Table Of ContentQuantumMechanics
Daniel R. Bes
Quantum Mechanics
A Modern and Concise Introductory Course
Second,RevisedEdition
With 57 Figures, 1 in Color and 10 Tables
123
Daniel R. Bes
PhyiscsDepartment,CAC, CNEA
Av.General Paz 1499
San Martin, Prov. de Buenos Aires
Argentina 1650
[email protected]
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ISBN-103-540-46215-5SpringerBerlinHeidelbergNewYork
ISBN-13978-3-540-46215-6SpringerBerlinHeidelbergNewYork
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A primeval representation of the hydrogen atom
This beautiful mandala is displayed at the temple court of Paro Dzong, the
monumental fortress of Western Bhutan [1]. It may be a primeval representa-
tionofthehydrogenatom:theouterredcircleconveysameaningofstrength,
which may correspond to the electron binding energy. The inner and spheri-
calnucleusissurroundedbylarge,osculatingcirclethatrepresentthemotion
of the electron: the circles do not only occupy a finite region of space (as in
Fig.6.4),butarealsoassociatedwithtrajectoriesofdifferentenergies(colours)
and/or with radiation transitions of different colours (wavelengths). At the
center, within the nucleus, there are three quarks.
Foreword
Quantum mechanics is undergoing a revolution. Not that its substance is
changing, but two major developments are placing it in the focus of renewed
attention, both within the physics community and among the scientifically
interested public. First, wonderfully clever table-top experiments involving
the manipulation of single photons, atomic particles, and molecules are reve-
aling in an ever-more convincing manner theoretically predicted facts about
the counterintuitive and sometimes ‘spooky’ behavior of quantum systems.
Second, the prospect of building quantum computers with enormously
increased capacity of information-processing is fast approaching reality. Both
developments demandmoreandbettertraining inquantum mechanics atthe
universities, with emphasis on a clear and solid understanding of the subject.
Cookbook-style learning of quantum mechanics, in which equations and
methods for their solution are memorized rather than understood, may help
the students to solve some standard problems and pass multiple-choice tests,
but it will not enable them to break new ground in real life as physicists.
On the other hand, some ‘Mickey Mouse courses’ on quantum mechanics for
engineers, biologists, and computer analysts may give an idea of what this
disciplineisabout,buttoooftenthestudentendsupwithanincorrectpicture
or, at best, a bunch of uncritical, blind beliefs. The present book represents
a fresh start toward helping achieve a deep understanding of the subject. It
presents the material with utmost rigor and will require from the students
ironclad, old-fashioned discipline in their study.
Toofrequently,intoday’suniversities,wehearthedemandthatthecourses
offered be “entertaining,” in response to which some departmental brochures
declare that “physics is fun”! Studying physics requires many hours of hard
work, deep concentration, long discussions with buddies, periodic consulta-
tion with faculty, and tough self-discipline. But it can, and should, become a
passion: the passion to achieve a deep understanding of how Nature works.
This understanding usually comes in discrete steps, and students will expe-
rience such a step-wise mode of progress as they work diligently through the
VIII Foreword
present book. The satisfaction of having successfully mastered each step will
then indeed feel very rewarding!
The “amount of information per unit surface” of text is very high in this
book – its pages cover all the important aspects of present-day quantum
mechanics, from the one-dimensional harmonic oscillator to teleportation.
Starting from a few basic principles and concentrating on the fundamental
behavior of systems (particles) with only a few degrees of freedom (low-
dimension Hilbert space), allows the author to plunge right into the core of
quantummechanics.Italsomakesitpossibletointroducefirst theHeisenberg
matrixapproach–inmyopinion,apedagogicallyrewardingmethodthathelps
sharpen the mental and mathematical tools needed in this discipline right at
thebeginning.Forinstance,tosolvethequantization oftheharmonicoscilla-
tor without the recourse of a differential equation is illuminating and teaches
dexterity in handling the vector and matrix representation of states and
operators, respectively.
Daniel Bes is a child of the Copenhagen school. Honed in one of the cra-
dlesofquantummechanicsby˚AgeBohr,sonofthegreatmaster,andbyBen
Mottelson, he developed an unusually acute understanding of the subject,
which after years of maturing has been projected into a book by him. The
emphasis given throughout the text to the fundamental role and meaning of
the measurement process, and its intimate connection to Heisenberg’s prin-
ciple of uncertainty and noncommutativity, will help the student overcome
the initial reaction to the counterintuitive aspects of quantum mechanics and
to better comprehend the physical meaning and properties of Schro¨dinger’s
wave function. The human brain is an eminently classical system (albeit the
most complex one in the Universe as we know it), whose phylo- and ontoge-
neticevolutionweredrivenbyclassical physicalandinformationalinteractions
between organism and the environment. The neural representations of envi-
ronmental and ontological configurations and events, too, involve eminently
classical entities. It is therefore only natural that when this classical brain
looks into the microscopic domain using human-designed instruments which
musttranslatequantumhappeningsintoclassical,macroscopicallyobservable
effects, strange things with unfamiliar behavior may be seen! And it is only
natural that, thus, the observer’s intentions and his instruments cannot be
left outside the framework of quantum physics! Bes’ book helps to recognize,
understand, and accept quantum “paradoxes” not as such but as the facts of
“Nature under observation.” Once this acceptance has settled in the mind,
the student will have developed a true intuition or, as the author likes to call
it, a “feeling” for quantum mechanics.
Chapter 2 contains the real foundation on which quantum mechanics is
built; it thus deserves, in my opinion, repeated readings – not just at the
beginning, but after each subsequent chapter. With the exception of the dis-
cussion of two additional principles, the rest of the book describes the math-
ematical formulations of quantum mechanics (both the Heisenberg matrix
mechanics, most suitable for the treatment of low-dimension state vectors,
Foreword IX
and Schro¨dinger’s wave mechanics for continuous variables) as well as many
applications.Theexamplescoverawidevarietyoftopics,fromthesimplehar-
monic oscillator mentioned above, to subjects in condensed matter physics,
nuclearphysics,electrodynamics,andquantumcomputing.Itisinterestingto
note,regardingthelatter,thattheconceptofqubitappearsinamostnatural
way in the middle of the book, almost in passing, well before the essence of
quantum computing is discussed towards the end. Of particular help are the
carefullythought-outproblemsattheendofeachchapter,aswellastheocca-
sional listings of “common misconceptions.” A most welcome touch is the in-
clusion of a final chapter on the history of theoretical quantum mechanics
– indeed, it is regrettable that so little attention is given to it in university
physicscurricula:muchadditionalunderstandingcanbegainedfromlearning
how ideas have matured (or failed) during the historical development of a
given discipline!
Let me conclude with a personal note. I have known Daniel Bes for over
60 years. As a matter of fact, I had known of him even before we met in
person: our fathers were “commuter-train acquaintances” in Buenos Aires,
andbothservedinthePTAoftheprimaryschoolthatDanielandIattended
(in different grades). Daniel and I were physics students at the University of
BuenosAires(again,atdifferentlevels),thenonthephysicsfaculty,andyears
later, visiting staff members of Los Alamos. We were always friends, but we
neverworkedtogether–Danielwasatheoreticianalmostfromthebeginning,
whereasIstartedasacosmic-rayandelementary-particleexperimentalist(see
Fig.2.5!). It gives me a particular pleasure that now, after so many years and
despite residing at opposite ends of the American continent, we have become
professionally “entangled” through this wonderful textbook!
University of Alaska-Fairbanks, Juan G. Roederer
January 2004 Professor of Physics Emeritus
Preface to the First Edition
Thistextfollowsthetraditionofstartinganexpositionofquantummechanics
with the presentation of the basic principles. This approach is logically
pleasing and it is easy for students to comprehend. Paul Dirac, Richard
Feynman and, more recently, Julian Schwinger, have all written texts which
are epitomes of this approach.
However, up to now, texts adopting this line of presentation cannot be
considered as introductory courses. The aim of the present book is to make
thisapproachtoquantummechanicsavailabletoundergraduateandfirstyear
graduate students, or their equivalent.
A systematic dual presentation of both the Heisenberg and Schro¨dinger
procedures is made, with the purpose of getting as quickly as possible to
concrete and modern illustrations. As befits an introductory text, the tradi-
tional material on one- and three-dimensional problems, many-body systems,
approximationmethodsandtime-dependenceisincluded.Inaddition,modern
examples are also presented. For instance, the ever-useful harmonic oscillator
is applied not only to the description of molecules, nuclei, and the radiation
field,butalsotorecentexperimentalfindings,likeBose–Einsteincondensation
and the integer quantum Hall effect.
This approach also pays dividends through the natural appearance of the
most quantum of all operators: the spin. In addition to its intrinsic concep-
tual value, spin allows us to simplify discussions on fundamental quantum
phenomena like interference and entanglement; on time-dependence (as in
nuclear magnetic resonance); and on applications of quantum mechanics in
the field of quantum information.
This text permits two different readings: one is to take the shorter path
to operating with the formalism within some particular branch of physics
(solidstate,molecular,atomic,nuclear,etc.)byprogressingstraightforwardly
from Chaps.2 to 9. The other option, for computer scientists and for those
readers more interested in applications like cryptography and teleportation,
istoskipChaps.4,6,7,and8,inordertogettoChap.10assoonaspossible,
which starts with a presentation of the concept of entanglement. Chapter 12
XII Preface
is devoted to a further discussion of measurements and interpretations in
quantum mechanics. A brief history of quantum mechanics is presented in
order to acquaint the newcomer with the development of one of the most
spectacular adventures of the human mind to date (Chap.13). It intends also
toconveythefeelingthat,farfrombeingfinished,thisenterpriseiscontinually
being updated.
Sectionslabeledbyanasteriskincludeeitherthemathematicalbackground
of material that has been previously presented, or display a somewhat more
advanceddegreeofdifficulty.Theselastonesmaybeleftforasecondreading.
Any presentation of material from many different branches of physics
requires the assistance of experts in the respective fields. I am most indebted
for corrections and/or suggestions to my colleagues and friends Ben Bayman,
Horacio Ceva, Osvaldo Civitarese, Roberto Liotta, Juan Pablo Paz, Alberto
Pignotti,JuanRoederer,MarcosSaraceno,NorbertoScoccola,andGuillermo
Zemba. However, none of the remaining mistakes can be attributed to them.
Civitarese and Scoccola also helped me a great deal with the manuscript.
Questions (and the dreaded absence of them) from students in courses
given at Universidad Favaloro (UF) and Universidad de Buenos Aires (UBA)
were another source of improvements. Sharing teaching duties with Guido
Berlin, Cecilia Lo´pez and Dar´ıo Mitnik at UBA was a plus. The interest of
Ricardo Pichel (UF) is fully appreciated.
Thanks are due to Peter Willshaw for correcting my English. The help of
Martin Mizrahi and Ruben Weht in drawing the figures is gratefully acknow-
ledged. Raul Bava called my attention to the mandala on p.V.
I like to express my appreciation to Arturo Lo´pez D´avalos for putting me
in contact with Springer-Verlag and to Angela Lahee and Petra Treiber of
Springer-Verlag for their help.
Mytrainingasaphysicistowesverymuchto˚AgeBohrandBenMottelson
ofNielsBohrInstitutetandNORDITA(Copenhagen).Duringthe1950sNiels
Bohr, in his long-standing tradition of receiving visitors from all over the
world, used his institute as an open place where physicists from East and
Westcouldworktogetherandunderstandeachother.From1956to1959,Iwas
thereasayoungrepresentativeoftheSouth.MywifeandImetMargretheand
NielsBohrattheirhomeinCarlsberg.Iremembergatheringtherewithother
visitors and listening to Bohr’s profound and humorous conversation. He was
a kind of father figure, complete with a pipe that would go out innumerable
times while he was talking. Years later I became a frequent visitor to the
Danish institute, but after 1962 Bohr was no longer there.
MywifeGladyscarriedthegreatestburdenwhileIwaswritingthisbook.
It must have been difficult to be married to a man who was mentally absent
for the better part of almost two years. I owe her much more than a mere
acknowledgment, because she never gave up in her attempts to change this
situation (as she never did on many other occasions in our life together).
My three sons, David, Martin, and Juan have been a permanent source of
strength and help. They were able to convey their encouragement even from
