# 量子演化系统微分几何概念札记(I)：几何相联络、规范势、度规与曲率张量A Note on the Differential Geometry Concepts in Quantum Evolutional Systems (I): Geometric-Phase Connection, Gauge Potentials, Metric and Curvature Tensor

DOI: 10.12677/MP.2019.96029, PDF, HTML, XML, 下载: 299  浏览: 482

Abstract: Quantum evolutional systems can be defined as systems/models of interacting Hamiltonian operators with certain evolutional parameters. Quantum evolutional systems carry global or topological characteristics, exhibiting some geometric effects and phenomena. Based on quantum mechanics and electromagnetic gauge theory, we study some topics of the applications of differential geometry concepts in such systems. The brief history of quantum evolutional systems and geometric effects are reviewed with emphasis on the geometric phase and gauge potential (affine connection). The properties of the unitary transformation (related to the Lewis-Riesenfeld invariant formalism) are compared with vielbein fields in a manifold of differential geometry, and it can be found that such a unitary transformation operator can be identified as a vielbein field in gauge group space. The methods of calculating the state functions and geometric phases of the quantum evolutional systems as well as the manifold metric of parameter space in the evolutional Hamiltonian systems will be addressed. As a pedagogical note, the content of this paper would find an application in understanding the topics relevant to classical electromagfnetism, quantum optics, constrained quantum system dynamics, various gauge field theories and interdisciplinary researches.

1. 引言

(dual mass)。我们把通常的具有质量的物质称为引力电性物质，质量就是引力电荷量，其引力场分布由爱因斯坦引力场方程决定。与此相对应,我们可以提出引力磁性物质和引力磁荷(对偶质量)的概念，并建立引力磁荷的引力场方程。从对引力场方程的弱场低速近似形式和对偶方程 ${\epsilon }_{\mu \lambda \rho \tau }{R}^{\lambda \rho \tau }{}_{\nu }={t}_{\mu \nu }$ 静态的球对称精确解的分析可以看出，引力磁荷的存在与引力场的度规 ${g}_{\mu \nu }$ 的畸点(非解析性)有关。引力磁荷(对偶质量)与引力电荷(质量)不同，它是时空的拓扑荷，因此引力磁荷的引力场方程和运动方程的弱场近似形式类似于电动力学中磁单极子的电磁场方程和磁单极子运动方程。与电动力学中的磁荷一样，引力磁荷在宇宙中倒底是否确实存在,其实难以回答。也许引力磁荷可能以点粒子形式存在，也有可能以一定体积(有限大小)的所谓拓扑性孤粒子(topological soliton) [7] [8] 形式存在(类似在非阿贝尔规范理论中’t Hooft-Polyakov拓扑磁荷孤粒子 [7] [8] )。

[11]。在Berry之前，很多人对这个相位不以为然，认为它不重要，可以略去 [11]。但Berry发现，这是一个非平庸的相位，它在循环(巡回)条件下，可以写为Aharonov-Bohm相位那种闭合环路积分形式 [9]。

$i〈\psi |\frac{\partial }{\partial l}|\psi 〉$ 的闭合路径积分之后，相位 $\gamma$ 被称呼为Berry相位(Berry相位是一种绝热循环几何相位 [9] [10]

[11]。所谓循环条件，是指哈密顿量算符经过一段时间演化后，最终可以复归为初始哈密顿量)。而

$\mathrm{exp}\left(-i\int E\text{d}t/\hslash \right)$ 中的 $\int E\text{d}t/\hslash$ 只是一个普通的动力学相位。

$\mathrm{exp}\left(i\alpha \right)$ 中的参量 $\alpha$ 是一个解析函数，根据Stokes定理， $\nabla \alpha$ 的闭合路径积分为零(即 ${\int }_{c}\text{d}l\cdot \nabla \alpha =0$ )，所以Berry相位 $\gamma =i{\int }_{c}〈\psi |\nabla |\psi 〉\cdot \text{d}l$ 具有规范不变性(只要有规范不变性，自然就有物理含义了，也即实验上

2. 演化系统理论研究方法

[9] [10] [11]，状态函数 $|\psi 〉$ 是瞬时定态Schrödinger方程 $\stackrel{^}{H}\left(t\right)|\psi 〉=E|\psi 〉$ 的瞬时本征态。但在这里要指出，“瞬时定态Schrödinger方程”和“瞬时本征态”这两个概念在本质上是错误的。它们是不应该存在的概念。实际上，一旦哈密顿量算符含时，瞬时定态Schrödinger方程 $\stackrel{^}{H}\left(t\right)|\psi 〉=E|\psi 〉$ 便不再严格成立，只有

$\stackrel{^}{H}\left(t\right)|\Psi 〉=i\hslash \frac{\partial }{\partial t}|\Psi 〉$，改求某个不变量 $\stackrel{^}{I}\left(t\right)$ 的本征值方程 $\stackrel{^}{I}\left(t\right)|\psi 〉=\sigma |\psi 〉$，而Schrödinger方程的解 $|\Psi 〉$

[24]。Lewis-Riesenfeld不变量 $\stackrel{^}{I}\left(t\right)$ 满足(或定义为)： $\frac{\partial \stackrel{^}{I}\left(t\right)}{\partial t}+\frac{1}{i\hslash }\left[\stackrel{^}{I},\stackrel{^}{H}\right]=0$ [24]。此方程右边为零，体现

$\stackrel{^}{H}\left(t\right)|\Psi 〉=i\hslash \frac{\partial }{\partial t}|\Psi 〉$ 的特解为 $|{\Psi }_{\sigma }\left(t\right)〉=\mathrm{exp}\left(\frac{1}{i}{\Phi }_{\sigma }\right)|{\psi }_{\sigma }\left(t\right)〉$，相位

${\Phi }_{\sigma }\left(t\right)=\frac{1}{\hslash }{\int }_{0}^{t}〈{\psi }_{\sigma }\left({t}^{\prime }\right)|\left[\stackrel{^}{H}\left({t}^{\prime }\right)-i\hslash \frac{\partial }{\partial {t}^{\prime }}\right]|{\psi }_{\sigma }\left({t}^{\prime }\right)〉\text{d}{t}^{\prime }$，其中 $\frac{1}{\hslash }{\int }_{0}^{t}〈{\psi }_{\sigma }\left({t}^{\prime }\right)|\stackrel{^}{H}\left({t}^{\prime }\right)|{\psi }_{\sigma }\left({t}^{\prime }\right)〉\text{d}{t}^{\prime }$ 是普通的动力学相位， $\frac{1}{\hslash }{\int }_{0}^{t}〈{\psi }_{\sigma }\left({t}^{\prime }\right)|\left[-i\hslash \frac{\partial }{\partial {t}^{\prime }}\right]|{\psi }_{\sigma }\left({t}^{\prime }\right)〉\text{d}{t}^{\prime }$ 是非循环、非绝热过程中呈现几何特性的相位 [24]。含时Schrödinger方程

$\stackrel{^}{I}\left(t\right)|\psi 〉=\sigma |\psi 〉$ [24] 合在一起时具有含时Schrödinger方程 $\stackrel{^}{H}\left(t\right)|\Psi 〉=i\hslash \frac{\partial }{\partial t}|\Psi 〉$ 的功能。态的演化，其实

$\stackrel{^}{I}\left(t\right)|\psi 〉=\sigma |\psi 〉$ 、不变量方程 $\frac{\partial \stackrel{^}{I}\left(t\right)}{\partial t}+\frac{1}{i\hslash }\left[\stackrel{^}{I},\stackrel{^}{H}\right]=0$ [24] 以及含时Schrödinger方程 $\stackrel{^}{H}\left(t\right)|\Psi 〉=i\hslash \frac{\partial }{\partial t}|\Psi 〉$ 三者

$|\Psi \left(t\right)〉=\mathrm{exp}\left(\frac{1}{i\hslash }\stackrel{^}{H}t\right)|\Psi \left(0\right)〉$。但是如果哈密顿量 $\stackrel{^}{H}$ 含时，有人如果要将Schrödinger方程的解写为 $|\Psi \left(t\right)〉=\mathrm{exp}\left(\frac{1}{i\hslash }\int \stackrel{^}{H}\left(t\right)\text{d}t\right)|\Psi \left(0\right)〉$，这就错误。这是因为不同时刻的哈密顿量如 $\stackrel{^}{H}\left({t}_{1}\right)$$\stackrel{^}{H}\left({t}_{2}\right)$ 不可对易，

3. 几个含时演化量子系统

$i\hslash \frac{\partial {\psi }_{1}}{\partial t}=\frac{\epsilon }{2}{\psi }_{1}+K{\psi }_{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}i\hslash \frac{\partial {\psi }_{2}}{\partial t}=-\frac{\epsilon }{2}{\psi }_{2}+{K}^{*}{\psi }_{1}$

$i\hslash \frac{\partial }{\partial t}\left(\begin{array}{c}{\psi }_{1}\\ {\psi }_{2}\end{array}\right)=\left(\begin{array}{cc}\frac{\epsilon }{2}& K\\ {K}^{*}& -\frac{\epsilon }{2}\end{array}\right)\left(\begin{array}{c}{\psi }_{1}\\ {\psi }_{2}\end{array}\right)\equiv H\left(\begin{array}{c}{\psi }_{1}\\ {\psi }_{2}\end{array}\right)$

$i\hslash \frac{\partial {\psi }_{1}}{\partial t}=\frac{\epsilon }{2}{\psi }_{1}+K{\text{e}}^{i\omega t}{\psi }_{2}$$i\hslash \frac{\partial {\psi }_{2}}{\partial t}=-\frac{\epsilon }{2}{\psi }_{2}+{K}^{*}{\text{e}}^{-i\omega t}{\psi }_{1}$

$\begin{array}{c}i\hslash \frac{\partial {\stackrel{˜}{\psi }}_{1}}{\partial t}=i\hslash \frac{\partial {\psi }_{1}}{\partial t}{\text{e}}^{i\alpha t}-\hslash \alpha {\stackrel{˜}{\psi }}_{1}=\left(\frac{\epsilon }{2}{\psi }_{1}+K{\psi }_{2}{\text{e}}^{i\omega t}\right){\text{e}}^{i\alpha t}-\hslash \alpha {\stackrel{˜}{\psi }}_{1}\\ =\left(\frac{\epsilon }{2}-\hslash \alpha \right){\stackrel{˜}{\psi }}_{1}+K{\psi }_{2}{\text{e}}^{i\left(\alpha t+\omega t\right)}\end{array}$

$\begin{array}{c}i\hslash \frac{\partial {\stackrel{˜}{\psi }}_{2}}{\partial t}=i\hslash \frac{\partial {\psi }_{2}}{\partial t}{\text{e}}^{i\beta t}-\hslash \beta {\stackrel{˜}{\psi }}_{2}=\left(-\frac{\epsilon }{2}{\psi }_{2}+{K}^{*}{\psi }_{1}{\text{e}}^{-i\omega t}\right){\text{e}}^{i\beta t}-\hslash \beta {\stackrel{˜}{\psi }}_{2}\\ =\left(-\frac{\epsilon }{2}-\hslash \beta \right){\stackrel{˜}{\psi }}_{2}+{K}^{*}{\psi }_{1}{\text{e}}^{i\left(\beta t-\omega t\right)}\end{array}$

$i\hslash \frac{\partial {\stackrel{˜}{\psi }}_{1}}{\partial t}=\left(\frac{\epsilon }{2}-\hslash \alpha \right){\stackrel{˜}{\psi }}_{1}+K{\stackrel{˜}{\psi }}_{2}$$i\hslash \frac{\partial {\stackrel{˜}{\psi }}_{2}}{\partial t}=\left(-\frac{\epsilon }{2}-\hslash \beta \right){\stackrel{˜}{\psi }}_{2}+{K}^{*}{\stackrel{˜}{\psi }}_{1}$

$\beta =-\alpha$。于是，我们可以取 $\beta =-\alpha =\omega /2$ (其它虽满足 $\beta -\alpha =\omega$ 但不满足 $\beta =-\alpha$$\alpha ,\beta$ 也是允许的。所有多种选择之间无非仅仅差了一个或几个幺正变换)。现在我们看到，原本耦合系数含有“时谐振荡因子” ${\text{e}}^{±i\omega t}$ 的哈密顿量，通过一个幺正变换，就变为了不含时的哈密顿量。由此说明，耦合系数含“时谐振荡因子”的哈密顿量系统，不带有几何相位。但是，如果哈密顿量的耦合系数含时因子不能写成时谐振荡因子 ${\text{e}}^{±i\omega t}$，那么就不存在这么一个简单的幺正变换，因此也就无法将含时的哈密顿量变换为定态(不含时)的哈密顿量，这样的系统肯定会携带几何相位。

$\begin{array}{c}\left(\begin{array}{cc}\frac{\epsilon }{2}& K\\ {K}^{*}& -\frac{\epsilon }{2}\end{array}\right)=K\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)+{K}^{*}\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)+\frac{\epsilon }{2}\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)=K{J}_{+}+{K}^{*}{J}_{-}+\epsilon {J}_{3}\\ =\hslash {\omega }_{0}\left\{\frac{1}{2}\mathrm{sin}\theta \mathrm{exp}\left[-i\phi \right]{J}_{+}+\frac{1}{2}\mathrm{sin}\theta \mathrm{exp}\left[i\phi \right]{J}_{-}+\mathrm{cos}\theta {J}_{3}\right\}\end{array}$

$\hslash {\omega }_{0}\mathrm{cos}\theta =\epsilon$$\hslash {\omega }_{0}\frac{1}{2}\mathrm{sin}\theta \mathrm{exp}\left[-i\phi \right]=K$$\hslash {\omega }_{0}\frac{1}{2}\mathrm{sin}\theta \mathrm{exp}\left[i\phi \right]={K}^{*}$${\left(\hslash {\omega }_{0}\right)}^{2}={\epsilon }^{2}+4{K}^{*}K$

${\stackrel{˙}{a}}_{1}=\frac{i}{2}{\Omega }^{*}{a}_{2}$${\stackrel{˙}{a}}_{2}=-i\Delta {a}_{2}+\frac{i}{2}\Omega {a}_{1}$

$\frac{\partial {\stackrel{˜}{a}}_{1}}{\partial t}={\stackrel{˙}{a}}_{1}\mathrm{exp}\left(i\Delta t/2\right)+\left(i\Delta /2\right){\stackrel{˜}{a}}_{1}=\frac{i}{2}{\Omega }^{*}{a}_{2}\mathrm{exp}\left(i\Delta t/2\right)+\left(i\Delta /2\right){\stackrel{˜}{a}}_{1}=i\frac{\Delta }{2}{\stackrel{˜}{a}}_{1}+\frac{i}{2}{\Omega }^{*}{\stackrel{˜}{a}}_{2}$

$\frac{\partial {\stackrel{˜}{a}}_{2}}{\partial t}={\stackrel{˙}{a}}_{2}\mathrm{exp}\left(i\Delta t/2\right)+\left(i\Delta /2\right){\stackrel{˜}{a}}_{2}=\left(-i\Delta {a}_{2}+\frac{i}{2}\Omega {a}_{1}\right)\mathrm{exp}\left(i\Delta t/2\right)+\left(i\Delta /2\right){\stackrel{˜}{a}}_{2}=-\frac{i}{2}\Delta {\stackrel{˜}{a}}_{2}+\frac{i}{2}\Omega {\stackrel{˜}{a}}_{1}$

$\frac{\partial {\stackrel{˜}{a}}_{1}}{\partial t}=i\frac{\Delta }{2}{\stackrel{˜}{a}}_{1}+\frac{i}{2}{\Omega }^{*}{\stackrel{˜}{a}}_{2}$$\frac{\partial {\stackrel{˜}{a}}_{2}}{\partial t}=-\frac{i}{2}\Delta {\stackrel{˜}{a}}_{2}+\frac{i}{2}\Omega {\stackrel{˜}{a}}_{1}$

$i\hslash \frac{\partial }{\partial t}\left(\begin{array}{c}{\stackrel{˜}{a}}_{1}\\ {\stackrel{˜}{a}}_{2}\end{array}\right)=\left(\begin{array}{cc}-\frac{\hslash \Delta }{2}& -\frac{1}{2}\hslash {\Omega }^{*}\\ -\frac{1}{2}\hslash \Omega & \frac{\hslash \Delta }{2}\end{array}\right)\left(\begin{array}{c}{\stackrel{˜}{a}}_{1}\\ {\stackrel{˜}{a}}_{2}\end{array}\right)\equiv H\left(\begin{array}{c}{\stackrel{˜}{a}}_{1}\\ {\stackrel{˜}{a}}_{2}\end{array}\right)$

$\frac{\partial }{\partial t}\left(\begin{array}{c}{a}_{1}\\ {a}_{2}\end{array}\right)=\left(\begin{array}{cc}0& \frac{i}{2}{\Omega }^{*}\\ \frac{i}{2}\Omega & -i\Delta \end{array}\right)\left(\begin{array}{c}{a}_{1}\\ {a}_{2}\end{array}\right)$

$H=\left(\begin{array}{cc}0& \frac{i}{2}{\Omega }^{*}\\ \frac{i}{2}\Omega & -i\Delta \end{array}\right)=\left(\begin{array}{cc}i\frac{\Delta }{2}& \frac{i}{2}{\Omega }^{*}\\ \frac{i}{2}\Omega & -i\frac{\Delta }{2}\end{array}\right)-\left(\begin{array}{cc}i\frac{\Delta }{2}& 0\\ 0& i\frac{\Delta }{2}\end{array}\right)$

$\frac{\partial }{\partial t}\left(\begin{array}{c}{a}_{1}\\ {a}_{2}\end{array}\right)=\left[\frac{\partial }{\partial t}\left(\begin{array}{c}{\stackrel{˜}{a}}_{1}\\ {\stackrel{˜}{a}}_{2}\end{array}\right)\right]\mathrm{exp}\left(-i\frac{\Delta }{2}t\right)+\left(-i\frac{\Delta }{2}\right)\left(\begin{array}{c}{\stackrel{˜}{a}}_{1}\\ {\stackrel{˜}{a}}_{2}\end{array}\right)\mathrm{exp}\left(-i\frac{\Delta }{2}t\right)$

$\begin{array}{l}\left[\frac{\partial }{\partial t}\left(\begin{array}{c}{\stackrel{˜}{a}}_{1}\\ {\stackrel{˜}{a}}_{2}\end{array}\right)\right]\mathrm{exp}\left(-i\frac{\Delta }{2}t\right)+\left(-i\frac{\Delta }{2}\right)\left(\begin{array}{c}{\stackrel{˜}{a}}_{1}\\ {\stackrel{˜}{a}}_{2}\end{array}\right)\mathrm{exp}\left(-i\frac{\Delta }{2}t\right)\\ =\left(\begin{array}{cc}i\frac{\Delta }{2}& \frac{i}{2}{\Omega }^{*}\\ \frac{i}{2}\Omega & -i\frac{\Delta }{2}\end{array}\right)\left(\begin{array}{c}{\stackrel{˜}{a}}_{1}\\ {\stackrel{˜}{a}}_{2}\end{array}\right)\mathrm{exp}\left(-i\frac{\Delta }{2}t\right)-\left(\begin{array}{cc}i\frac{\Delta }{2}& 0\\ 0& i\frac{\Delta }{2}\end{array}\right)\left(\begin{array}{c}{\stackrel{˜}{a}}_{1}\\ {\stackrel{˜}{a}}_{2}\end{array}\right)\mathrm{exp}\left(-i\frac{\Delta }{2}t\right)\end{array}$

$\frac{\partial }{\partial t}\left(\begin{array}{c}{\stackrel{˜}{a}}_{1}\\ {\stackrel{˜}{a}}_{2}\end{array}\right)=\left(\begin{array}{cc}i\frac{\Delta }{2}& \frac{i}{2}{\Omega }^{*}\\ \frac{i}{2}\Omega & -i\frac{\Delta }{2}\end{array}\right)\left(\begin{array}{c}{\stackrel{˜}{a}}_{1}\\ {\stackrel{˜}{a}}_{2}\end{array}\right)$

${B}^{\prime }=-\frac{1}{{c}^{2}}v×E$。于是自旋磁矩与磁感应强度的耦合哈密顿量是 $H=-\mu \cdot {B}^{\prime }=\mu \cdot \left(\frac{1}{{c}^{2}}v×E\right)=v\cdot \left(\frac{1}{{c}^{2}}E×\mu \right)$。 令 $\frac{1}{{c}^{2}}E×\mu =-gA$，那么自旋粒子与场的耦合哈密顿量是 $H=-gA\cdot v$ (其在数学结构上类似于电流与磁

$i\hslash \frac{\partial }{\partial t}{|\Psi \left(t\right)〉}_{s}=\stackrel{^}{H}\left(t\right){|\Psi \left(t\right)〉}_{s}$，它的解可以写为 ${|\Psi \left(t\right)〉}_{s}=\mathrm{exp}\left[\frac{1}{i\hslash }\varphi \left(t\right)\right]{|\Psi \left(t\right)〉}_{I}$，其中 ${|\Psi \left(t\right)〉}_{I}$

Lewis-Riesenfeld不变量算符 $I\left(t\right)$ 的本征态： $I\left(t\right){|\Psi \left(t\right)〉}_{I}=\sigma {|\Psi \left(t\right)〉}_{I}$。此不变量 $I\left(t\right)$ 满足方程 $\frac{\partial I\left(t\right)}{\partial t}+\frac{1}{i\hslash }\left[I\left(t\right),H\left(t\right)\right]=0$。含时Schrödinger方程解析解 ${|\Psi \left(t\right)〉}_{s}$ 中的相位为

$\varphi \left(t\right)=\frac{1}{\hslash }{\int }_{0}^{t}{}_{I}〈\Psi \left({t}^{\prime }\right)|H\left({t}^{\prime }\right)-i\hslash \frac{\partial }{\partial {t}^{\prime }}{|\Psi \left({t}^{\prime }\right)〉}_{I}\text{d}{t}^{\prime }$。这个相位为两部分之和。我们可以分别称呼其为动力学相位(与哈密顿量算符 $H\left({t}^{\prime }\right)$ 有关)和几何相位(与导数算符 $-i\hslash \frac{\partial }{\partial {t}^{\prime }}$ 有关)。我们的任务就落实于求解

Lewis-Riesenfeld不变量方程中的不变量算符 $I\left(t\right)$ 以及本征值方程( $I\left(t\right){|\Psi \left(t\right)〉}_{I}=\sigma {|\Psi \left(t\right)〉}_{I}$ )。

4. 几种计算态矢量与相位的方法

1) 使用不变量理论, 但不使用幺正变换方法：

$I\left(t\right)=l\left(t\right)\cdot J=\frac{1}{2}\mathrm{sin}\lambda \left(t\right)\mathrm{exp}\left[-i\gamma \left(t\right)\right]{J}_{+}+\frac{1}{2}\mathrm{sin}\lambda \left(t\right)\mathrm{exp}\left[i\gamma \left(t\right)\right]{J}_{-}+\mathrm{cos}\lambda \left(t\right){J}_{3}$

${|\Psi \left(t\right)〉}_{+}=\left(\begin{array}{c}\mathrm{cos}\frac{\lambda \left(t\right)}{2}\\ \mathrm{sin}\frac{\lambda \left(t\right)}{2}{\text{e}}^{i\gamma \left(t\right)}\end{array}\right)$${|\Psi \left(t\right)〉}_{-}=\left(\begin{array}{c}-\mathrm{sin}\frac{\lambda \left(t\right)}{2}{\text{e}}^{-i\gamma \left(t\right)}\\ \mathrm{cos}\frac{\lambda \left(t\right)}{2}\end{array}\right)$

$\omega \left(t\right)={\omega }_{0}\left(t\right)\left[\mathrm{sin}\theta \left(t\right)\mathrm{cos}\phi \left(t\right),\mathrm{sin}\theta \left(t\right)\mathrm{sin}\phi \left(t\right),\mathrm{cos}\theta \left(t\right)\right]$

$\begin{array}{l}{}_{+}〈\Psi \left(t\right)|H\left(t\right){|\Psi \left(t\right)〉}_{+}\\ =\frac{1}{2}\hslash {\omega }_{0}\left(t\right)\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda \left(t\right)}{2}& \mathrm{sin}\frac{\lambda \left(t\right)}{2}{\text{e}}^{-i\gamma \left(t\right)}\end{array}\right)\left(\begin{array}{cc}\mathrm{cos}\theta \left(t\right)& {\text{e}}^{-i\phi \left(t\right)}\mathrm{sin}\theta \left(t\right)\\ {\text{e}}^{i\phi \left(t\right)}\mathrm{sin}\theta \left(t\right)& -\mathrm{cos}\theta \left(t\right)\end{array}\right)\left(\begin{array}{c}\mathrm{cos}\frac{\lambda \left(t\right)}{2}\\ \mathrm{sin}\frac{\lambda \left(t\right)}{2}{\text{e}}^{i\gamma \left(t\right)}\end{array}\right)\\ =\frac{1}{2}\hslash {\omega }_{0}\left(t\right)\left[\mathrm{cos}\theta \left(t\right)\mathrm{cos}\lambda \left(t\right)+\mathrm{sin}\theta \left(t\right)\mathrm{sin}\lambda \left(t\right)\mathrm{cos}\left(\gamma -\phi \right)\right]\end{array}$

$\begin{array}{c}\frac{\partial }{\partial t}{|\Psi \left(t\right)〉}_{+}=\frac{\partial }{\partial t}\left(\begin{array}{c}\mathrm{cos}\frac{\lambda \left(t\right)}{2}\\ \mathrm{sin}\frac{\lambda \left(t\right)}{2}{\text{e}}^{i\gamma \left(t\right)}\end{array}\right)\\ =\left(\begin{array}{c}-\stackrel{˙}{\lambda }\left(t\right)\frac{1}{2}\mathrm{sin}\frac{\lambda \left(t\right)}{2}\\ \stackrel{˙}{\lambda }\left(t\right)\frac{1}{2}\mathrm{cos}\frac{\lambda \left(t\right)}{2}{\text{e}}^{i\gamma \left(t\right)}+i\stackrel{˙}{\gamma }\left(t\right)\mathrm{sin}\frac{\lambda \left(t\right)}{2}{\text{e}}^{i\gamma \left(t\right)}\end{array}\right)\end{array}$

$\begin{array}{l}{}_{+}〈\Psi \left(t\right)|\frac{\partial }{\partial t}{|\Psi \left(t\right)〉}_{+}\\ =\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda \left(t\right)}{2}& \mathrm{sin}\frac{\lambda \left(t\right)}{2}{\text{e}}^{-i\gamma \left(t\right)}\end{array}\right)\left(\begin{array}{c}-\stackrel{˙}{\lambda }\left(t\right)\frac{1}{2}\mathrm{sin}\frac{\lambda \left(t\right)}{2}\\ \stackrel{˙}{\lambda }\left(t\right)\frac{1}{2}\mathrm{cos}\frac{\lambda \left(t\right)}{2}{\text{e}}^{i\gamma \left(t\right)}+i\stackrel{˙}{\gamma }\left(t\right)\mathrm{sin}\frac{\lambda \left(t\right)}{2}{\text{e}}^{i\gamma \left(t\right)}\end{array}\right)\\ =i\stackrel{˙}{\gamma }\left(t\right){\mathrm{sin}}^{2}\frac{\lambda \left(t\right)}{2}=\frac{i}{2}\stackrel{˙}{\gamma }\left(t\right)\left(1-\mathrm{cos}\lambda \left(t\right)\right)\end{array}$

$\begin{array}{l}{}_{+}〈\Psi \left(t\right)|H\left(t\right){|\Psi \left(t\right)〉}_{+}-i{\hslash }_{+}〈\Psi \left(t\right)|\frac{\partial }{\partial t}{|\Psi \left(t\right)〉}_{+}\\ =\frac{1}{2}\hslash {\omega }_{0}\left(t\right)\left[\mathrm{cos}\theta \left(t\right)\mathrm{cos}\lambda \left(t\right)+\mathrm{sin}\theta \left(t\right)\mathrm{sin}\lambda \left(t\right)\mathrm{cos}\left(\gamma -\phi \right)\right]+\frac{1}{2}\hslash \stackrel{˙}{\gamma }\left(t\right)\left(1-\mathrm{cos}\lambda \left(t\right)\right)\end{array}$

2) 使用幺正变换方法 [26]，但避用Baker-Campbell-Hausdorff公式：

$\mathrm{exp}\left(in\cdot \sigma \theta \right)=\mathrm{cos}\theta +in\cdot \sigma \mathrm{sin}\theta =\left(\begin{array}{cc}\mathrm{cos}\theta +i{n}_{3}\mathrm{sin}\theta & i\left({n}_{1}-i{n}_{2}\right)\mathrm{sin}\theta \\ i\left({n}_{1}+i{n}_{2}\right)\mathrm{sin}\theta & \mathrm{cos}\theta -i{n}_{3}\mathrm{sin}\theta \end{array}\right)$

$\mathrm{exp}\left[\alpha \left({\sigma }_{1}+i{\sigma }_{2}\right)-{\alpha }^{*}\left({\sigma }_{1}-i{\sigma }_{2}\right)\right]=\mathrm{exp}\left\{i\left[\frac{1}{i}\left(\alpha -{\alpha }^{*}\right){\sigma }_{1}+\left(\alpha +{\alpha }^{*}\right){\sigma }_{2}\right]\right\}$。与上面的式子比较，我们可以得到 ${n}_{1}\theta =\frac{1}{i}\left(\alpha -{\alpha }^{*}\right)$${n}_{2}\theta =\alpha +{\alpha }^{*}$${n}_{3}=0$，其中 $\theta =\sqrt{-{\left(\alpha -{\alpha }^{*}\right)}^{2}+{\left(\alpha +{\alpha }^{*}\right)}^{2}}=2\sqrt{{\alpha }^{*}\alpha }$${n}_{1}=\frac{-i\left(\alpha -{\alpha }^{*}\right)}{2\sqrt{{\alpha }^{*}\alpha }}$${n}_{2}=\frac{\alpha +{\alpha }^{*}}{2\sqrt{{\alpha }^{*}\alpha }}$。于是我们得到如下两个关系：

${n}_{1}+i{n}_{2}=\frac{-i\left(\alpha -{\alpha }^{*}\right)}{2\sqrt{{\alpha }^{*}\alpha }}+i\frac{\alpha +{\alpha }^{*}}{2\sqrt{{\alpha }^{*}\alpha }}=\frac{i{\alpha }^{*}}{\sqrt{{\alpha }^{*}\alpha }}$${n}_{1}-i{n}_{2}=\frac{-i\left(\alpha -{\alpha }^{*}\right)}{2\sqrt{{\alpha }^{*}\alpha }}-i\frac{\alpha +{\alpha }^{*}}{2\sqrt{{\alpha }^{*}\alpha }}=-\frac{i\alpha }{\sqrt{{\alpha }^{*}\alpha }}$

$\begin{array}{l}\mathrm{exp}\left[\alpha \left({\sigma }_{1}+i{\sigma }_{2}\right)-{\alpha }^{*}\left({\sigma }_{1}-i{\sigma }_{2}\right)\right]=\left(\begin{array}{cc}\mathrm{cos}\theta & i\left({n}_{1}-i{n}_{2}\right)\mathrm{sin}\theta \\ i\left({n}_{1}+i{n}_{2}\right)\mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right)\\ =\left(\begin{array}{cc}\mathrm{cos}\left(2\sqrt{{\alpha }^{*}\alpha }\right)& \frac{\alpha }{\sqrt{{\alpha }^{*}\alpha }}\mathrm{sin}\left(2\sqrt{{\alpha }^{*}\alpha }\right)\\ -\frac{{\alpha }^{*}}{\sqrt{{\alpha }^{*}\alpha }}\mathrm{sin}\left(2\sqrt{{\alpha }^{*}\alpha }\right)& \mathrm{cos}\left(2\sqrt{{\alpha }^{*}\alpha }\right)\end{array}\right)\end{array}$

$\begin{array}{l}\mathrm{exp}\left[-\frac{\lambda }{2}{\text{e}}^{-i\gamma }\left(\frac{{\sigma }_{1}+i{\sigma }_{2}}{2}\right)-\left(-\frac{\lambda }{2}{\text{e}}^{i\gamma }\right)\left(\frac{{\sigma }_{1}-i{\sigma }_{2}}{2}\right)\right]\\ =\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& -{\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ {\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& \mathrm{cos}\frac{\lambda }{2}\end{array}\right)\end{array}$

$\begin{array}{c}I\left(t\right)=l\left(t\right)\cdot J\\ =\frac{1}{2}\mathrm{sin}\lambda \left(t\right)\mathrm{exp}\left[-i\gamma \left(t\right)\right]{J}_{+}+\frac{1}{2}\mathrm{sin}\lambda \left(t\right)\mathrm{exp}\left[i\gamma \left(t\right)\right]{J}_{-}+\mathrm{cos}\lambda \left(t\right){J}_{3}\\ =\frac{1}{2}\left(\begin{array}{cc}\mathrm{cos}\lambda & {\text{e}}^{-i\gamma }\mathrm{sin}\lambda \\ {\text{e}}^{i\gamma }\mathrm{sin}\lambda & -\mathrm{cos}\lambda \end{array}\right)\end{array}$

$V=\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& -{\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ {\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& \mathrm{cos}\frac{\lambda }{2}\end{array}\right)$${V}^{+}=\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& {\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ -{\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& \mathrm{cos}\frac{\lambda }{2}\end{array}\right)$

$\begin{array}{c}{V}^{+}IV=\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& {\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ -{\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& \mathrm{cos}\frac{\lambda }{2}\end{array}\right)\frac{1}{2}\left(\begin{array}{cc}\mathrm{cos}\lambda & {\text{e}}^{-i\gamma }\mathrm{sin}\lambda \\ {\text{e}}^{i\gamma }\mathrm{sin}\lambda & -\mathrm{cos}\lambda \end{array}\right)\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& -{\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ {\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& \mathrm{cos}\frac{\lambda }{2}\end{array}\right)\\ =\frac{1}{2}\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& {\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ -{\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& \mathrm{cos}\frac{\lambda }{2}\end{array}\right)\left(\begin{array}{cc}\mathrm{cos}\lambda \mathrm{cos}\frac{\lambda }{2}+\mathrm{sin}\lambda \mathrm{sin}\frac{\lambda }{2}& -{\text{e}}^{-i\gamma }\mathrm{cos}\lambda \mathrm{sin}\frac{\lambda }{2}+{\text{e}}^{-i\gamma }\mathrm{sin}\lambda \mathrm{cos}\frac{\lambda }{2}\\ {\text{e}}^{i\gamma }\mathrm{sin}\lambda \mathrm{cos}\frac{\lambda }{2}-{\text{e}}^{i\gamma }\mathrm{cos}\lambda \mathrm{sin}\frac{\lambda }{2}& -\mathrm{sin}\lambda \mathrm{sin}\frac{\lambda }{2}-\mathrm{cos}\lambda \mathrm{cos}\frac{\lambda }{2}\end{array}\right)\\ =\frac{1}{2}\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& {\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ -{\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& \mathrm{cos}\frac{\lambda }{2}\end{array}\right)\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& {\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ {\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& -\mathrm{cos}\frac{\lambda }{2}\end{array}\right)=\frac{1}{2}\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\end{array}$

$\begin{array}{c}{V}^{+}HV\\ =\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& {\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ -{\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& \mathrm{cos}\frac{\lambda }{2}\end{array}\right)\frac{1}{2}\hslash {\omega }_{0}\left(t\right)\left(\begin{array}{cc}\mathrm{cos}\theta & {\text{e}}^{-i\phi }\mathrm{sin}\theta \\ {\text{e}}^{i\phi }\mathrm{sin}\theta & -\mathrm{cos}\theta \end{array}\right)\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& -{\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ {\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& \mathrm{cos}\frac{\lambda }{2}\end{array}\right)\\ =\frac{1}{2}\hslash {\omega }_{0}\left(t\right)\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& {\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ -{\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& \mathrm{cos}\frac{\lambda }{2}\end{array}\right)\left(\begin{array}{cc}\mathrm{cos}\theta \mathrm{cos}\frac{\lambda }{2}+{\text{e}}^{-i\left(\phi -\gamma \right)}\mathrm{sin}\theta \mathrm{sin}\frac{\lambda }{2}& -{\text{e}}^{-i\gamma }\mathrm{cos}\theta \mathrm{sin}\frac{\lambda }{2}+{\text{e}}^{-i\phi }\mathrm{sin}\theta \mathrm{cos}\frac{\lambda }{2}\\ {\text{e}}^{i\phi }\mathrm{sin}\theta \mathrm{cos}\frac{\lambda }{2}-{\text{e}}^{i\gamma }\mathrm{cos}\theta \mathrm{sin}\frac{\lambda }{2}& -{\text{e}}^{i\left(\phi -\gamma \right)}\mathrm{sin}\theta \mathrm{sin}\frac{\lambda }{2}-\mathrm{cos}\theta \mathrm{cos}\frac{\lambda }{2}\end{array}\right)\\ =\frac{1}{2}\hslash {\omega }_{0}\left(t\right)\left(\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\end{array}\right)\end{array}$

$\begin{array}{l}{a}_{11}=\mathrm{cos}\frac{\lambda }{2}\left(\mathrm{cos}\theta \mathrm{cos}\frac{\lambda }{2}+{\text{e}}^{-i\left(\phi -\gamma \right)}\mathrm{sin}\theta \mathrm{sin}\frac{\lambda }{2}\right)+{\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\left({\text{e}}^{i\phi }\mathrm{sin}\theta \mathrm{cos}\frac{\lambda }{2}-{\text{e}}^{i\gamma }\mathrm{cos}\theta \mathrm{sin}\frac{\lambda }{2}\right),\\ {a}_{12}=\mathrm{cos}\frac{\lambda }{2}\left(-{\text{e}}^{-i\gamma }\mathrm{cos}\theta \mathrm{sin}\frac{\lambda }{2}+{\text{e}}^{-i\phi }\mathrm{sin}\theta \mathrm{cos}\frac{\lambda }{2}\right)+{\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\left(-{\text{e}}^{i\left(\phi -\gamma \right)}\mathrm{sin}\theta \mathrm{sin}\frac{\lambda }{2}-\mathrm{cos}\theta \mathrm{cos}\frac{\lambda }{2}\right),\\ {a}_{21}=-{\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}\left(\mathrm{cos}\theta \mathrm{cos}\frac{\lambda }{2}+{\text{e}}^{-i\left(\phi -\gamma \right)}\mathrm{sin}\theta \mathrm{sin}\frac{\lambda }{2}\right)+\mathrm{cos}\frac{\lambda }{2}\left({\text{e}}^{i\phi }\mathrm{sin}\theta \mathrm{cos}\frac{\lambda }{2}-{\text{e}}^{i\gamma }\mathrm{cos}\theta \mathrm{sin}\frac{\lambda }{2}\right),\\ {a}_{22}=-{\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}\left(-{\text{e}}^{-i\gamma }\mathrm{cos}\theta \mathrm{sin}\frac{\lambda }{2}+{\text{e}}^{-i\phi }\mathrm{sin}\theta \mathrm{cos}\frac{\lambda }{2}\right)+\mathrm{cos}\frac{\lambda }{2}\left(-{\text{e}}^{i\left(\phi -\gamma \right)}\mathrm{sin}\theta \mathrm{sin}\frac{\lambda }{2}-\mathrm{cos}\theta \mathrm{cos}\frac{\lambda }{2}\right)\end{array}$

$\begin{array}{l}{a}_{11}=\mathrm{cos}\theta \mathrm{cos}\lambda +\mathrm{sin}\theta \mathrm{sin}\lambda \mathrm{cos}\left(\phi -\gamma \right),\\ {a}_{12}=-{\text{e}}^{-i\gamma }\mathrm{cos}\theta \mathrm{sin}\lambda +{\text{e}}^{-i\gamma }\mathrm{sin}\theta \left({\text{e}}^{-i\left(\phi -\gamma \right)}{\mathrm{cos}}^{2}\frac{\lambda }{2}-{\text{e}}^{i\left(\phi -\gamma \right)}{\mathrm{sin}}^{2}\frac{\lambda }{2}\right),\\ {a}_{21}=-{\text{e}}^{i\gamma }\mathrm{cos}\theta \mathrm{sin}\lambda +{\text{e}}^{i\gamma }\mathrm{sin}\theta \left({\text{e}}^{i\left(\phi -\gamma \right)}{\mathrm{cos}}^{2}\frac{\lambda }{2}-{\text{e}}^{-i\left(\phi -\gamma \right)}{\mathrm{sin}}^{2}\frac{\lambda }{2}\right),\\ {a}_{22}=-\mathrm{cos}\theta \mathrm{cos}\lambda -\mathrm{sin}\theta \mathrm{sin}\lambda \mathrm{cos}\left(\phi -\gamma \right)\end{array}$

$\begin{array}{c}{a}_{12}=-{\text{e}}^{-i\gamma }\mathrm{cos}\theta \mathrm{sin}\lambda +{\text{e}}^{-i\gamma }\mathrm{sin}\theta \left({\text{e}}^{-i\left(\phi -\gamma \right)}{\mathrm{cos}}^{2}\frac{\lambda }{2}-{\text{e}}^{i\left(\phi -\gamma \right)}{\mathrm{sin}}^{2}\frac{\lambda }{2}\right)\\ =-{\text{e}}^{-i\gamma }\mathrm{cos}\theta \mathrm{sin}\lambda +{\text{e}}^{-i\gamma }\mathrm{sin}\theta \left(\mathrm{cos}\left(\phi -\gamma \right){\mathrm{cos}}^{2}\frac{\lambda }{2}-i\mathrm{sin}\left(\phi -\gamma \right){\mathrm{cos}}^{2}\frac{\lambda }{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\mathrm{cos}\left(\phi -\gamma \right){\mathrm{sin}}^{2}\frac{\lambda }{2}-i\mathrm{sin}\left(\phi -\gamma \right){\mathrm{sin}}^{2}\frac{\lambda }{2}\right)\\ =-{\text{e}}^{-i\gamma }\mathrm{cos}\theta \mathrm{sin}\lambda +{\text{e}}^{-i\gamma }\mathrm{sin}\theta \left(\mathrm{cos}\left(\phi -\gamma \right)\mathrm{cos}\lambda -i\mathrm{sin}\left(\phi -\gamma \right)\right)\end{array}$

$\begin{array}{c}{a}_{21}=-{\text{e}}^{i\gamma }\mathrm{cos}\theta \mathrm{sin}\lambda +{\text{e}}^{i\gamma }\mathrm{sin}\theta \left({\text{e}}^{i\left(\phi -\gamma \right)}{\mathrm{cos}}^{2}\frac{\lambda }{2}-{\text{e}}^{-i\left(\phi -\gamma \right)}{\mathrm{sin}}^{2}\frac{\lambda }{2}\right)\\ =-{\text{e}}^{i\gamma }\mathrm{cos}\theta \mathrm{sin}\lambda +{\text{e}}^{i\gamma }\mathrm{sin}\theta \left(\mathrm{cos}\left(\phi -\gamma \right)\mathrm{cos}\lambda +i\mathrm{sin}\left(\phi -\gamma \right)\right)\end{array}$

$\begin{array}{c}{V}^{+}i\frac{\partial }{\partial t}V=i\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& {\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ -{\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& \mathrm{cos}\frac{\lambda }{2}\end{array}\right)\frac{\partial }{\partial t}\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& -{\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ {\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& \mathrm{cos}\frac{\lambda }{2}\end{array}\right)\\ =i\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& {\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ -{\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& \mathrm{cos}\frac{\lambda }{2}\end{array}\right)\left(\begin{array}{cc}-\frac{\stackrel{˙}{\lambda }}{2}\mathrm{sin}\frac{\lambda }{2}& i\stackrel{˙}{\gamma }{\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}-{\text{e}}^{-i\gamma }\frac{\stackrel{˙}{\lambda }}{2}\mathrm{cos}\frac{\lambda }{2}\\ i\stackrel{˙}{\gamma }{\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}+{\text{e}}^{i\gamma }\frac{\stackrel{˙}{\lambda }}{2}\mathrm{cos}\frac{\lambda }{2}& -\frac{\stackrel{˙}{\lambda }}{2}\mathrm{sin}\frac{\lambda }{2}\end{array}\right)\\ =i\left(\begin{array}{cc}-\frac{\stackrel{˙}{\lambda }}{2}\mathrm{cos}\frac{\lambda }{2}\mathrm{sin}\frac{\lambda }{2}+\mathrm{sin}\frac{\lambda }{2}\left(i\stackrel{˙}{\gamma }\mathrm{sin}\frac{\lambda }{2}+\frac{\stackrel{˙}{\lambda }}{2}\mathrm{cos}\frac{\lambda }{2}\right)& {\text{e}}^{-i\gamma }\mathrm{cos}\frac{\lambda }{2}\left(i\stackrel{˙}{\gamma }\mathrm{sin}\frac{\lambda }{2}-\frac{\stackrel{˙}{\lambda }}{2}\mathrm{cos}\frac{\lambda }{2}\right)-{\text{e}}^{-i\gamma }\frac{\stackrel{˙}{\lambda }}{2}{\mathrm{sin}}^{2}\frac{\lambda }{2}\\ \frac{\stackrel{˙}{\lambda }}{2}{\text{e}}^{i\gamma }{\mathrm{sin}}^{2}\frac{\lambda }{2}+{\text{e}}^{i\gamma }\mathrm{cos}\frac{\lambda }{2}\left(i\stackrel{˙}{\gamma }\mathrm{sin}\frac{\lambda }{2}+\frac{\stackrel{˙}{\lambda }}{2}\mathrm{cos}\frac{\lambda }{2}\right)& -\mathrm{sin}\frac{\lambda }{2}\left(i\stackrel{˙}{\gamma }\mathrm{sin}\frac{\lambda }{2}-\frac{\stackrel{˙}{\lambda }}{2}\mathrm{cos}\frac{\lambda }{2}\right)-\frac{\stackrel{˙}{\lambda }}{2}\mathrm{cos}\frac{\lambda }{2}\mathrm{sin}\frac{\lambda }{2}\end{array}\right)\\ =i\left(\begin{array}{cc}i\stackrel{˙}{\gamma }{\mathrm{sin}}^{2}\frac{\lambda }{2}& i\stackrel{˙}{\gamma }{\text{e}}^{-i\gamma }\mathrm{cos}\frac{\lambda }{2}\mathrm{sin}\frac{\lambda }{2}-{\text{e}}^{-i\gamma }\frac{\stackrel{˙}{\lambda }}{2}\\ {\text{e}}^{i\gamma }\frac{\stackrel{˙}{\lambda }}{2}+i\stackrel{˙}{\gamma }{\text{e}}^{i\gamma }\mathrm{cos}\frac{\lambda }{2}\mathrm{sin}\frac{\lambda }{2}& -i\stackrel{˙}{\gamma }{\mathrm{sin}}^{2}\frac{\lambda }{2}\end{array}\right)\\ =\left(\begin{array}{cc}-\stackrel{˙}{\gamma }{\mathrm{sin}}^{2}\frac{\lambda }{2}& -\stackrel{˙}{\gamma }{\text{e}}^{-i\gamma }\mathrm{cos}\frac{\lambda }{2}\mathrm{sin}\frac{\lambda }{2}-i{\text{e}}^{-i\gamma }\frac{\stackrel{˙}{\lambda }}{2}\\ i{\text{e}}^{i\gamma }\frac{\stackrel{˙}{\lambda }}{2}-\stackrel{˙}{\gamma }{\text{e}}^{i\gamma }\mathrm{cos}\frac{\lambda }{2}\mathrm{sin}\frac{\lambda }{2}& \stackrel{˙}{\gamma }{\mathrm{sin}}^{2}\frac{\lambda }{2}\end{array}\right)\end{array}$

$\begin{array}{c}-{V}^{+}i\frac{\partial }{\partial t}V=\left(\begin{array}{cc}\stackrel{˙}{\gamma }\frac{1-\mathrm{cos}\lambda }{2}& \stackrel{˙}{\gamma }{\text{e}}^{-i\gamma }\frac{1}{2}\mathrm{sin}\lambda +i{\text{e}}^{-i\gamma }\frac{\stackrel{˙}{\lambda }}{2}\\ -i{\text{e}}^{i\gamma }\frac{\stackrel{˙}{\lambda }}{2}+\stackrel{˙}{\gamma }{\text{e}}^{i\gamma }\frac{1}{2}\mathrm{sin}\lambda & -\stackrel{˙}{\gamma }\frac{1-\mathrm{cos}\lambda }{2}\end{array}\right)\\ =\frac{1}{2}\left(\begin{array}{cc}\stackrel{˙}{\gamma }\left(1-\mathrm{cos}\lambda \right)& \stackrel{˙}{\gamma }{\text{e}}^{-i\gamma }\mathrm{sin}\lambda +i{\text{e}}^{-i\gamma }\stackrel{˙}{\lambda }\\ -i{\text{e}}^{i\gamma }\stackrel{˙}{\lambda }+\stackrel{˙}{\gamma }{\text{e}}^{i\gamma }\mathrm{sin}\lambda & -\stackrel{˙}{\gamma }\left(1-\mathrm{cos}\lambda \right)\end{array}\right)\end{array}$

$\begin{array}{l}\hslash \left[\frac{1}{2}{\omega }_{0}\left(t\right){a}_{12}+\frac{1}{2}\left(\stackrel{˙}{\gamma }{\text{e}}^{-i\gamma }\mathrm{sin}\lambda +i{\text{e}}^{-i\gamma }\stackrel{˙}{\lambda }\right)\right]\\ =\frac{1}{2}\hslash {\omega }_{0}\left(t\right)\left[-{\text{e}}^{-i\gamma }\mathrm{cos}\theta \mathrm{sin}\lambda +{\text{e}}^{-i\gamma }\mathrm{sin}\theta \left(\mathrm{cos}\left(\phi -\gamma \right)\mathrm{cos}\lambda -i\mathrm{sin}\left(\phi -\gamma \right)\right)\right]\\ +\frac{1}{2}\hslash \left(\stackrel{˙}{\gamma }{\text{e}}^{-i\gamma }\mathrm{sin}\lambda +i{\text{e}}^{-i\gamma }\stackrel{˙}{\lambda }\right)\end{array}$

$\stackrel{˙}{\lambda }={\omega }_{0}\left(t\right)\mathrm{sin}\theta \mathrm{sin}\left(\phi -\gamma \right)$$\stackrel{˙}{\gamma }={\omega }_{0}\left(t\right)\left[\mathrm{cos}\theta -\mathrm{sin}\theta \mathrm{cot}\lambda \mathrm{cos}\left(\phi -\gamma \right)\right]$

$\begin{array}{l}\stackrel{˙}{\gamma }{\text{e}}^{-i\gamma }\mathrm{sin}\lambda +i{\text{e}}^{-i\gamma }\stackrel{˙}{\lambda }\\ ={\omega }_{0}\left(t\right)\left[\mathrm{cos}\theta -\mathrm{sin}\theta \mathrm{cot}\lambda \mathrm{cos}\left(\phi -\gamma \right)\right]{\text{e}}^{-i\gamma }\mathrm{sin}\lambda +i{\text{e}}^{-i\gamma }{\omega }_{0}\left(t\right)\mathrm{sin}\theta \mathrm{sin}\left(\phi -\gamma \right)\\ ={\omega }_{0}\left(t\right)\mathrm{cos}\theta {\text{e}}^{-i\gamma }\mathrm{sin}\lambda -{\omega }_{0}\left(t\right){\text{e}}^{-i\gamma }\mathrm{sin}\theta \mathrm{cos}\lambda \mathrm{cos}\left(\phi -\gamma \right)+i{\text{e}}^{-i\gamma }{\omega }_{0}\left(t\right)\mathrm{sin}\theta \mathrm{sin}\left(\phi -\gamma \right)\end{array}$

$\begin{array}{l}\frac{1}{2}{\omega }_{0}\left(t\right){a}_{12}+\frac{1}{2}\left(\stackrel{˙}{\gamma }{\text{e}}^{-i\gamma }\mathrm{sin}\lambda +i{\text{e}}^{-i\gamma }\stackrel{˙}{\lambda }\right)\\ =\frac{1}{2}{\omega }_{0}\left(t\right)\left[-{\text{e}}^{-i\gamma }\mathrm{cos}\theta \mathrm{sin}\lambda +{\text{e}}^{-i\gamma }\mathrm{sin}\theta \left(\mathrm{cos}\left(\phi -\gamma \right)\mathrm{cos}\lambda -i\mathrm{sin}\left(\phi -\gamma \right)\right)\right]\\ +\frac{1}{2}\left[{\omega }_{0}\left(t\right)\mathrm{cos}\theta {\text{e}}^{-i\gamma }\mathrm{sin}\lambda -{\omega }_{0}\left(t\right){\text{e}}^{-i\gamma }\mathrm{sin}\theta \mathrm{cos}\lambda \mathrm{cos}\left(\phi -\gamma \right)+i{\text{e}}^{-i\gamma }{\omega }_{0}\left(t\right)\mathrm{sin}\theta \mathrm{sin}\left(\phi -\gamma \right)\right]\\ =0\end{array}$

${V}^{+}HV-{V}^{+}i\hslash \frac{\partial }{\partial t}V=\hslash \left\{{\omega }_{0}\left(t\right)\left[\mathrm{cos}\theta \mathrm{cos}\lambda +\mathrm{sin}\theta \mathrm{sin}\lambda \mathrm{cos}\left(\phi -\gamma \right)\right]+\stackrel{˙}{\gamma }\left(1-\mathrm{cos}\lambda \right)\right\}\frac{{\sigma }_{3}}{2}$

3) 既使用幺正变换方法，也使用Baker-Hausdorff公式：

${\text{e}}^{A}B{\text{e}}^{-A}=B+\left[A,B\right]+\frac{1}{2!}\left[A,\left[A,B\right]\right]+\frac{1}{3!}\left[A,\left[A,\left[A,B\right]\right]\right]+\cdots$

$F\left(\alpha \right)={\text{e}}^{\alpha A}B{\text{e}}^{-\alpha A}$$\alpha$ 为某个实参数。下面我们计算算符 $F\left(\alpha \right)$ 的一阶、二阶、三阶关于参数 $\alpha$ 的导数 [61]：

$\frac{\text{d}F\left(\alpha \right)}{\text{d}\alpha }=A{\text{e}}^{\alpha A}B{\text{e}}^{-\alpha A}-{\text{e}}^{\alpha A}B{\text{e}}^{-\alpha A}A=\left[A,F\left(\alpha \right)\right]$

$\frac{{\text{d}}^{2}F\left(\alpha \right)}{\text{d}{\alpha }^{2}}=\left[A,\frac{\text{d}F\left(\alpha \right)}{\text{d}\alpha }\right]=\left[A,\left[A,F\left(\alpha \right)\right]\right]$

$\frac{{\text{d}}^{3}F\left(\alpha \right)}{\text{d}{\alpha }^{3}}=\left[A,\left[A,\frac{\text{d}F\left(\alpha \right)}{\text{d}\alpha }\right]\right]=\left[A,\left[A,\left[A,F\left(\alpha \right)\right]\right]\right]$$\cdots$

$F\left(\alpha \right)=\underset{n=0}{\overset{\infty }{\sum }}\frac{{\alpha }^{n}}{n!}{\frac{{\text{d}}^{n}F\left(\alpha \right)}{\text{d}{\alpha }^{n}}|}_{\alpha =0}=B+\frac{\alpha }{1!}\left[A,B\right]+\frac{{\alpha }^{2}}{2!}\left[A,\left[A,B\right]\right]+\frac{{\alpha }^{3}}{3!}\left[A,\left[A,\left[A,B\right]\right]\right]+\cdots$

$V={\text{e}}^{L}$${V}^{+}={\text{e}}^{-L}$，我定义 $F\left(\alpha \right)={\text{e}}^{-\alpha L}\frac{\partial }{\partial t}{\text{e}}^{\alpha L}$ ( $\alpha$ 是参变量，本身不含时间，时间含在矩阵算符L

$\begin{array}{c}\frac{\text{d}F\left(\alpha \right)}{\text{d}\alpha }=-L{\text{e}}^{-\alpha L}\frac{\partial }{\partial t}{\text{e}}^{\alpha L}+{\text{e}}^{-\alpha L}\frac{\partial }{\partial t}\left({\text{e}}^{\alpha L}L\right)=-L{\text{e}}^{-\alpha L}\frac{\partial }{\partial t}{\text{e}}^{\alpha L}+\left({\text{e}}^{-\alpha L}\frac{\partial }{\partial t}{\text{e}}^{\alpha L}\right)L+\frac{\partial L}{\partial t}\\ =\left[{\text{e}}^{-\alpha L}\frac{\partial }{\partial t}{\text{e}}^{\alpha L},L\right]+\frac{\partial L}{\partial t}=\left[F\left(\alpha \right),L\right]+\frac{\partial L}{\partial t}\end{array}$

$\frac{{\text{d}}^{2}F\left(\alpha \right)}{\text{d}{\alpha }^{2}}=\left[\frac{\text{d}F\left(\alpha \right)}{\text{d}\alpha },L\right]=\left[\left[F\left(\alpha \right),L\right]+\frac{\partial L}{\partial t},L\right]=\left[\left[F\left(\alpha \right),L\right],L\right]+\left[\frac{\partial L}{\partial t},L\right]$

$\frac{{\text{d}}^{3}F\left(\alpha \right)}{\text{d}{\alpha }^{3}}=\left[\left[\frac{\text{d}F\left(\alpha \right)}{\text{d}\alpha },L\right],L\right]=\left[\left[\left[F\left(\alpha \right),L\right],L\right],L\right]+\left[\left[\frac{\partial L}{\partial t},L\right],L\right]$$\cdots$

$\alpha$ 趋近于零时， $F\left(\alpha \right)={\text{e}}^{-\alpha L}\frac{\partial }{\partial t}{\text{e}}^{\alpha L}\to 0$，上面结果变为

$\frac{\text{d}F\left(\alpha \right)}{\text{d}\alpha }\to \frac{\partial L}{\partial t}$$\frac{{\text{d}}^{2}F\left(\alpha \right)}{\text{d}{\alpha }^{2}}\to \left[\frac{\partial L}{\partial t},L\right]$$\frac{{\text{d}}^{3}F\left(\alpha \right)}{\text{d}{\alpha }^{3}}\to \left[\left[\frac{\partial L}{\partial t},L\right],L\right]$$\cdots$

$F\left(\alpha \right)=\underset{n=0}{\overset{\infty }{\sum }}\frac{{\alpha }^{n}}{n!}{\frac{{\text{d}}^{n}F\left(\alpha \right)}{\text{d}{\alpha }^{n}}|}_{\alpha =0}=\frac{\alpha }{1!}\frac{\partial L}{\partial t}+\frac{{\alpha }^{2}}{2!}\left[\frac{\partial L}{\partial t},L\right]+\frac{{\alpha }^{3}}{3!}\left[\left[\frac{\partial L}{\partial t},L\right],L\right]+\cdots$

${\text{e}}^{-L}\frac{\partial }{\partial t}{\text{e}}^{L}=\frac{\partial L}{\partial t}+\frac{1}{2!}\left[\frac{\partial L}{\partial t},L\right]+\frac{1}{3!}\left[\left[\frac{\partial L}{\partial t},L\right],L\right]+\cdots$

$\frac{\partial I\left(t\right)}{\partial t}+\frac{1}{i\hslash }\left[I\left(t\right),H\left(t\right)\right]=0$ [24] 得到的非线性辅助代数方程是 $\stackrel{˙}{\lambda }={\omega }_{0}\left(t\right)\mathrm{sin}\theta \mathrm{sin}\left(\phi -\gamma \right)$

$\stackrel{˙}{\gamma }={\omega }_{0}\left(t\right)\left[\mathrm{cos}\theta -\mathrm{sin}\theta \mathrm{cot}\lambda \mathrm{cos}\left(\phi -\gamma \right)\right]$。这实际上是将Lewis-Riesenfeld不变量算符方程代数化 [26]。该辅助代数方程是关于求解Lewis-Riesenfeld不变量算符 $I\left(t\right)$ 中的含时参数的方程 [24] [26] [27] [28]。

Schrödinger方程对于态矢量 $|\Psi 〉$ 而言，它是线性方程。但是对于态矢量 $|\Psi 〉$ 内的各个分量而言，它就不一定也能说是线性的，这是因为在二态和多态体系中，态矢量内诸分量之间有耦合，且态矢量还要满足归一化约束条件 $〈\Psi |\Psi 〉=1$ (从一定意义上讲，这种约束条件也代表一种耦合)。所以，线性的Schrödinger方程也可能表现非线性混沌特点，即态矢量对初始数值具有敏感性(微挠能指数发散)。从根本上讲，这种耦合效应，也与不同时刻的哈密顿量不可对易也有关系。确实如此，如上面由Lewis-Riesenfeld不变量方程得到的辅助代数方程 $\stackrel{˙}{\lambda }={\omega }_{0}\left(t\right)\mathrm{sin}\theta \mathrm{sin}\left(\phi -\gamma \right)$$\stackrel{˙}{\gamma }={\omega }_{0}\left(t\right)\left[\mathrm{cos}\theta -\mathrm{sin}\theta \mathrm{cot}\lambda \mathrm{cos}\left(\phi -\gamma \right)\right]$ 就是这样的非线性方程(值得一提的是，即使避用不变量算符方程 [24]，仅仅使用Schrödinger方程，也可以得到该组方程。下面一节“对简单系统避用不变量理论和么正变换方法”证明了这个结论)。但这组方程在一定条件下也有定态解，如假设哈密顿量参数 $\phi$ 线性( $\phi =\xi t$ )且 $\theta ={\theta }_{0}$ (常数)，那么我们就有解 $\gamma =\xi t$$\lambda =\mathrm{arctan}\left[\mathrm{sin}{\theta }_{0}/\left(\mathrm{cos}{\theta }_{0}-\xi /{\omega }_{0}\right)\right]$。但是一旦 $\theta \ne {\theta }_{0}$，情况就复杂，一般没有解析解，需要依靠数值计算。由于量子计算中涉及二态体系的演化，因此这种由于参数演化所带来的非线性效应值得研究。

5. 对简单系统避用不变量理论和么正变换方法

$\stackrel{^}{H}=\frac{\hslash \Omega }{2}\left(\begin{array}{cc}\mathrm{cos}\theta & {\text{e}}^{-i\varphi }\mathrm{sin}\theta \\ {\text{e}}^{+i\varphi }\mathrm{sin}\theta & -\mathrm{cos}\theta \end{array}\right)$

$|+〉=\left(\begin{array}{c}\mathrm{cos}\frac{\theta }{2}\\ {\text{e}}^{+i\varphi }\mathrm{sin}\frac{\theta }{2}\end{array}\right)$$|-〉=\left(\begin{array}{c}-{\text{e}}^{-i\varphi }\mathrm{sin}\frac{\theta }{2}\\ \mathrm{cos}\frac{\theta }{2}\end{array}\right)$

$i\hslash \frac{\partial }{\partial t}|\psi 〉=\stackrel{^}{H}|\psi 〉$。此时上面的 $|±〉$ 不再是Schrödinger方程的解。但是我们可以预测，含时Schrödinger方

$|+,t〉=\left(\begin{array}{c}\mathrm{cos}\frac{\lambda }{2}\\ {\text{e}}^{+i\gamma }\mathrm{sin}\frac{\lambda }{2}\end{array}\right)$$|-,t〉=\left(\begin{array}{c}-{\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ \mathrm{cos}\frac{\lambda }{2}\end{array}\right)$

$|\psi 〉={c}_{+}|+,t〉{\text{e}}^{i{\Phi }_{+}\left(t\right)}+{c}_{-}|-,t〉{\text{e}}^{i{\Phi }_{-}\left(t\right)}$

$i\hslash \frac{\partial }{\partial t}\left(|+,t〉{\text{e}}^{i{\Phi }_{+}\left(t\right)}\right)=\stackrel{^}{H}\left(t\right)|+,t〉{\text{e}}^{i{\Phi }_{+}\left(t\right)}$

$\stackrel{^}{H}|+,t〉=\frac{\hslash \Omega }{2}\left(\begin{array}{c}\mathrm{cos}\theta \mathrm{cos}\frac{\lambda }{2}+{\text{e}}^{i\left(\gamma -\varphi \right)}\mathrm{sin}\theta \mathrm{sin}\frac{\lambda }{2}\\ {\text{e}}^{i\varphi }\mathrm{sin}\theta \mathrm{cos}\frac{\lambda }{2}-{\text{e}}^{i\gamma }\mathrm{cos}\theta \mathrm{sin}\frac{\lambda }{2}\end{array}\right)$

$\frac{\partial }{\partial t}|+,t〉=\left(\begin{array}{c}-\frac{\stackrel{˙}{\lambda }}{2}\mathrm{sin}\frac{\lambda }{2}\\ i\stackrel{˙}{\gamma }{\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}+\frac{\stackrel{˙}{\lambda }}{2}{\text{e}}^{i\gamma }\mathrm{cos}\frac{\lambda }{2}\end{array}\right)$

$\begin{array}{l}-{\stackrel{˙}{\Phi }}_{+}\mathrm{cos}\frac{\lambda }{2}-i\frac{\stackrel{˙}{\lambda }}{2}\mathrm{sin}\frac{\lambda }{2}=\frac{\Omega }{2}\left(\mathrm{cos}\theta \mathrm{cos}\frac{\lambda }{2}+{\text{e}}^{i\left(\gamma -\varphi \right)}\mathrm{sin}\theta \mathrm{sin}\frac{\lambda }{2}\right),\\ -{\stackrel{˙}{\Phi }}_{+}{\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}+i\left(i\stackrel{˙}{\gamma }{\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}+\frac{\stackrel{˙}{\lambda }}{2}{\text{e}}^{i\gamma }\mathrm{cos}\frac{\lambda }{2}\right)=\frac{\Omega }{2}\left({\text{e}}^{i\varphi }\mathrm{sin}\theta \mathrm{cos}\frac{\lambda }{2}-{\text{e}}^{i\gamma }\mathrm{cos}\theta \mathrm{sin}\frac{\lambda }{2}\right)\end{array}$

$\begin{array}{l}{\stackrel{˙}{\Phi }}_{+}=-\frac{\Omega }{2}\left[\mathrm{cos}\theta +\mathrm{cos}\left(\gamma -\varphi \right)\mathrm{sin}\theta \mathrm{tan}\frac{\lambda }{2}\right],\\ \stackrel{˙}{\lambda }=\Omega \mathrm{sin}\left(\varphi -\gamma \right)\mathrm{sin}\theta \end{array}$

$\begin{array}{l}-{\stackrel{˙}{\Phi }}_{+}-\stackrel{˙}{\gamma }+i\frac{\stackrel{˙}{\lambda }}{2}\mathrm{cot}\frac{\lambda }{2}\\ =\frac{\Omega }{2}\left[\mathrm{cos}\left(\varphi -\gamma \right)\mathrm{sin}\theta \mathrm{cot}\frac{\lambda }{2}-\mathrm{cos}\theta +i\mathrm{sin}\left(\varphi -\gamma \right)\mathrm{sin}\theta \mathrm{cot}\frac{\lambda }{2}\right]\end{array}$

${\stackrel{˙}{\Phi }}_{+}+\stackrel{˙}{\gamma }=\frac{\Omega }{2}\left[\mathrm{cos}\theta -\mathrm{cos}\left(\varphi -\gamma \right)\mathrm{sin}\theta \mathrm{cot}\frac{\lambda }{2}\right]$

$\begin{array}{c}\stackrel{˙}{\gamma }=\frac{\Omega }{2}\left[\mathrm{cos}\theta -\mathrm{cos}\left(\varphi -\gamma \right)\mathrm{sin}\theta \mathrm{cot}\frac{\lambda }{2}\right]-{\stackrel{˙}{\Phi }}_{+}\\ =\frac{\Omega }{2}\left[\mathrm{cos}\theta -\mathrm{cos}\left(\varphi -\gamma \right)\mathrm{sin}\theta \mathrm{cot}\frac{\lambda }{2}\right]+\frac{\Omega }{2}\left[\mathrm{cos}\theta +\mathrm{cos}\left(\gamma -\varphi \right)\mathrm{sin}\theta \mathrm{tan}\frac{\lambda }{2}\right]\\ =\Omega \left[\mathrm{cos}\theta -\mathrm{cos}\left(\varphi -\gamma \right)\mathrm{sin}\theta \mathrm{cot}\lambda \right]\end{array}$

$\mathrm{tan}\frac{\lambda }{2}=\frac{2{\mathrm{sin}}^{2}\frac{\lambda }{2}}{\mathrm{sin}\lambda }=\frac{1-\mathrm{cos}\lambda }{\mathrm{sin}\lambda }=\mathrm{sin}\lambda -\frac{\mathrm{cos}\lambda }{\mathrm{sin}\lambda }+\frac{{\mathrm{cos}}^{2}\lambda }{\mathrm{sin}\lambda }=\mathrm{sin}\lambda -\mathrm{cot}\lambda \left(1-\mathrm{cos}\lambda \right)$。那么相位的时间变化率最终可以化为

$\begin{array}{c}{\stackrel{˙}{\Phi }}_{+}=-\frac{\Omega }{2}\left\{\mathrm{cos}\theta +\mathrm{cos}\left(\gamma -\varphi \right)\mathrm{sin}\theta \left[\mathrm{sin}\lambda -\mathrm{cot}\lambda \left(1-\mathrm{cos}\lambda \right)\right]\right\}\\ =-\frac{\Omega }{2}\left\{\mathrm{cos}\lambda \mathrm{cos}\theta +\mathrm{sin}\lambda \mathrm{sin}\theta \mathrm{cos}\left(\gamma -\varphi \right)+\left[\mathrm{cos}\theta -\mathrm{sin}\theta \mathrm{cot}\lambda \mathrm{cos}\left(\varphi -\gamma \right)\right]\left(1-\mathrm{cos}\lambda \right)\right\}\\ =-\frac{1}{2}\left\{\Omega \left[\mathrm{cos}\lambda \mathrm{cos}\theta +\mathrm{sin}\lambda \mathrm{sin}\theta \mathrm{cos}\left(\gamma -\varphi \right)\right]+\stackrel{˙}{\gamma }\left(1-\mathrm{cos}\lambda \right)\right\}\end{array}$

$\begin{array}{c}〈+,t|\stackrel{^}{H}|+,t〉\\ =\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& {\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\end{array}\right)\frac{\hslash \Omega }{2}\left(\begin{array}{cc}\mathrm{cos}\theta & {\text{e}}^{-i\varphi }\mathrm{sin}\theta \\ {\text{e}}^{+i\varphi }\mathrm{sin}\theta & -\mathrm{cos}\theta \end{array}\right)\left(\begin{array}{c}\mathrm{cos}\frac{\lambda }{2}\\ {\text{e}}^{+i\gamma }\mathrm{sin}\frac{\lambda }{2}\end{array}\right)\\ =\frac{\hslash \Omega }{2}\left[\mathrm{cos}\lambda \mathrm{cos}\theta +\mathrm{sin}\lambda \mathrm{sin}\theta \mathrm{cos}\left(\gamma -\varphi \right)\right]\end{array}$

$\begin{array}{c}〈+,t|i\hslash \frac{\partial }{\partial t}|+,t〉\\ =i\hslash \left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& {\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\end{array}\right)\left(\begin{array}{c}-\frac{\stackrel{˙}{\lambda }}{2}\mathrm{sin}\frac{\lambda }{2}\\ i\stackrel{˙}{\gamma }{\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}+\frac{\stackrel{˙}{\lambda }}{2}{\text{e}}^{i\gamma }\mathrm{cos}\frac{\lambda }{2}\end{array}\right)\\ =i\hslash \left[\mathrm{cos}\frac{\lambda }{2}\left(-\frac{\stackrel{˙}{\lambda }}{2}\mathrm{sin}\frac{\lambda }{2}\right)+{\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\left(i\stackrel{˙}{\gamma }{\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}+\frac{\stackrel{˙}{\lambda }}{2}{\text{e}}^{i\gamma }\mathrm{cos}\frac{\lambda }{2}\right)\right]\\ =-\hslash \stackrel{˙}{\gamma }{\mathrm{sin}}^{2}\frac{\lambda }{2}=-\hslash \stackrel{˙}{\gamma }\frac{1-\mathrm{cos}\lambda }{2}\end{array}$

$|1,t〉=\left(\begin{array}{c}\mathrm{cos}\frac{\lambda }{2}\\ {\text{e}}^{+i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ 0\end{array}\right)$$|2,t〉=\left(\begin{array}{c}-{\text{e}}^{-i\tau }\mathrm{sin}\frac{\sigma }{2}\left(-{\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\right)\\ -{\text{e}}^{-i\tau }\mathrm{sin}\frac{\sigma }{2}\mathrm{cos}\frac{\lambda }{2}\\ \mathrm{cos}\frac{\sigma }{2}\end{array}\right)$$|3,t〉=\left(\begin{array}{c}\mathrm{cos}\frac{\sigma }{2}\left(-{\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\right)\\ \mathrm{cos}\frac{\sigma }{2}\mathrm{cos}\frac{\lambda }{2}\\ {\text{e}}^{i\tau }\mathrm{sin}\frac{\sigma }{2}\end{array}\right)$

6. 不变量理论和么正变换的物理含义

$\frac{\partial I}{\partial t}+\frac{1}{i\hslash }\left[I,H\right]=0$ 类比于度规的metricity条件 ${\nabla }_{\lambda }{g}^{\mu \nu }=0$ 的展开 ${\partial }_{\lambda }{g}^{\mu \nu }+{\Gamma }^{\mu }{}_{\lambda \sigma }{g}^{\sigma \nu }+{\Gamma }^{\nu }{}_{\lambda \sigma }{g}^{\mu \sigma }=0$。从这个角

${H}_{V}={V}^{+}HV-{V}^{+}i\hslash \frac{\partial }{\partial t}V$，可以看出 ${H}_{V}$ 类比于自旋仿射联络(洛伦兹联络) ${\omega }_{\mu }{}^{pq}=i{e}^{p}{}_{\lambda }{\nabla }_{\mu }{e}^{q\lambda }$，它的具体展开式子是 $i{e}^{p}{}_{\lambda }\left({\partial }_{\mu }{e}^{q\lambda }+{\Gamma }^{\lambda }{}_{\mu \sigma }{e}^{q\sigma }\right)=i{e}^{p}{}_{\lambda }{\partial }_{\mu }{e}^{q\lambda }+i{e}^{p}{}_{\lambda }{\Gamma }^{\lambda }{}_{\mu \sigma }{e}^{q\sigma }$。(iv) ${H}_{V}={V}^{+}HV-{V}^{+}i\hslash \frac{\partial }{\partial t}V$ 可以化为

${H}_{V}{V}^{+}={V}^{+}H+i\hslash \frac{\partial }{\partial t}{V}^{+}$$i\hslash \frac{\partial }{\partial t}{V}^{+}+{V}^{+}H-{H}_{V}{V}^{+}=0$，此式其实类比于标架场(vierbein) ${e}^{p}{}_{\mu }$ 的协变导数为

${|\Psi \left(t\right)〉}_{+}=\left(\begin{array}{c}\mathrm{cos}\frac{\lambda \left(t\right)}{2}\\ \mathrm{sin}\frac{\lambda \left(t\right)}{2}{\text{e}}^{i\gamma \left(t\right)}\end{array}\right)$

${|\Psi \left(t\right)〉}_{-}=\left(\begin{array}{c}-\mathrm{sin}\frac{\lambda \left(t\right)}{2}{\text{e}}^{-i\gamma \left(t\right)}\\ \mathrm{cos}\frac{\lambda \left(t\right)}{2}\end{array}\right)$

$\begin{array}{l}{|\Psi \left(t\right)〉}_{+}{}_{+}〈\Psi \left(t\right)|+{|\Psi \left(t\right)〉}_{-}{}_{-}〈\Psi \left(t\right)|\\ =\left(\begin{array}{c}\mathrm{cos}\frac{\lambda \left(t\right)}{2}\\ \mathrm{sin}\frac{\lambda \left(t\right)}{2}{\text{e}}^{i\gamma \left(t\right)}\end{array}\right)\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda \left(t\right)}{2}& \mathrm{sin}\frac{\lambda \left(t\right)}{2}{\text{e}}^{-i\gamma \left(t\right)}\end{array}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+\left(\begin{array}{c}-\mathrm{sin}\frac{\lambda \left(t\right)}{2}{\text{e}}^{-i\gamma \left(t\right)}\\ \mathrm{cos}\frac{\lambda \left(t\right)}{2}\end{array}\right)\left(\begin{array}{cc}-\mathrm{sin}\frac{\lambda \left(t\right)}{2}{\text{e}}^{i\gamma \left(t\right)}& \mathrm{cos}\frac{\lambda \left(t\right)}{2}\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\end{array}$

$\begin{array}{c}{|\Psi \left(t\right)〉}_{+}{}_{+}〈\Psi \left(t\right)|=\left(\begin{array}{c}\mathrm{cos}\frac{\lambda \left(t\right)}{2}\\ \mathrm{sin}\frac{\lambda \left(t\right)}{2}{\text{e}}^{i\gamma \left(t\right)}\end{array}\right)\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda \left(t\right)}{2}& \mathrm{sin}\frac{\lambda \left(t\right)}{2}{\text{e}}^{-i\gamma \left(t\right)}\end{array}\right)\\ =\left(\begin{array}{cc}{\mathrm{cos}}^{2}\frac{\lambda \left(t\right)}{2}& \mathrm{cos}\frac{\lambda \left(t\right)}{2}\mathrm{sin}\frac{\lambda \left(t\right)}{2}{\text{e}}^{-i\gamma \left(t\right)}\\ \mathrm{cos}\frac{\lambda \left(t\right)}{2}\mathrm{sin}\frac{\lambda \left(t\right)}{2}{\text{e}}^{i\gamma \left(t\right)}& {\mathrm{sin}}^{2}\frac{\lambda \left(t\right)}{2}\end{array}\right)\\ =\frac{1}{2}\left(\begin{array}{cc}1+\mathrm{cos}\lambda & {\text{e}}^{-i\gamma }\mathrm{sin}\lambda \\ {\text{e}}^{i\gamma }\mathrm{sin}\lambda & 1-\mathrm{cos}\lambda \end{array}\right)\end{array}$

$\begin{array}{c}I\left(t\right)=l\left(t\right)\cdot J=\frac{1}{2}\mathrm{sin}\lambda \left(t\right)\mathrm{exp}\left[-i\gamma \left(t\right)\right]{J}_{+}+\frac{1}{2}\mathrm{sin}\lambda \left(t\right)\mathrm{exp}\left[i\gamma \left(t\right)\right]{J}_{-}+\mathrm{cos}\lambda \left(t\right){J}_{3}\\ =\frac{1}{2}\left(\begin{array}{cc}\mathrm{cos}\lambda & {\text{e}}^{-i\gamma }\mathrm{sin}\lambda \\ {\text{e}}^{i\gamma }\mathrm{sin}\lambda & -\mathrm{cos}\lambda \end{array}\right)\end{array}$

$\begin{array}{c}{|\Psi \left(t\right)〉}_{-}{}_{-}〈\Psi \left(t\right)|=\left(\begin{array}{c}-\mathrm{sin}\frac{\lambda \left(t\right)}{2}{\text{e}}^{-i\gamma \left(t\right)}\\ \mathrm{cos}\frac{\lambda \left(t\right)}{2}\end{array}\right)\left(\begin{array}{cc}-\mathrm{sin}\frac{\lambda \left(t\right)}{2}{\text{e}}^{i\gamma \left(t\right)}& \mathrm{cos}\frac{\lambda \left(t\right)}{2}\end{array}\right)\\ =\left(\begin{array}{cc}{\mathrm{sin}}^{2}\frac{\lambda \left(t\right)}{2}& -{\text{e}}^{-i\gamma \left(t\right)}\mathrm{sin}\frac{\lambda \left(t\right)}{2}\mathrm{cos}\frac{\lambda \left(t\right)}{2}\\ -{\text{e}}^{i\gamma \left(t\right)}\mathrm{sin}\frac{\lambda \left(t\right)}{2}\mathrm{cos}\frac{\lambda \left(t\right)}{2}& {\mathrm{cos}}^{2}\frac{\lambda \left(t\right)}{2}\end{array}\right)\\ =\frac{1}{2}\left(\begin{array}{cc}1-\mathrm{cos}\lambda & -{\text{e}}^{-i\gamma }\mathrm{sin}\lambda \\ -{\text{e}}^{i\gamma }\mathrm{sin}\lambda & 1+\mathrm{cos}\lambda \end{array}\right)\end{array}$

${e}^{aN}=\left(\begin{array}{cc}{|\Psi \left(t\right)〉}_{+}& {|\Psi \left(t\right)〉}_{-}\end{array}\right)$${\vartheta }^{Na}=\left(\begin{array}{c}{}_{+}〈\Psi \left(t\right)|\\ {}_{-}〈\Psi \left(t\right)|\end{array}\right)$

${e}^{aN}=\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& -{\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ {\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& \mathrm{cos}\frac{\lambda }{2}\end{array}\right)$${\vartheta }^{Na}=\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& {\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ -{\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& \mathrm{cos}\frac{\lambda }{2}\end{array}\right)$

$\begin{array}{c}\left[{g}^{MN}\right]=\left[{\vartheta }^{M}{}_{a}{e}^{aN}\right]=\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& {\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ -{\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& \mathrm{cos}\frac{\lambda }{2}\end{array}\right)\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& -{\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ {\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& \mathrm{cos}\frac{\lambda }{2}\end{array}\right)\\ =\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\end{array}$

$\begin{array}{c}\left[{\eta }^{ab}\right]=\left[{e}^{aN}{\vartheta }_{N}{}^{b}\right]=\left[{e}^{aN}{g}_{NM}{\vartheta }^{Mb}\right]\\ =\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& -{\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ {\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& \mathrm{cos}\frac{\lambda }{2}\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& {\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ -{\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& \mathrm{cos}\frac{\lambda }{2}\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\end{array}$

${I}^{a}{}_{b}{e}^{bN}=\sigma {e}^{aN}$${I}^{aN}=\sigma {e}^{aN}$

$I\left(t\right)=l\left(t\right)\cdot J=\frac{1}{2}\left(\begin{array}{cc}\mathrm{cos}\lambda & {\text{e}}^{-i\gamma }\mathrm{sin}\lambda \\ {\text{e}}^{i\gamma }\mathrm{sin}\lambda & -\mathrm{cos}\lambda \end{array}\right)$${e}^{aN}=\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& -{\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ {\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& \mathrm{cos}\frac{\lambda }{2}\end{array}\right)$

$\begin{array}{l}I\left(\begin{array}{cc}{|\Psi \left(t\right)〉}_{+}& {|\Psi \left(t\right)〉}_{-}\end{array}\right)\\ =\frac{1}{2}\left(\begin{array}{cc}\mathrm{cos}\lambda & {\text{e}}^{-i\gamma }\mathrm{sin}\lambda \\ {\text{e}}^{i\gamma }\mathrm{sin}\lambda & -\mathrm{cos}\lambda \end{array}\right)\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& -{\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ {\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& \mathrm{cos}\frac{\lambda }{2}\end{array}\right)\\ =\frac{1}{2}\left(\begin{array}{cc}\mathrm{cos}\lambda \mathrm{cos}\frac{\lambda }{2}+\mathrm{sin}\lambda \mathrm{sin}\frac{\lambda }{2}& {\text{e}}^{-i\gamma }\left(-\mathrm{cos}\lambda \mathrm{sin}\frac{\lambda }{2}+\mathrm{sin}\lambda \mathrm{cos}\frac{\lambda }{2}\right)\\ {\text{e}}^{i\gamma }\left(\mathrm{sin}\lambda \mathrm{cos}\frac{\lambda }{2}-\mathrm{cos}\lambda \mathrm{sin}\frac{\lambda }{2}\right)& -\left(\mathrm{sin}\lambda \mathrm{sin}\frac{\lambda }{2}+\mathrm{cos}\lambda \mathrm{cos}\frac{\lambda }{2}\right)\end{array}\right)\\ =\frac{1}{2}\left(\begin{array}{cc}\mathrm{cos}\frac{\lambda }{2}& {\text{e}}^{-i\gamma }\mathrm{sin}\frac{\lambda }{2}\\ {\text{e}}^{i\gamma }\mathrm{sin}\frac{\lambda }{2}& -\mathrm{cos}\frac{\lambda }{2}\end{array}\right)\end{array}$

$\begin{array}{c}I\left(\begin{array}{cc}{|\Psi \left(t\right)〉}_{+}& {|\Psi \left(t\right)〉}_{-}\end{array}\right)=\left(\begin{array}{cc}I{|\Psi \left(t\right)〉}_{+}& I{|\Psi \left(t\right)〉}_{-}\end{array}\right)\\ =\left(\begin{array}{cc}\frac{1}{2}{|\Psi \left(t\right)〉}_{+}& -\frac{1}{2}{|\Psi \left(t\right)〉}_{-}\end{array}\right)\\ =\frac{1}{2}\left(\begin{array}{cc}{|\Psi \left(t\right)〉}_{+}& -{|\Psi \left(t\right)〉}_{-}\end{array}\right)\end{array}$

7. 有关几何相位的讨论

1) 所谓绝热(绝热近似、绝热条件)，对一个二态体系而言，就是指它的哈密顿量算符内的演化参数如 $\omega$ (哈密顿量在参数空间内的进动频率)远远小于 ${\omega }_{0}$ (二态跃迁频率) [9] [10] [11]，那么我们就有 $\lambda -\theta \to 0$。所以，在绝热情形的例子中，量子系统的哈密顿量算符 $H\left(t\right)$ 与Lewis-Riesenfeld不变量算符 $I\left(t\right)$ [24] 是十分接近的、几乎相等的(因此我们可以用量子系统的哈密顿量算符 $H\left(t\right)$ 代替不变量算符 $I\left(t\right)$，也就是说， $H\left(t\right)$ 可以近似允许有瞬时本征态，那么自然Lewis-Riesenfeld不变量算符 $I\left(t\right)$ 也就不再需要了) [26]。但是，对于非绝热情形，哈密顿量 $H\left(t\right)$ 与不变量算符 $I\left(t\right)$ 相差太大 [26]，不变量算符 $I\left(t\right)$ 有本征态，但哈密顿量算符 $H\left(t\right)$ 没有(瞬时)本征态(即认为“哈密顿量 $H\left(t\right)$ 有瞬时本征态”的观点是错误的，这是因为如果有瞬时本征态，那么这就与含时Schrödinger方程相悖) [9] [10] [11] [26] [27] [28]。还有一种情形也值得考虑：那就是哈密顿量中耦合项比较小，即 $\mathrm{sin}\theta$ 接近于零。此时从不变量辅助方程可以看出， $\stackrel{˙}{\lambda }\to 0$$\stackrel{˙}{\gamma }\to {\omega }_{0}\mathrm{cos}\theta$

$\begin{array}{c}{V}^{+}HV-{V}^{+}i\frac{\partial }{\partial t}V=\left[{\omega }_{0}\left(\mathrm{cos}\theta \mathrm{cos}\lambda +\mathrm{sin}\theta \mathrm{sin}\lambda \mathrm{cos}\left(\phi -\gamma \right)\right)+\stackrel{˙}{\gamma }\left(1-\mathrm{cos}\lambda \right)\right]\frac{{\sigma }_{3}}{2}\\ \to \left[{\omega }_{0}\mathrm{cos}\theta \mathrm{cos}\lambda +{\omega }_{0}\mathrm{cos}\theta \left(1-\mathrm{cos}\lambda \right)\right]\frac{{\sigma }_{3}}{2}={\omega }_{0}\mathrm{cos}\theta \frac{{\sigma }_{3}}{2}\end{array}$

2) 下面讨论一下含时Schrödinger方程的解的幺正变换(规范变换)：

$i\frac{\partial }{\partial t}\left[\mathrm{exp}\left(\frac{1}{i}\phi \left(t\right)\right)|\psi \left(t\right)〉\right]=H\left(t\right)\left[\mathrm{exp}\left(\frac{1}{i}\phi \left(t\right)\right)|\psi \left(t\right)〉\right]$

$\begin{array}{l}i\frac{\partial }{\partial t}\left[\mathrm{exp}\left(\frac{1}{i}\stackrel{˜}{\phi }\left(t\right)\right)|\Psi \left(t\right)〉\right]=i\frac{\partial }{\partial t}\left[\mathrm{exp}\left(\frac{1}{i}\left(\stackrel{˜}{\phi }\left(t\right)-\phi \left(t\right)\right)\right)U\underset{_}{\mathrm{exp}\left(\frac{1}{i}\phi \left(t\right)\right)|\psi \left(t\right)〉}\right]\\ =i\left\{\frac{\partial }{\partial t}\left[\mathrm{exp}\left(\frac{1}{i}\left(\stackrel{˜}{\phi }\left(t\right)-\phi \left(t\right)\right)\right)U\right]\right\}\mathrm{exp}\left(\frac{1}{i}\phi \left(t\right)\right)|\psi \left(t\right)〉+\mathrm{exp}\left(\frac{1}{i}\left(\stackrel{˜}{\phi }-\phi \left(t\right)\right)\right)Ui\frac{\partial }{\partial t}\left[\underset{_}{\mathrm{exp}\left(\frac{1}{i}\phi \left(t\right)\right)|\psi \left(t\right)〉}\right]\\ =i\left\{\frac{\partial }{\partial t}\left[\mathrm{exp}\left(\frac{1}{i}\left(\stackrel{˜}{\phi }\left(t\right)-\phi \left(t\right)\right)\right)\right]\right\}\mathrm{exp}\left(\frac{1}{i}\phi \left(t\right)\right)U|\psi \left(t\right)〉+\mathrm{exp}\left(\frac{1}{i}\stackrel{˜}{\phi }\left(t\right)\right)i\frac{\partial U}{\partial t}{U}^{-1}\left(U|\psi \left(t\right)〉\right)\\ +\mathrm{exp}\left(\frac{1}{i}\left(\stackrel{˜}{\phi }\left(t\right)-\phi \left(t\right)\right)\right)UH\mathrm{exp}\left(\frac{1}{i}\phi \left(t\right)\right)|\psi \left(t\right)〉\end{array}$

$\begin{array}{l}i\frac{\partial }{\partial t}\left[\mathrm{exp}\left(\frac{1}{i}\stackrel{˜}{\phi }\left(t\right)\right)|\Psi \left(t\right)〉\right]\\ =i\left\{\frac{\partial }{\partial t}\left[\mathrm{exp}\left(\frac{1}{i}\left(\stackrel{˜}{\phi }\left(t\right)-\phi \left(t\right)\right)\right)\right]\right\}\mathrm{exp}\left(\frac{1}{i}\phi \left(t\right)\right)U|\psi \left(t\right)〉+\mathrm{exp}\left(\frac{1}{i}\stackrel{˜}{\phi }\left(t\right)\right)i\frac{\partial U}{\partial t}{U}^{-1}\left(U|\psi \left(t\right)〉\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\mathrm{exp}\left(\frac{1}{i}\left(\stackrel{˜}{\phi }\left(t\right)-\phi \left(t\right)\right)\right)\mathrm{exp}\left(\frac{1}{i}\phi \left(t\right)\right)UH{U}^{-1}\left(U|\psi \left(t\right)〉\right)\\ =i\left\{\frac{\partial }{\partial t}\left[\mathrm{exp}\left(\frac{1}{i}\left(\stackrel{˜}{\phi }\left(t\right)-\phi \left(t\right)\right)\right)\right]\right\}\mathrm{exp}\left(\frac{1}{i}\phi \left(t\right)\right)U|\psi \left(t\right)〉+i\frac{\partial U}{\partial t}{U}^{-1}\left[\mathrm{exp}\left(\frac{1}{i}\stackrel{˜}{\phi }\left(t\right)\right)|\Psi \left(t\right)〉\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+UH{U}^{-1}\left[\mathrm{exp}\left(\frac{1}{i}\stackrel{˜}{\phi }\left(t\right)\right)|\Psi \left(t\right)〉\right]\end{array}$

${H}_{U}=UH{U}^{-1}+i\frac{\partial U}{\partial t}{U}^{-1}$$\frac{\partial }{\partial t}\left[\mathrm{exp}\left(\frac{1}{i}\left(\stackrel{˜}{\phi }\left(t\right)-\phi \left(t\right)\right)\right)\right]=0$

3) 下面我们来研究SU(2)含时三生成元体系的不变量算符与哈密顿量算符的数学关系(相关的性质可能在一些文献内已经有隐现或交代 [2] [24] [26] [27] [28] )，这可以让读者进一步理解不变量算符和哈密顿量算符之间的依赖关系。Lewis-Riesenfeld不变量算符和量子系统的哈密顿量算符的生成元系数各自可以写为三维空间的单位矢量 $l\left(t\right)$$h\left(t\right)$ 形式，即

$l\left(t\right)=\left[\mathrm{sin}\lambda \left(t\right)\mathrm{cos}\gamma \left(t\right),\mathrm{sin}\lambda \left(t\right)\mathrm{sin}\gamma \left(t\right),\mathrm{cos}\lambda \left(t\right)\right]$

$h\left(t\right)=\left[\mathrm{sin}\theta \left(t\right)\mathrm{cos}\phi \left(t\right),\mathrm{sin}\theta \left(t\right)\mathrm{sin}\phi \left(t\right),\mathrm{cos}\theta \left(t\right)\right]$

$\begin{array}{c}h\cdot l=\mathrm{sin}\theta \mathrm{cos}\phi \mathrm{sin}\lambda \mathrm{cos}\gamma +\mathrm{sin}\theta \mathrm{sin}\phi \mathrm{sin}\lambda \mathrm{sin}\gamma +\mathrm{cos}\theta \mathrm{cos}\lambda \\ =\mathrm{cos}\theta \mathrm{cos}\lambda +\mathrm{sin}\theta \mathrm{sin}\lambda \left(\mathrm{cos}\phi \mathrm{cos}\gamma +\mathrm{sin}\phi \mathrm{sin}\gamma \right)\\ =\mathrm{cos}\theta \mathrm{cos}\lambda +\mathrm{sin}\theta \mathrm{sin}\lambda \mathrm{cos}\left(\phi -\gamma \right)\end{array}$

${V}^{+}HV-{V}^{+}i\frac{\partial }{\partial t}V=\left\{{\omega }_{0}\left(t\right)\left[\mathrm{cos}\theta \mathrm{cos}\lambda +\mathrm{sin}\theta \mathrm{sin}\lambda \mathrm{cos}\left(\phi -\gamma \right)\right]+\stackrel{˙}{\gamma }\left(1-\mathrm{cos}\lambda \right)\right\}\frac{{\sigma }_{3}}{2}$

4) 什么样的量子系统含有几何相位呢？当然，含时系统含有几何相位。但并非所有含时系统都含有几何相位，根据分析，必须要排除两种情形：(1) 不同时刻的哈密顿量对易的系统。这样的系统其哈密顿量只能是这样的形式： $H\left(t\right)={\omega }_{0}\left(t\right)\left[\mathrm{sin}\theta \mathrm{cos}\phi {J}_{1}+\mathrm{sin}\theta \mathrm{sin}\phi {J}_{2}+\mathrm{cos}\theta {J}_{3}\right]$，其中参量 $\theta$$\phi$ 不含时，仅仅 ${\omega }_{0}\left(t\right)$ 含时。从Lewis-Riesenfeld不变量辅助方程可以看出， $\stackrel{˙}{\lambda }={\omega }_{0}\left(t\right)\mathrm{sin}\theta \mathrm{sin}\left(\phi -\gamma \right)$

$\stackrel{˙}{\gamma }={\omega }_{0}\left(t\right)\left[\mathrm{cos}\theta -\mathrm{sin}\theta \mathrm{cot}\lambda \mathrm{cos}\left(\phi -\gamma \right)\right]$，参量 $\theta$$\phi$ 不含时，那么 $\lambda$$\gamma$ 也为常数( $\lambda =\theta$$\gamma =\phi$ )，这样几何相位( $\stackrel{˙}{\gamma }\left(1-\mathrm{cos}\lambda \right)$ 的时间积分)为零；(2) 哈密顿量中的非对角项(与 ${J}_{1}$有关的项)含有时谐振荡因子，其可以通过幺正变换，将时谐振荡因子变换掉(这在上面已经有论证)。此外，为了几何相位是可测相位，还要求哈密顿量(与不变量)满足循回条件( $H\left(0\right)=H\left(T\right)$$I\left(0\right)=I\left(T\right)$ ) [2] [26]。在绝热条件下， $I\left(0\right)=I\left(T\right)$$H\left(0\right)=H\left(T\right)$ 是等价的。但在非绝热条件下， $I\left(0\right)=I\left(T\right)$$H\left(0\right)=H\left(T\right)$ 更基本、更重要 [2] [26]。这一点从与不变量 [24] 有关的幺正变换方法 [26] 得到的新哈密顿量(可以直接用于计算几何相位)

${V}^{+}HV-{V}^{+}i\frac{\partial }{\partial t}V=\left\{{\omega }_{0}\left(t\right)\left[\mathrm{cos}\theta \mathrm{cos}\lambda +\mathrm{sin}\theta \mathrm{sin}\lambda \mathrm{cos}\left(\phi -\gamma \right)\right]+\stackrel{˙}{\gamma }\left(1-\mathrm{cos}\lambda \right)\right\}\frac{{\sigma }_{3}}{2}$

8. 本专题的衍伸意义

${\omega }_{\mu }{}^{ab}=i{e}^{aN}{\partial }_{\mu }{\vartheta }_{N}{}^{b}$ 的类比，我们可以知道 $-i\hslash {\stackrel{^}{V}}^{+}\frac{\partial }{\partial t}\stackrel{^}{V}$ 中的么正变换 $\stackrel{^}{V}$ 就是一种“标架场”。引力理论中

${e}_{p}{}^{\lambda }\left(\left[{D}_{\mu },{D}_{\nu }\right]{\varphi }^{p}\right)=-i{e}_{p}{}^{\lambda }{\Omega }_{\mu \nu }{{}^{p}}_{q}{\varphi }^{q}$，其中左边为 ${e}_{p}{}^{\lambda }\left(\left[{D}_{\mu },{D}_{\nu }\right]{\varphi }^{p}\right)=\left[{\nabla }_{\mu },{\nabla }_{\nu }\right]{\varphi }^{\lambda }={R}^{\lambda }{}_{\sigma \mu \nu }{\varphi }^{\sigma }$ ( ${R}^{\lambda }{}_{\sigma \mu \nu }$ 为黎曼曲率张量)。将以上两式右边作比较，我们就得到了自旋联络规范场张量 ${\Omega }_{\mu \nu }{{}^{p}}_{q}$ 与黎曼曲率张量 ${R}^{\lambda }{}_{\sigma \mu \nu }$ 之间的关系： $-i{e}_{p}{}^{\lambda }{\Omega }_{\mu \nu }{{}^{p}}_{q}{e}^{q}{}_{\sigma }={R}^{\lambda }{}_{\sigma \mu \nu }$。这一关系颇让人有所启发：由 ${D}_{\mu }{e}_{p}{}^{\lambda }={\nabla }_{\mu }{e}_{p}{}^{\lambda }-i{\omega }_{\mu p}{}^{q}{e}_{q}{}^{\lambda }=0$，我们可以得到自旋联络 ${\omega }_{\mu p}{}^{q}=i{e}_{p}{}^{\tau }{\nabla }_{\mu }{e}^{q}{}_{\tau }$，但这里要注意： ${\nabla }_{\mu }$ 是Levi-Civita联络协变导数算符。如果假设Levi-Civita联络为零(要么是全局为零，即恒为零；要么局域为零，即只在局域惯性系内为零)，那么上面自旋联络将退化为 ${\omega }_{\mu p}{}^{q}=i{e}_{p}{}^{\tau }{\partial }_{\mu }{e}^{q}{}_{\tau }$ 以及 ${D}_{\mu }{e}_{p}{}^{\lambda }={\partial }_{\mu }{e}_{p}{}^{\lambda }-i{\omega }_{\mu p}{}^{q}{e}_{q}{}^{\lambda }=0$。但是，这样一来，由 $\left[{D}_{\mu },{D}_{\nu }\right]{\varphi }^{p}=-i{\Omega }_{\mu \nu }{{}^{p}}_{q}{\varphi }^{q}$ 仍旧可以得到正确的自旋联络规范场张量 ${\Omega }_{\mu \nu }{{}^{p}}_{q}={\partial }_{\mu }{\omega }_{\nu }{{}^{p}}_{q}-{\partial }_{\nu }{\omega }_{\mu }{{}^{p}}_{q}-i{\left[{\omega }_{\mu },{\omega }_{\nu }\right]}^{p}{}_{q}$，似乎当我们假设Levi-Civita联络为零，并不导致麻烦。其实不然，如果Levi-Civita联络全局为零，那么黎曼曲率张量 ${R}^{\lambda }{}_{\sigma \mu \nu }$ 恒为零，这样一来上面的关系 $-i{e}_{p}{}^{\lambda }{\Omega }_{\mu \nu }{{}^{p}}_{q}{e}^{q}{}_{\sigma }={R}^{\lambda }{}_{\sigma \mu \nu }$ 就不再成立(左边非零，右边为零)，导致矛盾。那么问题出在哪里呢？原来只要存在自旋联络规范场张量 ${\Omega }_{\mu \nu }{{}^{p}}_{q}$，黎曼曲率张量也必须存在。虽然可以假设Levi-Civita联络局域为零，即只在局域惯性系内为零，但是Levi-Civita联络的导数必不为零(对应于潮汐力即引力落差必然存在)。例如，在 ${\Omega }_{\mu \nu }{{}^{p}}_{q}$ 内，有 ${\partial }_{\mu }{\omega }_{\nu }{{}^{p}}_{q}-{\partial }_{\nu }{\omega }_{\mu }{{}^{p}}_{q}$，在它里面，自旋联络携带有Levi-Civita联络，即使它局域为零，但是其导数却不为零，最终要求黎曼曲率张量也必须存在。如此说来，一旦存在自旋联络或者Yang-Mills场(Yang-Mills场是高维Lorentz转动群规范场)，其内黎曼曲率张量就为非零，如高维空间(也即Yang-Mills规范群空间)必须是弯曲的。所以，最终要求自旋联络(Lorentz转动对称性联

$\Gamma$ 被隐去了，其实它是可以显示出来的。证明如下：我们从Schrödinger方程 $\stackrel{^}{H}|\Psi 〉=i\hslash \frac{\partial }{\partial t}|\Psi 〉$ 出发，将其化为 ${V}^{+}\stackrel{^}{H}\left(V{V}^{+}|\Psi 〉\right)={V}^{+}i\hslash \frac{\partial }{\partial t}\left(V{V}^{+}|\Psi 〉\right)$。再展开为

${V}^{+}\stackrel{^}{H}V\left({V}^{+}|\Psi 〉\right)=i\hslash \frac{\partial }{\partial t}\left({V}^{+}|\Psi 〉\right)+{V}^{+}i\hslash \frac{\partial V}{\partial t}\left({V}^{+}|\Psi 〉\right)$

$\left({V}^{+}\stackrel{^}{H}V-{V}^{+}i\hslash \frac{\partial V}{\partial t}\right)\left({V}^{+}|\Psi 〉\right)=i\hslash \frac{\partial }{\partial t}\left({V}^{+}|\Psi 〉\right)$

$\frac{\partial }{\partial t}\left(V{V}^{+}|\Psi 〉\right)=\frac{\partial V}{\partial t}\left({V}^{+}|\Psi 〉\right)+V\frac{\partial }{\partial t}\left({V}^{+}|\Psi 〉\right)$，但是实际上此式也可以写为

$\begin{array}{c}\frac{\partial }{\partial t}\left(V{V}^{+}|\Psi 〉\right)=\left(\frac{\partial }{\partial t}V+\Gamma V\right)\left({V}^{+}|\Psi 〉\right)+V\left[\frac{\partial }{\partial t}\left({V}^{+}|\Psi 〉\right)-{V}^{+}\Gamma |\Psi 〉\right]\\ =\left({D}_{t}V\right)\left({V}^{+}|\Psi 〉\right)+V{D}_{t}\left({V}^{+}|\Psi 〉\right)\end{array}$

$\left({V}^{+}\stackrel{^}{H}V-{V}^{+}i\hslash {D}_{t}V\right)\left({V}^{+}|\Psi 〉\right)=i\hslash {D}_{t}\left({V}^{+}|\Psi 〉\right)$

9. 结论