcal H$ is specified only up to a scalar: multiplying a vector in $\\mathcal H$ by a scalar does not change the [eigenvalues](../../../../../concepts/linear-algebra/eigenvalue) of operators or the relative probabilities of observing those eigenvalues. Consequently, the sign ambiguity above has no physical effect; one can regard states as defined modulo this scalar ambiguity. It seems more straightforward, however, to avoid projective representations and use the larger group $\\mathrm{Spin}(3)$, accepting it as the group that correctly describes the action of rotations on three-dimensional quantum systems.",
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"quote": "一般地，对于反自伴算符（即从酉李代数表示中得到的算符，也是指数化后得到酉算符的算符），我们将通过乘以 $i$ 来关联一个自伴算符。这些自伴算符的[特征值(eigenvalue)](../../../../../concepts/linear-algebra/eigenvalue)均为实数（此处为 $\\pm \\frac{1}{2}$），因此物理学家倾向于将它们作为可观测量，因为测量结果是实数。在相反方向上，给定物理学家的可观测量自伴算符，我们将乘以 −$i$ 以得到一个反自伴算符（它可能是某个酉李代数表示的算符）。",
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