Singer S.F. Linearity, Symmetry, and Prediction in the Hydrogen Atom

Singer, Stephanie Frank, 2005, – 404 p. – ISBN10: 0-387-24637-1.Reading this book, you will learn about one of the great successes of 20th century mathematics—its predictive power in quantum physics. In the process, you will see three core mathematical subjects (linear algebra, analysis and abstract algebra) combined to great effect. In particular, you will see how to make predictions about the dimensions of the basic states of a quantum system from the only two ingredients: the symmetry and the linear model of quantum mechanics. This method, known as representation theory to mathematicians and group theory to physicists and chemists, has a wide range of applications: atomic structure, crystallography, classification of manifolds with symmetry, etc.Setting the Stage
Linear Algebra over the Complex Numbers
Complex Scalar Product Spaces (a.k.a. Hilbert Spaces)
Lie Groups and Lie Group Representations
New Representations from Old
Irreducible Representations and Invariant Integration
Representations and the Hydrogen Atom
The Algebra so(4) Symmetry of the Hydrogen Atom
The Group SO(4) Symmetry of the Hydrogen Atom
Projective Representations and Spin
Independent Events and Tensor Products