论文标题
将翻译运算符转换为平面极性和球形坐标，以及它们在以表示无关的方式确定量子机械波函数中的使用
Converting translation operators into plane polar and spherical coordinates and their use in determining quantum-mechanical wavefunctions in a representation-independent fashion
论文作者
论文摘要
量子力学通常是在位置表示中开发的，但这不是必需的，即使对于波形，也可以以代表独立的方式执行计算。在这项工作中，我们说明了人们如何仅使用操作员以及这些操作员如何对状态向量作用，才能确定波形。要在平面极性和球形坐标中进行此操作，需要一个将翻译运算符转换为这些坐标。作为这种方法的示例，我们在两个和三个维度中说明了库仑问题的解决方案，而无需在位置空间中表达任何操作员。
Quantum mechanics is often developed in the position representation, but this is not necessary, and one can perform calculations in a representation-independent fashion, even for wavefunctions. In this work, we illustrate how one can determine wavefunctions, aside from normalization, using only operators and how those operators act on state vectors. To do this in plane polar and spherical coordinates requires one to convert the translation operator into those coordinates. As examples of this approach, we illustrate the solution of the Coulomb problem in two and three dimensions without needing to express any operators in position space.
