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"quote": "> [!example] 例 2.15 \n>\n> 复矩阵 $\\mathsf{A}$ 的伴随矩阵(adjoint)（共轭转置）定义为 $\\mathsf{A}^{\\dagger}=\\overline{\\mathsf{A}^{T}}$，分量为 $a_{ij}^{\\dagger}=\\overline{a_{ji}}$；复数上的横线表示复共轭。若 $n\\times n$ 复矩阵 $\\mathsf{U}$ 满足\n>\n> \n>\n> $$\n> \\mathsf{U}\\mathsf{U}^{\\dagger}=\\mathsf{I},\\tag{2.12}\n> $$\n>\n> 则称它为酉矩阵(unitary matrix)。立即可得\n>\n> $$\n> \\det\\mathsf{U}\\,\\overline{\\det\\mathsf{U}}=|\\det\\mathsf{U}|^{2}=1.\n> $$\n>\n> 因而存在实数 $\\phi\\in[0,2\\pi)$，使 $\\det\\mathsf{U}=e^{i\\phi}$。所有酉矩阵都非奇异，群性质也容易验证。全体 $n\\times n$ 酉矩阵组成$n$ 阶酉群(unitary group)，记作 $U(n)$。其中行列式为 $1$ 的酉矩阵组成$n$ 阶特殊酉群(special unitary group)，记作 $SU(n)$。",
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"quote": "在本章中，我们将分析具有 $\\mathcal { H } = \\mathbf { C } ^ { 2 }$ 的一般量子系统会发生什么，首先找出可能的可观测量。将这些可观测量取指数，将得到作用在 $\\mathcal { H } = \\mathbf { C } ^ { 2 }$ 上的 $2\\times2$ 酉矩阵群 $U(2)$。这是 $U(2)$ 的一个具体表示，即“定义”表示。将范围限制到 $U(2)$ 中行列式为 $1$ 的子群[特殊酉群(special unitary group)](../../../../../concepts/linear-algebra/special-unitary-group) $SU(2)$，就得到 $SU(2)$ 在 $\\mathbf { C } ^ { 2 }$ 上的一个表示，通常称为“自旋 $\\frac{1}{2}$”表示。",
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"quote": "In this chapter we will analyze what happens for general quantum systems with $\\mathcal { H } = \\mathbf { C } ^ { 2 }$ by first finding the possible observables. Exponentiating these will give the group $U ( 2 )$ of unitary 2 by 2 matrices acting on $\\mathcal { H } = \\mathbf { C } ^ { 2 }$ . This is a specific representation of $U ( 2 )$ , the “defining” representation. By restricting to the subgroup [special unitary group](../../../../../concepts/linear-algebra/special-unitary-group) $SU( 2 ) \\subset U ( 2 )$ of elements of determinant one, one gets a representation of $SU(2)$ on $\\mathbf { C } ^ { 2 }$ often called the “spin $\\frac{1}{2}$” representation.",
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"quote": "对于另外两种情形 $j=1$ 和 $j=2$，通过基变换，二者之一都可以化为相同的对角形式；但对某个 $j$ 做此变换，会使另外两种情形不再对角。要理解[特殊酉群(special unitary group)](../../../../../concepts/linear-algebra/special-unitary-group) $SU(2)$ 在 $\\mathcal{H}$ 上的作用，不能只考虑 $U(1)$ 子群，还要考虑通过将泡利矩阵的一般线性组合取指数而得到的完整三维 $SU(2)$ 群。",
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"quote": "For the other two cases $j = 1$ and $j = 2$, a change of basis can put either one in the same diagonal form, but doing this for one value of $j$ makes the other two no longer diagonal. To understand the action of the [special unitary group](../../../../../concepts/linear-algebra/special-unitary-group) $SU( 2 )$ on $\\mathcal { H }$, we must consider not just the $U ( 1 )$ subgroups but the full three-dimensional $SU( 2 )$ group obtained by exponentiating general linear combinations of Pauli matrices.",
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"quote": "因此，对自伴泡利矩阵的实线性组合乘以 $i$ 后取指数，会得到行列式为 1 的酉矩阵。它们可逆，并构成一个名为[特殊酉群(special unitary group)](../../../../../concepts/linear-algebra/special-unitary-group)（记作 $SU(2)$）的群，也就是由行列式为 1 的 $2\\times2$ 酉矩阵构成的群。若对 $i(\\phi\\mathbf 1+\\theta\\mathbf v\\cdot\\boldsymbol{\\sigma})$ 取指数，其中 $\\phi$ 为实数（当且仅当 $\\phi=0$ 时，该矩阵的迹为零），则会得到行列式为 $e^{i2\\phi}$ 的酉矩阵。全体 $2\\times2$ 酉矩阵构成二阶酉群(unitary group in dimension two)（记作 $U(2)$）。它包含 $SU(2)$ 以及本节开头介绍的 $U(1)$ 作为子群。$U(2)$ 与这两个子群的直积略有不同，因为 $-\\mathbf 1$ 同时属于两个子群：",
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"quote": "So, we see that by exponentiating $i$ times linear combinations of the self-adjoint Pauli matrices (which all have trace zero), we get unitary matrices of determinant one. These are invertible, and form the [special unitary group](../../../../../concepts/linear-algebra/special-unitary-group) $SU(2)$ , the group of unitary 2 by 2 matrices of determinant one. If we exponentiated not just $i \\theta \\mathbf { v } \\cdot { \\pmb \\sigma } .$ , but $i ( \\phi \\mathbf { 1 } + \\theta \\mathbf { v } \\cdot \\pmb { \\sigma } )$ for some real constant $\\phi$ (such matrices will not have trace zero unless $\\phi = 0$) , we would get a unitary matrix with determinant $e ^ { i 2 \\phi }$ . The group of all unitary 2 by 2 matrices is called $U ( 2 )$ . It contains as subgroups $SU( 2 )$ as well as the $U ( 1 )$ described at the beginning of this section. $U ( 2 )$ is slightly different from the product of these two subgroups, since the group element",
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"quote": "$H$ 中的 $h_0\\mathbf{1}$ 项贡献一个整体相位因子(overall phase factor) $e^{-ih_0t}$。提出这一相位后，$U(t)$ 的剩余部分属于[特殊酉群(special unitary group)](01-the-two-state-quantum-system#concept-3-special-unitary-group) $SU(2)$；它是更大的[二阶酉群(unitary group in dimension two)](01-the-two-state-quantum-system#concept-3-unitary-group-in-dimension-two) $U(2)$ 的子群，而 $U(2)$ 由所有 $2\\times2$ 酉矩阵构成。",
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"quote": "The $h_0\\mathbf{1}$ term in $H$ contributes an overall phase factor $e^{-ih_0t}$. After this phase is factored out, the remaining part of $U(t)$ lies in the [special unitary group](01-the-two-state-quantum-system#source-block-3071) $SU(2)$, a subgroup of the larger [unitary group in dimension two](01-the-two-state-quantum-system#source-block-3071) $U(2)$, whose elements are all $2\\times2$ unitary matrices.",
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"quote": "> [!recall]- 回顾：特殊酉群 → [原定义](../chapter-03/01-the-two-state-quantum-system#concept-3-special-unitary-group) \n> - [特殊酉群(special unitary group)](../../../../../concepts/linear-algebra/special-unitary-group) $SU(n)\\subset SL(n,\\mathbf{C})$ 是 $U(n)$ 中行列式为 $1$ 的矩阵构成的子群。其李代数 $\\mathfrak{su}(n)$ 由迹为零的 $n\\times n$ 斜厄米矩阵构成。\n>\n> \n>\n> ",
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"quote": "> [!recall]- Recall: special unitary group → [original definition](../chapter-03/01-the-two-state-quantum-system#concept-3-special-unitary-group) \n> - The [special unitary group](../../../../../concepts/linear-algebra/special-unitary-group) $SU(n)\\subset SL(n,\\mathbf{C})$ is the subgroup of $U(n)$ consisting of matrices with determinant one. Its Lie algebra $\\mathfrak{su}(n)$ consists of $n\\times n$ skew-Hermitian matrices with trace zero.\n>\n> \n>\n> ",
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