在量子相变领域，系统在量子相变点附近的动力学行为、尤其是其跃迁几率的性质，近年来受到越来越多的重视。现阶段，可用于研究这类动力学行为、且较为普适而又有效的解析工具并不多。本申请拟研究与量子相变点附近的基态性质有关的跃迁几率、及相关物理量随时间的变化，探究半经典理论适用于此类研究的条件，并在可能的情况下推导半经典描述公式。具体而言，针对固定不变外控制参数、与外参数随时间缓慢变化这两种情况，我们将在一些具体模型中，利用数值计算与解析推导的方法，研究上述物理量的性质。同时，我们将分析半经典理论的适用条件，并在条件满足的情况下推导上述跃迁几率的半经典表达式；通过与具体模型的计算结果相比较，我们将检验半经典理论预言的适用程度。

In the field of quantum phase transition, in recent years, more and more attention has been paid to systems' dynamical behaviors in the vicinity of critical poionts, in particular, properties of transition probabilities. At the present stage, not many analytical tools， which are both generally applicable and efficient, have been developed for studying this type of dynamical behavior. This proposal plans to study time variation of transition probabilities as well as relevant physical quantities, which are related to ground states in the neighborhood of quantum phase transitions, find condition for applicability of semiclassical theory to this type of problem, and derive semiclassical descriptions when possible. Specifically, in the case of fixed external controlling parameter, as well as in the case of slowly varying controlling parameter, we'll study the above discussed physical quantities in several models, utilizing the methods of numerical simulation and analytical derivation. Meanwhile, we'll analyze condition for the applicability of semicalssical theory and, in the case of being applicable, derive semicalssical expressions of the above discussed transition probabilities; then, we'll test the semiclassical predictions, by comparison with numerical results obtained in concrete models.