Group Theory: The Application to Quantum Mechanics

Many books explore group theory’s connection with physics, but few of them offer an introductory approach. This text provides upper-level undergraduate and graduate students with a foundation in problem solving by means of eigenfunction transformation properties. This study focuses on eigenvalue problems in which differential equations or boundaries are unaffected by certain rotations or translations. Its explanation of transformations induced in function space by rotations (or translations) in configuration space has numerous practical applications — not only to quantum mechanics but also to any other eigenvalue problems, including those of vibrating systems (molecules or lattices) or waveguides.
Points of special interest include the development of Schur's lemma, which features a proof illustrated with a symbolic diagram. The text places particular emphasis on the geometric representation of ideas: for instance, the similarity transformation is characterized as a rotation in multidimensional function space and the reduction is described in terms of mutual orthogonal spaces. General references provide suggestions for further study, citing works of particular clarity and readability.