固態物理群論運用 Group theory in Solid-state physics

開課單位

物理學系

授課教師

黃信銘

時數

8

學分

0.4

課程場次

其他報名

https://forms.gle/umXQaeqpHwrqnhZZ8

地點

PHYS 5009

人數限制

20

課程目標

本課程旨在介紹固態物理中常用的基礎群論與對稱性概念。學生僅需具備線性代數及量子力學中特徵值問題的基礎知識，即可循序理解群、子群、共軛類、陪集及群表示等數學結構。課程將以晶格與晶體系統為主要對象，探討點群、空間群及其對稱操作，並培養學生閱讀點群特徵標表與空間群表的能力。完成課程後，學生將能運用群論辨識物理系統的對稱性、分析不可約表示與能階簡併，並理解這些工具在固態物理能帶理論中的基本應用。

This course introduces the fundamental concepts of group theory and symmetry commonly used in solid-state physics. Only a basic knowledge of linear algebra and eigenvalue problems in quantum mechanics is required. Students will develop an understanding of the mathematical structures of groups, subgroups, conjugacy classes, cosets, and group representations. With an emphasis on lattices and crystalline systems, the course covers symmetry operations, point groups, and space groups, while developing students’ ability to interpret character tables and space-group tables. By the end of the course, students will be able to identify the symmetries of physical systems, analyze irreducible representations and energy-level degeneracies, and understand how group-theoretical methods are applied to the fundamentals of electronic band theory.

課程內容

(本課程若有外籍生選修，將以英語講授。)
本課程以固態物理中的晶格與晶體對稱性為主軸，介紹群論的基本概念及其物理應用。課程內容包括群、子群、共軛類、陪集與商群等數學結構，以及群表示、可約與不可約表示、特徵標表、正交定理和投影算符。學生將學習辨識旋轉、鏡射、反演、旋轉反演、螺旋軸及滑移面等對稱操作，並閱讀點群與空間群的符號及相關表格。課程最後將說明如何利用對稱性分析量子態、能階簡併及選擇定則，並初步連結群論與固態物理中的能帶理論。
本課程透過數學結構、物理概念與晶體實例的整合，培養學生運用基礎科學知識分析固態物理問題的能力。學生將線性代數及量子力學中的特徵值概念應用於群表示與對稱性分析，可強化數理推導、抽象思考及物理建模能力。

透過點群特徵標表與空間群表的閱讀和實作，學生將培養查找、判讀及整合專業資料的能力，並能依據系統的對稱性判斷物理量、量子態與能階可能具有的性質。課程亦藉由不可約表示、投影算符及能帶簡併等例題，訓練學生將理論方法應用於具體問題，建立進一步研習固態物理、凝態物理、材料科學及相關領域所需的基礎專業能力。

(The course will be taught in English if any international students enroll.)
This course focuses on lattice and crystal symmetries in solid-state physics and introduces the fundamental concepts of group theory and their physical applications. Topics include groups, subgroups, conjugacy classes, cosets, quotient groups, group representations, reducible and irreducible representations, character tables, orthogonality theorems, and projection operators. Students will learn to identify symmetry operations such as rotations, reflections, inversion, rotoinversion, screw rotations, and glide reflections, and to interpret the notation and tables associated with point groups and space groups. The course concludes with applications of symmetry to quantum states, energy-level degeneracies, and selection rules, providing an introduction to the role of group theory in electronic band theory.

The course supports the development of the following professional competencies:
1. Applying mathematics and fundamental physics to describe and analyze physical systems.
2. Identifying symmetry elements, point groups, and space groups in lattice and crystalline systems.
3. Using group representations, character tables, and projection operators to analyze quantum states and physical quantities.
4. Reading, interpreting, and integrating information from point-group tables, space-group tables, and related references.
5. Developing the capacity for independent study of advanced theoretical tools in condensed-matter physics and materials science.

限修條件

無

參考/指定用書

Dresselhaus, Group Theory (Springer, 2008)

聯絡資訊

聯絡人:黃信銘

信箱:shinming@mail.nsysu.edu.tw

電話:3729

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