Linearity, symmetry, and prediction in the hydrogen atom在氢原子中的线性,对称和预测
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作者： Stephanie Frank Singer著
出 版 社：
出版时间： 2005-8-1字数：版次：页数： 396印刷时间： 2005/08/01开本： 16开印次：纸张： 胶版纸I S B N ： 9780387246376包装： 精装内容简介
The predictive power of mathematics in quantum phenomena is one of the great intellectual successes of the 20th century. This textbook, aimed at undergraduate or graduate level students (depending on the college or university), concentrates on how to make predictions about the numbers of each kind of basic state of a quantum system from only two ingredients: the symmetry and the linear model of quantum mechanics. This method, involving the mathematical area of representation theory or group theory, combines three core mathematical subjects, namely, linear algebra, analysis and abstract algebra. Wide applications of this method occur in crystallography, atomic structure, classification of manifolds with symmetry, and other areas.
The topics unfold systematically, introducing the reader first to an important example of a quantum system with symmetry, the single electron in a hydrogen atom. Then the reader is given just enough mathematical tools to make predictions about the numbers of each kind of electronic orbital based solely on the physical spherical symmetry of the hydrogen atom. The final chapters address the related ideas of quantum spin, measurement and entanglement.
This user-friendly exposition, driven by numerous examples and exercises, requires a solid background in calculus and familiarity with either linear algebra or advanced quantum mechanics. Linearity, Symmetry, and Prediction in the Hydrogen Atom will benefit students in mathematics, physics and chemistry, as well as a literate general readership.
A separate solutions manual is available to instructors.
作者简介：
Stephanie Frank Singer received her Ph.D. in Mathematics from the Courant Institute in 1991. In 2002 she resigned her tenured professorship at Haverford College. Since then she has been writing and consulting independently. Her first book was Symmetry In Mechanics: A Gentle, Modern Introduction.
目录
Preface
1 Setting the Stage
1.1 Introduction
1.2 Fundamental Assumptions of Quantum Mechanics
1.3 The Hydrogen Atom
1.4 The Periodic Table
1.5 Preliminary Mathematics
1.6 Spherical Harmonics
1.7 Equivalence Classes
1.8 Exercises
2 Linear Algebra over the Complex Numbers
2.1 Complex Vector Spaces
2.2 Dimension
2.3 Linear Transformations
2.4 Kernels and Images of Linear Transformations
2.5 Linear Operators
2.6 Cartesian Sums and Tensor Products
2.7 Exercises
3 Complex Scalar Product Spaces (a.k.a. Hilbert Spaces)
3.1 Lebesgue Equivalence and L2(N3)
3.2 Complex Scalar Products
3.3 Euclidean-style Geometry in Complex Scalar Product Spaces
3.4 Norms and Approximations
3.5 Useful Spanning Subspaces
3.6 Exercises
4 Lie Groups and Lie Group Representations
4.1 Groups and Lie Groups
4.2 The Key Players: SO(3), SU(2) and SO(4)
4.3 The Spectral Theorem for SU(2) and the Double Cover of SO(3)
4.4 Representations: Definition and Examples
4.5 Representations in Quantum Mechanics
4.6 Homogeneous Polynomials in Two Variables
4.7 Characters of Representations
4.8 Exercises
5New Representations from Old
5.1 Subrepresentations
5.2 Cartesian Sums of Representations
5.3 Tensor Products of Representations
5.4 Dual Representations
5.5 The Representation Horn
5.6 Pullback and Pushforward Representations
5.7 Exercises
6 Irreducible Representations and Invariant Integration
6.1 Definitions and Schur's Lemma
6.2 Elementary States of Quantum Mechanical Systems
6.3 Invariant Integration and Characters of Irreducible Representations
6.4 Isotypic Decompositions (Optional)
6.5 Classification of the Irreducible Representations of SU(2)
6.6 Classification of the Irreducible Representations of SO(3)
6.7 Exercises
7 Representations and the Hydrogen Atom
8 The Algebra so(4)Symmetry of the Hydrogen Atom
9 The Group So(4)Symmetry of the Hydrogen Atom
10 Projective Representations and Spin
11 Independent Events and Tensor Products
A Spherical Harmonics
B Proof of the Correspondence Between Irreducible Linear Representations of SU(2)and
C Suggested Paper Topics
Bibliography
Glossary of Symbols and Notation
Index
