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Linearity, Simmetry, and Prediction in The Hydrogen Atom

Stephanie Frank Singer
Springer Verlag
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
[Reviewed by
William J. Satzer
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Linearity, Symmetry, and Prediction in the Hydrogen Atom, a volume in Springer’s series of Undergraduate Texts in Mathematics, is about representation theory and the fundamental role it plays in quantum mechanics and the prediction of quantum phenomena.  The author concentrates on one example of a quantum system with symmetry, namely a hydrogen atom with a single electron.  There are digressions aplenty, but the main goal is to make predictions of the numbers that characterize the basic state of a quantum system using only symmetry and the linear model of quantum mechanics.

The intended audience for this text includes advanced undergraduates or beginning graduate students in mathematics, physics or chemistry.  For physics and chemistry students, the author offers mathematical foundations for calculations that they may have already learned to do.  Mathematics students get to see a powerful application of “pure” mathematics in an extraordinary story of a mathematical success.

After some preliminary background material on quantum mechanics and linear algebra over the complex numbers, we are introduced to Lie groups and Lie group representations.  Right away we meet the major players: the special orthogonal groups SO(3) and SO(4) and the special unitary group SU(2).   The next couple of chapters describe how new representations are created from old ones, what irreducible representations are, and why we should care about invariant integration.  The book culminates with three chapters on the hydrogen atom, its symmetry and representations.  A key piece of this is a presentation of Fock’s original construction of a representation of SO(4) on L2 (R3), the phase space of the hydrogen atom.  Two final chapters look at projective representations, electron spin and quantum computing.

This is an attractive text with much to offer anyone interested in the interplay of mathematics, physics and chemistry.  The author suggests that the book is ideally suited for mixed groups of science and mathematics students who can benefit by sharing and pooling their expertise.  Nonetheless, the approach is rigorous and most results are proved, with the exception of a few things like the Stone-Weierstrass and Fubini theorems.  Prerequisites include a strong background in calculus and linear algebra; a nodding acquaintance with differentiable manifolds and the Lebesgue integral would also be useful.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.




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.- Represenatations 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.- A. Spherical Harmonics.- B. Proof of the Correspondence.- C. Suggested Paper Topics.- Bibliography.- Index.- References.