标题： 量子力学强微扰理论的高阶与世俗性问题 显示英文标题

标题： The Higher Orders of the Theory of Strong Perturbations in Quantum Mechanics and the Secularity Problem

摘要： 我们解决了M. Frasca在Phys. Rev. A 45, 43 (1992)中提出的量子力学强扰动理论中的高阶方程，通过假设在主阶上波函数是绝热的。这是通过推导主阶的绝热演化幺正算符来实现的。这样可以证明，相混合问题的一个至少原因之一，即扰动项通常称为secularities（久期项）的时间多项式增长效应，来源于哈密顿量的非摄动部分导致的扰动能级移动。给出一个两能级系统的例子，尽管如此，在小扰动的标准理论中，该系统在二阶时也表现出secularity（久期性）。该理论应用于经典问题的量子对应物，这个问题在受到不同振幅、频率和波数的两个波的影响下可能变得混沌。 显示英文摘要

摘要： We solve the higher order equations of the theory of the strong perturbations in quantum mechanics given in M. Frasca, Phys. Rev. A 45, 43 (1992), by assuming that, at the leading order, the wave function goes adiabatically. This is accomplished by deriving the unitary operator of adiabatic evolution for the leading order. In this way it is possible to show that at least one of the causes of the problem of phase-mixing, whose effect is the polynomial increase in time of the perturbation terms normally called secularities, arises from the shifts of the perturbation energy levels due to the unperturbed part of the hamiltonian. An example is given for a two-level system that, anyway, shows a secularity at second order also in the standard theory of small perturbations. The theory is applied to the quantum analog of a classical problem that can become chaotic, a particle under the effect of two waves of different amplitudes, frequencies and wave numbers.