论文标题
磁场中的圆柱类型的可集成和可整合系统
Integrable and superintegrable systems of cylindrical type in magnetic fields
论文作者
论文摘要
本论文的目的是寻找具有磁场的可集成和可整合系统。我们制定了圆柱坐标中运动二阶积分的量子机械确定方程式，并发现圆柱类型的所有四四次集成系统。其中，我们搜索承认其他运动积分的系统。我们发现所有具有经典和量子力学中具有额外一阶组成部分的系统。事实证明，所有这些系统都已经知道，没有其他这些系统。我们还找到了所有具有$ l^2+\ ldots $类型的集成的系统，分别为$ l_y p_y p_y-l_x p_y+\ ldots $，其中大多数是文献的新事物。所有人都发现可促进的系统接纳了一阶积分$ L_Z $，我们通过在圆柱坐标中的变量分离来解决他们的汉密尔顿 - 雅各布和施罗丁方程，以便在笛卡尔坐标中的一阶系统。
The goal of this thesis is the search for integrable and superintegrable systems with magnetic field. We formulate the quantum mechanical determining equations for second order integrals of motion in the cylindrical coordinates and we find all quadratically integrable systems of the cylindrical type. Among them we search for systems admitting additional integrals of motion. We find all systems with an additional first order integral both in classical and quantum mechanics. It turns out that all these systems have already been known and no other exist. We also find all systems with an additional integral of type $L^2+\ldots$, respectively $L_y p_y-L_x p_y+\ldots$, of which the majority is new to the literature. All found superintegrable systems admit the first order integral $L_z$ and we solve their Hamilton-Jacobi and Schrödinger equations by separation of variables in the cylindrical coordinates, for the first order systems in the Cartesian coordinates as well.
