## Quantum Mechanics

Ernest  Abers

Preface

The principles of quantum mechanics  were formulated by many people during a short period of time at the beginning of the twentieth century. Max Planck wrote down his formula for the spectrum of blackbody radiation and introduced the constant that now bears his name in 1900. By 1924, through the work of Einstein, Rutherford and Bohr, Schrödinger and Heisenberg, Born, Dirac, and many others, the principles of quantum mechanics were discovered much as we know them today. They have become the framework for thinking about  most of the phenomena that physicists study, from simple systems like atoms, molecules, and nuclei to more exotic ones like neutron stars, superfluids, and elementary particles.

This book is a text for an advanced course in quantum mechanics and, indeed, started out as notes for a graduate course at UCLA. Usually students in any field of physics must study quantum mechanics at this level before undertaking more specialized subjects.

The first part  covers some of the formalism of quantum mechanics, especially the mathematics of rotations and other symmetries.  It begins with a brief review of the Hamiltonian formulation of classical mechanics, which has become a trustworthy guide to finding the form of the quantum rules.  The second chapter   explains how the canonical quantum rules follow from  the superposition principle and some form of the correspondence principle. It ends with the Schrödinger equation and the uncertainty principle.

The third chapter is about stationary states and the energy eigenvalue problem, with particular emphasis on spherical symmetry. It includes the theory of orbital angular momentum and the famous hydrogen atom problem. The latter will serve  as a wonderful example over and over again.

The next two chapters are about the role of symmetry transformations in quantum mechanics, and how they restrict the possible values of some observables. There is a detailed discussion of three-dimensional rotations, the general theory of angular momentum, addition of angular momentum and selection rules. A good understanding of rotations in quantum mechanical systems is important for what follows. Rotations are an example for all sorts of other symmetries   we have discovered or invented. The techniques learned in this context can be recycled many times.

These first five chapters contain the mathematical foundation of our subject. I have tried to be fairly rigorous, understanding that this is the students' second course in quantum mechanics.

There follows a brief interlude containing a miscellany of short subjects: magnetic field interactions, measurement and probability, the density matrix, and a recently discovered example of a simple quantum system,  neutrino oscillations.

The rest is application. There is a section on bound-state perturbation theory, with the hydrogen atom as an example.  There is a brief discussion of the variational principle, important in the theory of atomic and molecular structure, and of the WKB method.  Transitions are  introduced next in the context of potential scattering, with some applications to atoms and nuclei.

Next I have chosen a topic that students seem to enjoy learning about but which is hard to find in much detail in most textbooks at this level. This is  the theory of transitions in general and, in particular, decay rates for excited states. There is an introduction to  path integration and a section on geometric phases.

Then comes the theory of photons, the quantized electromagnetic field. Historically, this subject came first. The blackbody spectrum and  the photoelectric effect were explained in terms of photons,  quanta of the electromagnetic field,  more than two decades  before a real theory was available. Now, with the full  power of the machinery of quantum mechanics in hand, we can understand  completely those observations that puzzled Planck and his contemporaries. The quantum theory of the electromagnetic field is a useful subject to learn in its own right,  and it is a good introduction to the methods used in both  many body physics and elementary particle physics.

Next there is a chapter on relativistic wave equations, developed in the spirit of the earlier discussion of rotational symmetry, but here the symmetry is Lorentz invariance. I conclude with the occupation number space  description of systems of identical particles, with a few applications.

I have tried to show the details of most mathematical calculations, and tried not to claim that one line follows easily from another unless my experience  is that an average student will actually find this to be true. For the same reason I have included in the appendix derivations of many mathematical formulas even though most of them can be found in standard works.

If you want to learn quantum mechanics from this book, you need some preparation. Only the most extraordinary student could be expected to get through this material  without the benefit of an  introductory course, though in principle it is possible.   You should also have studied classical mechanics and some mathematical methods at an introductory level.

The quantum mechanics ``prerequisite''  is to know what Schrödinger's wave equation is and how to use it. That means knowing  how to find the bound states of a given potential in one and three dimensions, about tunneling problems, transmission and reflection coefficients, momentum and energy eigenfunctions, the elementary theory of the  harmonic oscillator and  the hydrogen atom.  You can learn about them in more detail in some  of the books listed among the references at the back.

The mathematical prerequisite is minimal. The quantum chapter of the book of nature is written in the language of linear algebra, which is the mathematical formulation of  the superposition principle. I do not expect you to have studied Hilbert spaces or group theory previously.  Pieces of the  mathematics of linear vector spaces are presented as the need arises. But you should  already know a little about vector calculus in curvilinear coordinates, and  elementary concepts of vector space methods such as eigenvalues, Hermitean and unitary matrices, changes of basis, eigenfunction expansions, and so forth. I shall repeat the definitions of these tools, but this  is not the place to learn them for the first time. A  nodding acquaintance with complex numbers is also useful. More advanced parts of complex analysis, including the residue theorem, will be touched on only in the later parts.

You should also know some undergraduate-level classical mechanics, in particular  the central force problem and the Lagrangian and Hamiltonian formalisms.  I avoid mentioning Poisson brackets in the body of the text, but as they provide an important insight into the structure of quantum mechanics, several problems are devoted to them.

Over the years I have assembled  a collection of problems for the graduate quantum mechanics course. Some of the problems fill in gaps in the exposition. Most are the way to learn the tools of our trade. Occasionally the problems develop some themes not explained  thoroughly in the body of the text. For a few of the problems you need to have access to a computer and know how to use it. Most are to be done analytically. You must work out many of the problems if you want to understand what is going on.

I have enjoyed collaborating with the many people at Prentice Hall and their associates who worked to turn the  manuscript into this book. My thanks to Erik Fahlgren, my acquisitions editor, to Debra Wechsler, the production editor, to Daphne Hougham, who copyedited the manuscript, to Andrew Sobel, Bayani DeLeon, Adam Lewenberg, and many others whose names I do not know.

I am indebted to those who read earlier drafts with  care and made suggestions for improvement, almost all of which I have included in the final version. Many thanks especially to Mike Berger (Indiana University), John Donoghue (University of Massachusetts), Colin Gay (Yale University), Maarten Golterman (San Francisco State University), Herbert Hamber (University of California,  Irvine), Thomas Mehen (Duke University), Chandra Raman (Georgia Institute of Technology), Serge Rudaz (University of Minnesota), and several anonymous reviewers.  My special thanks to Kenneth Lane and his students at Boston University.

I thank the Department of Physics and Astronomy at UCLA for granting me the time to complete this manuscript, and the very many students over the years who suggested improvements or corrections in earlier versions. Finally, I thank my colleagues at UCLA and elsewhere for their criticism, advice, encouragement, and conversations about quantum mechanics.  I am particular grateful for the discussions I have had with Sudip Chakravarty, John M. Cornwall, Robert Cousins, Carlos A. A. de Carvalho, Eric d'Hoker, Robert Finkelstein, Graciela Gelmini, Noah Graham, Alex Kusenko, Richard Norton, Shmuel Nussinov, Silvia Pascoli, Hidenori Sonoda, and  E. Terry Tomboulis.

Ernest S. Abers
University of California,
Los Angeles