论文标题
电场和磁场中的氢原子：动态对称性，可整合和可集成的系统，精确的溶液
Hydrogen Atom in Electric and Magnetic Fields: Dynamical Symmetries, Superintegrable and Integrable Systems, Exact Solutions
论文作者
论文摘要
纯氢原子的哈密顿量具有由运动积分产生的SO（4）对称组：角动量和runge-lenz载体。纯氢原子是一个超对称和可整合的系统，因为在几个不同的坐标系统中，汉密尔顿 - 雅各布基和Schrödinger方程都是可分离的，并且具有精确的分析溶液。在均匀电场中的氢原子的Schrödinger方程（Stark效应）在抛物线坐标中可分离。该系统具有两个保守的量：广义runge-lenz载体和角动量的z射击。该问题是可以集成的，并且具有对称组SO（2）XSO（2）。氢分子的离子（两个库仑中心的问题）具有相似的对称组SO（2）XSO（2）（2）由两种保守的renge-lenz的保守Z-projentions和核轴上的角动量产生的。相应的schrödinger方程在椭圆坐标中是可分离的。对于均匀磁场中的氢原子，相应的schrödinger方程是不可分的。该问题是不可分割的，不可分割的，被认为是无法通过任何分析方法解决的量子混乱的代表性示例。然而，可以在均匀磁场中描述氢原子的量子状态的精确分析解决方案，作为两个变量的收敛功率序列，分别是两个变量，即半径和千角的正弦。对于磁场的任意强度，可以精确地计算地面和激发态的能级和波动函数。因此，尽管它没有超对称性和其他运动积分，但该问题可以视为可整合。
The Hamiltonian of a pure hydrogen atom possesses the SO(4) symmetry group generated by the integrals of motion: the angular momentum and the Runge-Lenz vector. The pure hydrogen atom is a supersymmetric and superintegrable system, since the Hamilton-Jacobi and the Schrödinger equations are separable in several different coordinate systems and has an exact analytical solution. The Schrödinger equation for a hydrogen atom in a uniform electric field (Stark effect) is separable in parabolic coordinates. The system has two conserved quantities: z-projections of the generalized Runge-Lenz vector and of the angular momentum. The problem is integrable and has the symmetry group SO(2)xSO(2). The ion of the hydrogen molecule (problem of two Coulomb centers) has similar symmetry group SO(2)xSO(2) generated by two conserved z-projections of the generalized Runge-Lenz and of the angular momentum on the internuclear axis. The corresponding Schrödinger equation is separable in the elliptical coordinates. For the hydrogen atom in a uniform magnetic field, the respective Schrödinger equation is not separable. The problem is non-separable and non-integrable and is considered as a representative example of quantum chaos that cannot be solved by any analytical method. Nevertheless, an exact analytical solution describing the quantum states of a hydrogen atom in a uniform magnetic field can be obtained as a convergent power series in two variables, the radius and the sine of the polar angle. The energy levels and wave functions for the ground and excited states can be calculated exactly, with any desired accuracy, for an arbitrary strength of the magnetic field. Therefore, the problem can be considered superintegrable, although it does not possess supersymmetry and additional integrals of motion.
