论文标题
多维氢态：位置和动量期望值
Multidimensional hydrogenic states: Position and momentum expectation values
论文作者
论文摘要
多维量子系统的位置和动量概率密度分别通过径向预期值$ \ langle r^α\ rangle $和$ \ weft \ langle p^α\ right \ rangle $。这些数量描述和/或与现实系统的各种基本特性密切相关，直到现在迄今尚未以分析有效的方式计算，除了许多三维氢态。在这项工作中，我们明确地显示了所有离散固定$ d $ d $维氢状态的期望值，就尺寸$ d $，库仑电势的强度（即核电）和$ d $ nate的超级量子数量而言。重点放在以封闭的紧凑形式获得的动量期望值（主要是未知的，特别是具有奇数的阶数）。应用于圆形，$ S $波，高能（Rydberg）和三维和多维氢原子的高维（伪经典）状态。这是可能的，因为特殊函数的分析代数和渐近性能（正交多项式，超声谐波）控制状态的波形。最后，还给出并讨论了一些因这些分散量所满足的Heisenberg样不确定性不平等。
The position and momentum probability densities of a multidimensional quantum system are fully characterized by means of the radial expectation values $\langle r^α\rangle$ and $\left\langle p^α\right\rangle$, respectively. These quantities, which describe and/or are closely related to various fundamental properties of realistic systems, have not been calculated in an analytical and effective manner up until now except for a number of three-dimensional hydrogenic states. In this work we explicitly show these expectation values for all discrete stationary $D$-dimensional hydrogenic states in terms of the dimensionality $D$, the strength of the Coulomb potential (i.e., the nuclear charge) and the $D$ state's hyperquantum numbers. Emphasis is placed on the momentum expectation values (mostly unknown, specially the ones with odd order) which are obtained in a closed compact form. Applications are made to circular, $S$-wave, high-energy (Rydberg) and high-dimensional (pseudo-classical) states of three- and multidimensional hydrogenic atoms. This has been possible because of the analytical algebraic and asymptotical properties of the special functions (orthogonal polynomials, hyperspherical harmonics) which control the states' wavefunctions. Finally, some Heisenberg-like uncertainty inequalities satisfied by these dispersion quantities are also given and discussed.
