分子点群特征标空间基于狄拉克符号的一种规范化表示

A normalized representation of character space based on Dirac symbols in molecular point group

由不可约表示(IRs)构建的特征标表是群论的基础,其中不可约表示的正交性已经由大正交定理(GOT)证明.本文将量子力学中的狄拉克符号引入群论教学,旨在用狄拉克符号代表的不可约表示为大正交定理提供简洁且容易理解的表述.由此可以用狄拉克符号写出由不可约表示展开的特征标空间的完备性的表达式.在特征标空间完备性的基础上,对两个分子(H_(2)O、NH_(3))的简正振动模式的可约表示用不可约表示展开,以说明狄拉克符号在群论中的应用.该教学方案可望帮助学生更好地理解分子点群、对称操作、不可约表示、大正交定理、特征标空间的完备性,以及可约表示的分解,进而提升教学质量.

Character table consisting of irreducible representations(IRs)is the basis of the group theory,and the orthonormality in irreducible representations has been proved by the great orthogonality theorem(GOT).In teaching group theory,we introduce Dirac symbols to represent irreducible representations,in order to provide a concise and easily understanding expression of the GOT.Consequently,the completeness of the character space,expanded by the irreducible representations,is expressed accordingly.On the basis of the completeness of character space,the reducible representations of the normal mode of vibrations of two molecules(H_(2)O and NH_(3))are expanded by the irreducible expansions,in order to illustrate the applications of Dirac symbols in group theory.Such teaching plan is expected to help students to better understand molecular point group,symmetry operations,irreducible representations,the GOT,completeness of character space,and the decomposition of reducible representation,toward improving the quality of teaching.