Quantum Mechanics

Abstract

This book is an outgrowth of lectures given at the University of Michigan at various times from 1966-1996 in a first-year graduate course on quantum mechanics. It is meant to be at a fairly high level. On the one hand, it should provide future research workers with the tools required to solve real problems in the field. On the other hand, the beginning graduate courses at the University of Michigan should be self-contained. Although most of the students will have had an undergraduate course in quantum mechanics, the lectures are intended to be such that a student with no previous background in quantum mechanics (perhaps an undergraduate mathematics or engineering major) can follow the course from beginning to end. Part I of the course, Introduction to Quantum Mechanics, thus begins with a brief background chapter on the duality of nature, which hopefully will stimulate students to take a closer look at the two references given there. These references are recommended for every serious student of quantum mechanics. Chapter 1 is followed by a review of Fourier analysis before we meet the SchrMinger equation and its interpretation. The dual purpose of the course can be seen in Chapters 4 and 5, where an introduction to simple square well problems and a first solution of the one-dimensional harmonic oscillator by Fuchsian differential equation techniques are followed by an introduction to the Bargmann transform, which gives us an elegant tool to show the completeness of the harmonic oscillator eigenfunctions and enables us to solve some challenging harmonic oscillator problems, (e.g., the case of general n for problem 11). Early chapters (7 through 12) on the eigenvalue problem are based on the coordinate representation and include detailed solutions of the spherical harmonics and radial functions of the hydrogen atom, as well as many of the soluble, one-dimensional potential problems. These chapters are based on the factorization method. It is hoped the ladder step-up and step-down operator approach of this method will help to lead the student naturally from the SchrOdinger equation approach to the more modem algebraic techniques of quantum mechanics, which are introduced in Chapters 13 to 19. The full Dirac bra, ket notation is introduced in Chapter 13. These chapters also give the full algebraic approach to the general angular momentum problem, SO(3) or SU(2), the harmonic oscillator algebra, and the SO(2, 1) algebra. The solution for the latter is given in problem 23, which is used in considerable detail in later chapters. The problems often amplify the material of the course. Part II of the course, Chapters 20 to 26, on time-independent perturbation theory, is based on Fermi's view that most of the important problems of quantum mechanics can be solved by perturbative techniques. This part of the course shows how various types of degeneracies can be handled in perturbation theory, particularly the case in which a degeneracy is not removed in lowest order of perturbation theory so that the lowest order perturbations do not lead naturally to the symmetryadapted basis; a case ignored in many books on quantum mechanics and perhaps particularly important in the case of accidental near-degeneracies. Chapters 25 and 26 deal with magnetic-field perturbations, including a short section on the Aharanov-Bohm effect, and a treatment on fine structure and Zeeman perturbations in one-electron atoms.