Linearity, Symmetry, and Prediction in the Hydrogen Atom (eBook)

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.

---

This is a textbook for a senior-level undergraduate course for math, physics and chemistry majors. This one course can play two different but complimentary roles: it can serve as a capstone course for students finishing their education, and it can serve as motivating story for future study of mathematics. Some textbooks are like a vigorous regular physical training program, preparing people for a wide range of challenges by honing their basic skills thoroughly. Some are like a series of day hikes. This book is more like an intended trek to a particularly beautiful goal. We'll take the easiest route to the top, and we'll stop to appreciate local flora as well as distant peaks worthy of the vigorous training one would need to scale them. Advice to the Student: This book was written with many different readers in mind. Some will be mathematics students interested to see a beautiful and powerful application of a "e;pure"e; mathematical subject. Some will be students of physics and chemistry curious about the mathematics behind some tools they use, such as spherical harmonics. Because the readership is so varied, no single reader should be put off by occasional digressions aimed at certain other readers.