ESSENTIAL QUANTUM MECHANICS

While still a relatively new graduate student, I once remarked to my advi
sor, Jim Cushing, that I still didn’t understand quantum mechanics. To this
he promptly replied: “You’ll spend the rest of your life trying to understand
quantum mechanics!” Despite countless books that the subject has spawned
since it first assumed a coherent form in the 1920s, quantum mechanics
remains notoriously, even legendarily, difficult. Some may believe students
should be told that physics really isn’t that hard, presumably so as not to
intimidate them. I disagree: what can be more demoralizing than struggling
mightily with a subject, only to be told that it’s really not that difficult?
Let me say it outright, then: quantum mechanics is hard. In writing
this book, I have not found any “magic bullet” by which I can render
the subject easily digestible. I have, however, tried to write a book that is
neither a popularization nor a “standard” text; a book that takes a modern
approach, rather than one grounded in pedagogical precedent; a book that
focuses on elucidating the structure and meaning of quantum mechanics,
leaving comprehensive treatments to the standard texts.
Above all, I have tried to write with the student in mind. The pri
mary target audience is undergraduates about to take, or taking, their first
quantum course. But my hope is that the book will also serve biologists,
philosophers, engineers, and other thoughtful people—people who are fasci
nated by quantum physics, but find the popularizations too simplistic, and
the textbooks too advanced and comprehensive—by providing a foothold
on “real” quantum mechanics, as used by working scientists.
Popularizations of quantum mechanics are intended not to expound
the subject as used by working scientists, but rather to discuss “quantum
weirdness,” such as Bell’s theorem and the measurement problem, in terms
palatable to interested non-scientists. As such, the mathematical level of
such books ranges from very low to essentially nonexistent.
In contrast, the comprehensive texts used in advanced courses often
make daunting conceptual and mathematical demands on the reader.
Preparation for such courses typically consists of a modern physics course,
but these tend to be rather conceptual. Modern physics texts generally
take a semi-historical approach, discussing topics such as the Bohr atom
and the Compton effect. Formalism is minimized and description empha
sized; the highly abstract mathematical and physical concepts of quantum
mechanics remain largely untouched. There is thus a rather large gap to be
bridged, and students in advanced courses may find that they must solve

problems and learn new applications even while the framework of quantum
mechanics remains unclear.
Neither popularization nor standard text, this book is intended to serve
in a variety of settings: as a primary text in a short course, a supple
mentary text in a standard course, a vehicle for independent study, or
a reference work. Knowledge of elementary calculus and basic complex
analysis should provide sufficient mathematical background (a condensed
discussion of these topics appears in Appendix A).
The book’s modernity is reflected in its overall style and tenor, but
also in some broad themes, such as the early and extensive use of Dirac
notation, and the fact that neither wavefunctions nor the time-independent
Schr¨ odinger equation are granted privileged status. Another such theme is
the adoption of the “statistical interpretation,” a very useful and lucid way
to understand how quantum mechanics works in actual practice. Because
the statistical interpretation is really a broad framework rather than an
interpretation per se, it is easily “imported” into other approaches as the
student may find necessary.
Notable by their absence from the book are many standard topics, such
as perturbation theory, scattering, and the Hydrogen atom. This is in keep
ing with a central motivating idea: that to properly understand the many
and varied applications of quantum mechanics, one must first properly
understand its overall structure. This implies a focus on fundamentals, such
as superposition and time evolution, with the result that they may then be
developed in a more detailed and explanatory style than in advanced texts.
Some authors seem to believe that if they provide a clear, elegant, terse
explanation, one time, any remaining confusion is the student’s responsi
bility. I disagree. Having taught (and learned) physics for many years at
many levels, I find that there are myriad ways to misunderstand the sub
ject, so I have tried to make this book especially explanatory and useful
for the student. Common variations in terminology and notation are clar
ified (e.g., the terms quantum state, state vector, and wavefunction). And
I discuss not only what is right, but what is wrong. For example, although
position-space and momentum-space are standard topics, students often
fail to realize that there is but one quantum state, which may be cast
into various representations. Such potential stumbling blocks are explicitly
pointed out and explained.
The great majority of problems are, to my knowledge, new. Most are
intended to help develop conceptual understanding. A vast array of addi
tional problems may be found in other quantum texts. The time-honored
physics dictum—that one doesn’t understand the physics unless one can
solve problems—bears repeating here. But so does its lesser-known cousin:
just solving problems, without the capacity to lucidly discuss those prob
lems and the attendant concepts and ideas, may also indicate insufficient

In part because this book is intended to transcend the traditional
physics audience, a few words about studying the subject are in order.
Much of our intellectual heritage–from art and music to social, political,
and historical thought–concerns our human experience of the world. By its
very nature, physics does not, and it is now clear that at the fundamental
level the physical world doesn’t conform to our preconceived ideas. The
concepts of physics, particularly quantum mechanics, can be exceedingly
abstract, their connections to our everyday experiences tenuous at best.
Because of this physical abstraction, and the requisite mathematical
sophistication, understanding can be hard to achieve in quantum mechan
ics. Nevertheless, I believe that understanding (not memorization) must be
the goal. To reach it, however, you may need to read more carefully, and
think more carefully, than ever before. This is an acquired skill! For most
humans it simply isn’t natural to exert the degree of concentration that
physics demands–you didn’t think quantum mechanics would be easy, did
you? The payoff for this hard work, to borrow Victor Weisskopf’s phrase,
is the joy of insight.
Essential Quantum Mechanics would not have become a reality absent
the freedom and support granted me by Northern Arizona University.
This includes a sabbatical spent, in part, developing the book at Loyola
University Chicago. Professor Ralph Baierlein generously and critically
read the manuscript and, as always, provided much wise and deeply
appreciated counsel. Professor Peter Kosso offered useful comments and
early encouragement. Sonke Adlung, of Oxford University Press, displayed
abundant patience, kindness, and professionalism in helping me through
the publishing process. Oxford’s Chloe Plummer endured my repeated
underestimates of the time required to correct the manuscript.
The influence of my late, great, Ph.D. advisor, Jim Cushing—whose life
put the lie to the notion that scientists are not real intellectuals—permeates
this book. My wife Katherine has been, and remains, a source of encourage
ment andforbearance through thick and thin. She also provided motivation,
often by asking: When are you going to finish that #&!* book? Finally, I
must thank my parents. Neither will see this book in print, yet both have
indelibly impacted my life, and continue to do so, regardless of my age.
After more than a few years on the planet, it sometimes seems to me
that there is but one great lesson to be learned. That is that the real worth
of a life is in contributing to the welfare of others. It is my hope that, in
some sense, and in some measure, I have done so with this book.
Flagstaff, Arizona
The true value of a human being is determined primarily by the mea
sure and the sense in which he has attained liberation from the self.