Operator Methods in Quantum Mechanics by Martin Schechter (PDF)

Description

Operator Methods in Quantum Mechanics demonstrates the power of operator theory as a tool in the study of quantum mechanics. More specifically, it shows how to use algebraic, representation-independent methods to solve one- and three-dimensional problems, including certain relativistic problems. It explains the applications of commutation relations, shift operators, and the virial, hypervirial, and Hellman-Feyman theorems to the calculation of eigenvalues, matrix elements, and wave functions. Organized into 16 chapters, this book begins by presenting a few simple postulates describing quantum theory and looking at a single particle moving along a straight line. Then, it introduces mathematical techniques that answer questions about the particle. It also discusses the use of spectral theorem in answering various questions concerning observables, along with negative eigenvalues and methods of determining parts of the spectrum or estimating lower bounds. Moreover, it explains the time-independent or stationary-state scattering theory and states, long-range potentials, and completeness and strong completeness. Oscillating potentials, eigenfunction expansions, restricted particles, hard-core potentials, the invariance principle, and the use of trace class operators to treat scattering theory are also described in this book. This volume is a valuable resource for physicists, as well as students of intermediate quantum mechanics and postgraduate students who want to be acquainted with the algebraic method of solving quantum mechanical problems.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The book is about the mathematics of a single particle in 1-D space, the line, written for math people. Yeah, it’s 300 pages for a little guy moving in a straight line! Isn’t that enough?It’s about mathematical modeling, power, and beauty. Physics background is not required. Maybe no idea is even better! Prof. Schechter has spent much time preparing the contents, so you don’t have to look around.This book could be used as a first course in mathematical physics, or a supplement of mathematical modeling or functional analysis. Prof. Schechter teaches us about translating QM postulates into math. Starts off very simple 1-D case for a particle, and no more. Again, isn’t that enough? I think “encouraging” is the best word to describe his style for this book. A very gentle math teacher. The quality of math is as good as von Neumann or Weyl, and the representation is better. He even tells you what probability really is. (Do you know?)It’s more user-friendly than many other mathematical physics books. And the cover is maybe one of the most beautiful among Dover paperbacks.Yet another “Mathematical Foundations of Quantum Mechanics” — a very unique one!I was very lucky to encounter this book which was almost hidden in a pile of books. Attracted by the cover design, so I pulled it out hard. Then found that I was right.If you are kind-of math guy and interested in QM or modeling as well, check this book out. It’s hard to find in bookstores, though.___________________________________”Though this be madness, yet there’s method in it.” — Polunius, in Hamlet.

⭐This book is excellent. I don’t pretend it should be used as a course for either quantum mechanics or functional analysis. It is at a level way above that necessary for quantum mechanics and it does not cover enough material to be a serious textbook for functional analysis. That said at 15$ it is a great addition to any library. A reader can move through the book and really get an understanding of the topics presented. The books by Reed and Simon are a must for any mathematical physicist but they can sometimes be intimidating. The material presented here is relevant and presented in a very straightforward manner. If you have a background in analysis and wish to learn more about quantum mechanics I would definitely recommend this book. Another possible use is as an additional text in a functional analysis course so students can see direct applications.

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