Mathematics of Classical and Quantum Physics - Dover Books on Physics | Essential Textbook for Physics Students & Researchers | Study Quantum Mechanics & Classical Physics Theories

This book is nothing short of a mathematical symphony! One of the best books on the subject of Hilbert Spaces and Inner Product Spaces (in addition to Fourier series and transform). The subject of inner product spaces is a subtle one and the authors do a superb job in developing it piece by piece in the most elegant way I've seen (so far). Chapters 3, 4, and 5 are devoted to the development of the subjects of inner product spaces, Hilbert Spaces, and related concepts such as Linear Transformations, Eigenvalues and Eigenvectors, and Orthogonality and Completeness. For a physics student, the materials in the these three chapters is all they'd need to manage through a Quantum Mechanics course at the level of both Griffiths' Introduction to Quantum Mechanics and Shankar's Principles of Quantum Mechanics and similar books (such as my favorite book on quantum mechanics: A Modern Approach to Quantum Mechanics by John Townsend). All the polynomials such Hermite, Legendre, and Laguerre polynomials which are necessary to understand the derivations of the various eigen-functions in QM are nicely developed early on in the book. I should also mention that the first two chapters of the book dealing with mathematics of Classical Mechanics are also extremely useful. Chapter two introduces the student to Calculus of Variation and related applications in Classical Mechanics. The book is divided into two volumes: chapters 1 through 5 together form the first volume and the rest (chapters 6 to 10) form the second volume. Chapter 7 is devoted to Green's function while chapter 10 is devoted to Group Theory.The book isn't easy! For a gentler introduction to the subjects of inner product spaces and Hilbert spaces, I would highly recommend Linear Algebra and Matrix Theory by Gilbert and Gilbert. It's an excellent book on the subject and a really easy book to read. The chapters are short, the problems and examples are sufficient, and the development is complete. If you're a physics undergrad, then this is really all you'd need.