几何相位、动力学不变量的对称性和薛定谔方程的精确解
Geometric Phases, Symmetries of Dynamical Invariants, and Exact Solution of the Schrödinger Equation
摘要：
我们引入了几何等价量子系统（GEQS）的概念，即对于给定的一组完整的初始状态向量，导致相同的几何相位的量子系统。我们给出了GEQS的一个特征。这些系统有一个共同的动力学不变量，它们的哈密顿量和演化算子通过不变量的对称变换联系在一起。如果不变量是$T$ -周期的，则相应的GEQS类包括具有$T$ -周期哈密顿量的系统。我们应用我们的一般结果来研究类的GEQS，包括一个系统的曲柄哈密顿$H(t)=e^{-iKt}H_0e^{iKt}$。我们证明了推转算子$K$也属于这一类。因此，尽管它是与时间无关的，但它导致了非平凡的循环演化和几何相位。我们的分析允许一个明确的建设一套完整的非定常循环状态的任何时间无关的简谐振子。这些循环状态的周期是振荡器特征周期的一半。
英文摘要：
We introduce the notion of the geometrically equivalent quantum systems (GEQS) as quantum systems that lead to the same geometric phases for a given complete set of initial state vectors. We give a characterization of the GEQS. These systems have a common dynamical invariant, and their Hamiltonians and evolution operators are related by symmetry transformations of the invariant. If the invariant is $T$-periodic, the corresponding class of GEQS includes a system with a $T$-periodic Hamiltonian. We apply our general results to study the classes of GEQS that include a system with a cranked Hamiltonian $H(t)=e^{-iKt}H_0e^{iKt}$. We show that the cranking operator $K$ also belongs to this class. Hence, in spite of the fact that it is time-independent, it leads to nontrivial cyclic evolutions and geometric phases. Our analysis allows for an explicit construction of a complete set of nonstationary cyclic states of any time-independent simple harmonic oscillator. The period of these cyclic states is half the characteristic period of the oscillator.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
