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
Return to Quantum Mechanics main page.