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氢原子量子力学系统可用动力学对称性代数方法求解。本文首先介绍了泡利的结果，这一结果给出了具有相同能量的各种本征态之间用算子联系的完整关系。其次用普通的代数运算，利用连带勒让德函数之间的关系也导出了同样的关系。作为这些运算的一个附带结果，本文还导出了径向波函数的一个递推关系，这一关系使我们能够从最简单的只有最高次的一个单项的径向波函数导出最复杂的径向波函数，类似于本文通常从最高权态利用阶梯算子得到任意态。
Abstract：The quantum mechanics of the hydrogen atom can be solved algebraically by the dynamical symmetry. In this note we introduced firstly Pauli's derivation of the complete relations among the various basic states with the same eigenvalue of the Hamiltonian. Secondly we also derived the same results by using algebraic method and some relations among the associated Legendre functions. As a byproduct, we also obtained a recursive relation of the radial wave functions. By using this recursive relation, we can derive the general radial wave function from the most simple radial wave function which has only an monomial term, much like deriving a general state from the highest weight state by using ladder operators.
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(1)虽然一般的初级量子力学课本如[1]并没有引入角动量的升降算符,但只要一小步就可以得到。给出这一个关系的量子力学课本可见如[2,3]。
基本信息:
中图分类号:O413.1
引用信息:
[1]朱传界.谈谈氢原子的动力学对称性[J].物理与工程,2025,35(01):5-9+20.
基金信息:
2020年湖南省普通高等学校教学改革研究项目“2十4物理师范生培养方案探索”(HNJG-2020-0737); 2020年湖南文理学院校级教改项目“2十4物理师范生培养方案探索”(JGZD2009); 湖南文理学院高层次人才引进启动基金