Conceptual Basis of Quantum Mechanics

349 Pages·2015·3.744 MB·English

by  Jan-Markus Schwindt

Undergraduate Lecture Notes in Physics
Jan-Markus Schwindt
Conceptual
Basis of
Quantum
Mechanics
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Jan-Markus Schwindt
Conceptual Basis
of Quantum Mechanics
123
Jan-Markus Schwindt
Dossenheim
Germany
ISSN 2192-4791 ISSN 2192-4805 (electronic)
Undergraduate Lecture Notesin Physics
ISBN978-3-319-24524-9 ISBN978-3-319-24526-3 (eBook)
DOI 10.1007/978-3-319-24526-3
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Preface
Quantum mechanics (QM) is still quite a mysterious theory. In contrast to the
theory of relativity, for example, it did not evolve out of some basic idea, some
physical principle. Only reluctantly, physicists had to accept its claims which are
completely counterintuitive. Many aspects regarding its interpretation are still a
matter of debate nowadays.
I tried to write a textbook I would have liked to read as a student; a book that
avoids many misunderstandings, unclear and vague ideas, and the questions that
havesettledinmymindforquiteawhile,atthattime;orconfirmsthatsomeofmy
questionsbelongtotheissuesofinterpretationwhichareconsideredunclearbythe
entirephysicscommunityandareunderheavydebate.Theresultingbookdiffersin
some points from most of the other QM textbooks.
First, the mysterious properties of the theory are emphasized and discussed in
detail. For that reason, it starts with Bell’s inequalities as a foretaste of what we
embark on. Other QM books often treat QM as something that became
common-place and self-evident, just as a part of the physics canon. But this is not
the case. The conceptual basis of QM has been under steady research and fiery
debatesthroughoutallthedecadesfrom1900untiltoday(incontrasttospecialand
generalrelativitywhosefoundationshavebeenunderstoodfor90years),withnew
andsurprisinginsightscomingupagainandagain.Bynow,thereisawholebunch
of interpretations of the theory which fundamentally differ in their world view.
A separate chapter is dedicated to these interpretations.
Second,thebookdevelopsitsmatterfromthegeneraltothespecific.Atfirst,the
general postulatesofQM arepresentedanddiscussed indetail. The wave function
as a special case of a quantum state follows later. This has the advantage that the
main hurdle is taken right in the beginning, and a double run-up is avoided. For
many books develop the matter twice: first by means of Schrödinger’s wave
mechanics and later again under the title of “abstract formulation.” From my
point of view, it is better to begin with the abstract part and thereby avoid the
misunderstanding that QM is only a theory of wave functions.
v
vi Preface
Third,thegeneralpostulatesandbasicnotionsareallexplainedbymeansofthe
simplestnontrivialquantumsystem:thetwo-dimensionalstatespaceoftheelectron
spin, also called qubit.
Also for wave functions we stick to our principle of the simplest example.
Insteadoffollowingmostbookswhichdirectlyjumpfromonetothreedimensions,
wetakeastopoverintwodimensions.Forthereitiseasiertoexplaintheseparation
of variables and in particular some features of a rotation symmetric potential
(namely without spherical harmonics).
Fourth,thereisastrongfocusontheclarityofnotionsandunderstandingofthe
mathematical background. This clarity helps to avoid misunderstandings from the
beginning.
Forexample,alargepartofQMtakesplaceintheinfinite-dimensionalspaceof
wave functions. Many theorems known from linear algebra for finite-dimensional
vector spaces are no longer valid there. These peculiarities are discussed in detail.
Tensorproductsalsogetenoughspace,sincetheyplayanenormouslyimportant
role in QM, e.g., for the notion of an entangled state, and for the combination of
wave function and spin state.
Fifth,theexercisesarenotgatheredattheendofeachchapter,butareembedded
in the main text. This is supposed to encourage the reader to think about and to
recalculatethingsduringhisread.Thesolutionstotheexercisesaregatheredatthe
end of the book.
The focus of this book is on the general postulates of QM, its interpretation, its
basic notions, and its mathematical formulation. The first and most extensive part
of the book is dedicated to this topic.
In the second part, an important special case is treated: the QM of wave func-
tions in one, two, and three spatial dimensions, under the assumption that the
Hamiltonian operator consists only of a kinetic term and a time-independent
potential. Here, the most important examples are the harmonic oscillator and the
hydrogen atom. Scattering theory is also discussed within this context.
Thethirdpartencompassesfurthertopicswhichbelongtothecanonicalmaterial
of a lecture on QM: combination of spin and angular momentum, QM with elec-
tromagnetism,perturbation theory,andsystemswithmany particles.Hereweonly
develop the basic ideas and methods, as well as some simple examples. For
applications like the fine and hyperfine structure of hydrogen, or the theory of
atomic transitions, we refer the reader to the literature. Finally, we provide a short
explanationofthenotionofapathintegralanddiscusstherelativistictheoryofthe
electron (Dirac equation).
The target audience of this book is, of course, in the first place students of
physicswhostudyQMinthecontextofacourseontheoreticalphysics.Butdueto
the axiomatic, deductive approach and the detailed discussion of the mathematical
background,it isalso verywell suitedformathematicianswhowant tounderstand
QM. In some places we explicitly try to overcome the “cultural barrier” between
mathematicians and physicists.
Preface vii
Some topics which are exciting but not mandatory for the further reading (and
for exams) are placed in a dedicated text environment—the so-called “nerd’s cor-
ner”—for curious readers.
Acknowledgments
Thisbookdidnotresultfrommyworkalone.Iwanttoexpressmygratitudetothe
people who have enabled, supported, and contributed to this project:
(cid:129) VeraSpillneratSpringerfortheopportunitytowritethisbook;fortheinspiring
discussions;forthesupplywithreadingmaterial;andforproofreading,acrucial
contribution to the quality of this book;
(cid:129) Bianca Alton at Springer for the organizational supervision of the project; she
has helped me with many questions and formalities;
(cid:129) ClausAscheronatSpringerfortheopportunitytotranslateandpublishthisbook
also in English;
(cid:129) Kristin Riebe for the creation of two images, once again shining light on her
graphical talent;
(cid:129) Bernhard Brosda for the nice idea with the cartoons;
(cid:129) Anja Stemme and Jörg Kügler for reading material about Bell’s inequalities;
(cid:129) Andreas Rüdinger for mentioning the Gelfand triple;
(cid:129) Michael Doran for pointing me to Shankar;
(cid:129) My mother for her moral support.
ThefactthatIamabletowritesuchabookIowetothemanyteachers,friends,
andsourcesofinspirationonmyway.AtthispointIwanttomentiontwoofthem
explicitly: I thank
(cid:129) MymathteacherHanspeterEichhorn,whoalongtimeagonurturedmyinterest
in mathematics and gave it, so to speak, legs to walk.
(cid:129) My Ph.D. supervisor Prof. Christof Wetterich for inspiration and support over
many years.
Frankfurt, June 2013 Jan-Markus Schwindt
Heidelberg, July 2015
Contents
Part I Formalism and Interpretation
1 Introduction: Nonlocal or Unreal?. . . . . . . . . . . . . . . . . . . . . . . . 3
2 Formalism I: Finite-Dimensional Hilbert Spaces. . . . . . . . . . . . . . 9
2.1 The Postulates of Quantum Mechanics—Overview . . . . . . . . . 9
2.2 States in Hilbert Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Linear Hermitian Operators . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Eigenvalues and Eigenvectors. . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Projection and Measurement. . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6 Unitary Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.7 Time Evolution and Schrödinger Equation. . . . . . . . . . . . . . . 40
2.8 Commutator and Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . 47
2.9 Schrödinger Picture and Heisenberg Picture . . . . . . . . . . . . . . 54
2.10 Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3 Formalism II: Infinite-Dimensional Hilbert Spaces. . . . . . . . . . . . 71
3.1 Sets of Functions as Vector Spaces. . . . . . . . . . . . . . . . . . . . 72
3.2 Scalar Product and Orthonormal Basis. . . . . . . . . . . . . . . . . . 75
3.3 Pseudo-Vectors and Fourier Transformation. . . . . . . . . . . . . . 89
3.4 Position and Momentum Operator,
Correspondence Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.5 Matter Waves and Wave-Particle Duality. . . . . . . . . . . . . . . . 103
3.6 Schrödinger Equation in One-Dimensional Position Space . . . . 109
3.7 Several Space Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.8 Several Particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4 Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.1 The Problem of Interpreting QM. . . . . . . . . . . . . . . . . . . . . . 127
4.2 Many Worlds Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.3 Copenhagen Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.4 De Broglie–Bohm Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.5 Collapse Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
ix
x Contents
4.6 New Age Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Part II A Single Scalar Particle in an External Potential
5 One-Dimensional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.1 Dissolving of a Gaussian Wave Packet . . . . . . . . . . . . . . . . . 149
5.2 Piecewise Constant Potentials. . . . . . . . . . . . . . . . . . . . . . . . 153
5.2.1 General Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.2.2 Potential Step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.2.3 Potential Well. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.2.4 Potential Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.3 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6 Two-Dimensional Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.1 Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.2 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7 Three-Dimensional Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7.1 Angular Momentum Algebra . . . . . . . . . . . . . . . . . . . . . . . . 182
7.2 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7.3 Spherically Symmetric Potential . . . . . . . . . . . . . . . . . . . . . . 197
7.4 Free Particle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
7.5 Coulomb Potential and Hydrogen Atom. . . . . . . . . . . . . . . . . 206
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
8 Scattering Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
8.1 Scattering Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
8.2 Born Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
8.3 Partial Wave Expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Part III Advanced Topics
9 Spin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
9.1 Spin 1/2 and Spin 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
9.2 Addition of Angular Momenta . . . . . . . . . . . . . . . . . . . . . . . 237
9.3 SO(3) and SU(2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
10 Electromagnetic Interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
10.1 Hamiltonian Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
10.2 Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
10.3 Magnetic Moment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
10.4 Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
10.4.1 Normal Zeeman Effect. . . . . . . . . . . . . . . . . . . . . . . 261
10.4.2 Stern-Gerlach Experiment. . . . . . . . . . . . . . . . . . . . . 263
10.4.3 Aharanov-Bohm Effect . . . . . . . . . . . . . . . . . . . . . . 264