Introductory Quantum Mechanics

Abstract

This book is based on junior and senior level undergraduate courses that I have given at both New York University and the University of Michigan. You might ask, in heavens name, why anyone would want to write yet another introductory text on quantum mechanics. And you would not be far off base with this assessment. There are many excellent introductory quantum mechanics texts. Moreover, with the material available on the internet, you can access almost any topic of your choosing. Therefore, I must agree that there are probably no compelling reasons to publish this text. I have undertaken this task mainly at the urging of my students, who felt that it would be helpful to students studying quantum mechanics. For the most part, the approach taken is a traditional one. I have tried to emphasize the relationship of the quantum results with those of classical mechanics and classical electromagnetism. In this manner, I hope that students will be able to gain physical insight into the nature of the quantum results. For example, in the study of angular momentum, you will see that the absolute squares of the spherical harmonics can be given a relatively simple physical interpretation. Moreover, by using the effective potential in solving problems with spherical symmetry, I am able to provide a physical interpretation of the probability distributions associated with the eigenfunctions of such problems and to interpret the structures seen in scattering cross sections. I also try to stress the time-dependent aspects of problems in quantum mechanics, rather than focus simply on the calculation of eigenvalues and eigenfunctions. The book is intended to be used in a year-long introductory course. Chapters 1– 13 or 1–14 can serve as the basis for a one-semester course. I do not introduce Dirac notation until Chap. 11. I do this so students can try to master the wave function approach and its implications before engaging in the more abstract Dirac formalism. Dirac notation is developed in the context of a more general approach in which different representations, such as the position and momentum representations, appear on an equal footing. Most topics are treated at a level appropriate to an undergraduate course. Some topics, however, such as the hyperfine interactions described in the appendix of Chap. 21, are at a more advanced level. These are included for reference purposes, since they are not typically included in undergraduate (or graduate) texts. There is a web site for this book, http://wwwpersonal. umich.edu/~pberman/qmbook.html, that contains an Errata, Mathematica subroutines, and some additional material. The problems form an integral part of the book. Many are standard problems, but there are a few that might be unique to this text. Quantum mechanics is a difficult subject for beginning students. I often tell them that falling behind in a course such as this is a disease from which it is difficult to recover. In writing this book, my foremost task has been to keep the students in mind. On the other hand, I know that no textbook is a substitute for a dedicated instructor who guides, excites, and motivates students to understand the material. I would like to thank Bill Ford, Aaron Leanhardt, Peter Milonni,Michael Revzen, Alberto Rojo, and Robin Shakeshaft for their insightful comments. I would also like to acknowledge the many discussions I had with Duncan Steel on topics contained in this book. Finally, I am indebted to my students for their encouragement and positive (as well as negative) feedback over the years. I am especially grateful to the Fulbright foundation for having provided the support that allowed me to offer a course in quantum mechanics to students at the College of Science and Technology at the University of Rwanda. My interactions with these students will always remain an indelible chapter of my life.