论文标题
物理中的计算机代数：隐藏的氢原子对称性
Computer Algebra in Physics: The hidden SO(4) symmetry of the hydrogen atom
论文作者
论文摘要
Pauli首先注意到量子力学早期阶段的氢原子的隐藏SO（4）对称性[1]。偏离了这种对称性，可以恢复无旋转氢原子的频谱以及其状态的变性，而无需明确求解Schrödinger方程[2]。在本文中，我们使用计算机代数系统（CAS）得出（4）对称和频谱。虽然这个问题是众所周知的[3，4]，但其解决方案涉及使用张力量子操作员来操纵表达式的几个步骤，从而通过考虑换向器规则和爱因斯坦的重复指标的总和规则来简化它们。因此，这是一个出色的模型，用于测试CAS与这种量子和敏感的代数计算有关的当前状态。一般而言，CAS能够大大帮助进行操纵，这些操纵像受代数规则的非交流性张量子微积分一样，是乏味，耗时且容易出错的。演示文稿还显示了计算机代数操作的模式，该模式对于系统地解决此类象征性问题更复杂。
Pauli first noticed the hidden SO(4) symmetry for the Hydrogen atom in the early stages of quantum mechanics [1]. Departing from that symmetry, one can recover the spectrum of a spinless hydrogen atom and the degeneracy of its states without explicitly solving Schrödinger's equation [2]. In this paper, we derive that SO(4) symmetry and spectrum using a computer algebra system (CAS). While this problem is well known [3, 4], its solution involves several steps of manipulating expressions with tensorial quantum operators, simplifying them by taking into account a combination of commutator rules and Einstein's sum rule for repeated indices. Therefore, it is an excellent model to test the current status of CAS concerning this kind of quantum-and-tensor-algebra computations. Generally speaking, when capable, CAS can significantly help with manipulations that, like non-commutative tensor calculus subject to algebra rules, are tedious, time-consuming and error-prone. The presentation also shows a pattern of computer algebra operations that can be useful for systematically tackling more complicated symbolic problems of this kind.
