Basic Quantum Mechanics

516 Pages·2019·6.302 MB·English

by  Kyriakos Tamvakis

Undergraduate Texts in Physics
Kyriakos Tamvakis
Basic Quantum
Mechanics
Undergraduate Texts in Physics
Series Editors
Kurt H. Becker, NYU Polytechnic School of Engineering, Brooklyn, NY, USA
Jean-Marc Di Meglio, Matière et Systèmes Complexes, Université Paris Diderot,
Bâtiment Condorcet, Paris, France
SadriD.Hassani,DepartmentofPhysics,LoomisLaboratory,UniversityofIllinois
at Urbana-Champaign, Urbana, IL, USA
Morten Hjorth-Jensen, Department of Physics, Blindern, University of Oslo, Oslo,
Norway
Michael Inglis, Patchogue, NY, USA
Bill Munro, NTT Basic Research Laboratories, Optical Science Laboratories,
Atsugi, Kanagawa, Japan
Susan Scott, Department of Quantum Science, Australian National University,
Acton, ACT, Australia
Martin Stutzmann, Walter Schottky Institute, Technical University of Munich,
Garching, Bayern, Germany
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Kyriakos Tamvakis
Basic Quantum Mechanics
123
Kyriakos Tamvakis
Department ofPhysics
University of Ioannina
Ioannina, Greece
ISSN 2510-411X ISSN 2510-4128 (electronic)
Undergraduate Texts inPhysics
ISBN978-3-030-22776-0 ISBN978-3-030-22777-7 (eBook)
https://doi.org/10.1007/978-3-030-22777-7
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To my daughters Anna and Maria
Preface
This textbook on Quantum Mechanics has been designed for use in two-semester
undergraduatecoursesandasasupplementaltextbookingraduatecourses.Itistrue
that there are many excellent books on the subject, like the classic examples of
Baym,Gottfried,Messiah,orSchiff,which,althoughwrittenhalfacenturyago,are
stillfittocovermostofthebasicmaterial.Nevertheless,newtopicshaveenteredinto
therealmofinterestofthepresent-daystudentofphysicsthathavetobeincludedin
the basic material. In addition to that, each new book represents a challenge in the
selection of topics, in the emphasis on each of them and the overall organization
ofthematerial,whichistoalargeextentsubjective.Thepresentbookhascomeout
asaresultofteachingQuantumMechanicssince1982attheUniversityofIoannina.
It aims to describe the basic concepts of Quantum Mechanics, to explain the use
ofthemathematicalformalismandtoprovideillustrativeexamplesofbothconcepts
and methods. In that sense, its purpose is quite conventional as to the training of
physicsstudents.Althoughitisintendedtoprovideamasteryoftheuseofquantum
mechanics as a tool, it also provides some discussion on the meaning of quantum
concepts, despite the fact that no general consensus as to what its fundamental
principles mean has been reached. After a brief discussion of the basic features of
Quantum Mechanics in the framework of a simplified version of the two-slit
experiment,thereaderisintroducedtotheSchroedingerequationandisfamiliarized
withitsapplicationonanumberofsimpleone-dimensionalsystems.Thefollowing
chaptersintroducethefullmathematicalformalismandbasicpostulatesofQuantum
Mechanics.Furtherapplicationsfollowonanumberofsimplesystemsamongwhich
two-statesystems,the one-dimensionallattice,periodicpotentials,etc.Thefollowing
chaptersaredevotedtoangularmomentumandspin.Next,three-dimensionalsystems
are considered with a number of applications in central potentials. The following
chaptersdealwithmany-particlesystems,atoms,andmolecules.Achapteronparticle
interactionswithanexternalelectromagneticfieldisdevotedtoLandaulevelsandthe
Bohm–Aharonov phenomena. The next chapter on approximation methods includes
theWKBmethod,theadiabaticapproximation,variationalmethods,andperturbation
theory with a number of applications. A chapter on symmetries deals with rotations,
tensor operators, and the Wigner–Eckart theorem, discrete symmetries as well as
vii
viii Preface
dynamical symmetries. Scattering theory is considered next with a number of appli-
cations. A chapter on quantum behavior deals with the measurement process, the
conceptsofmixedstatesanddensitymatrices,entanglement,theEPRissue,andBell’s
theorem.Next,achapterisdevotedtothequantizationoftheelectromagneticfieldand
its interactionwith matter. Finally,the last chapter consists ofan introduction on the
pathintegralformulationofQuantumMechanics.
It is important to stress that the material of this book should not be approached
passively, a very important factor in the learning process being the initiative
exercised by the prospecting student. To this end, apart from about 60 worked out
examples within each chapter, there are around 200 problems and exercises at the
endofeachchapter,whichcanbequitehelpful,ifnotnecessary,towardsachieving
a command of the subject.
Ioannina, Greece Kyriakos Tamvakis
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 A Thought Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The de Broglie Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 The Wave Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 A Free Particle of Definite Momentum. . . . . . . . . . . . . . . . . . 8
1.5 The Schroedinger Equation for a Free Particle . . . . . . . . . . . . 11
1.6 Wave Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 The Schroedinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1 The Fundamental Time-Evolution Equation . . . . . . . . . . . . . . 19
2.2 Conservation of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 The Hamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Stationary States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 General Solution of the Schroedinger Equation. . . . . . . . . . . . 26
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Some Simple Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 The Infinite Square Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Piecewise Constant Potentials . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.1 The One-Dimensional Potential Step . . . . . . . . . . . . . 38
3.2.2 The Square Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.3 The Square Well . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.4 The Delta Function Potential. . . . . . . . . . . . . . . . . . . 52
3.2.5 Scattering by Two Delta Functions
and Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4 Physical Observables as Operators . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1 Physical Quantities and Operators . . . . . . . . . . . . . . . . . . . . . 63
4.2 Eigenvalues and Eigenfunctions. . . . . . . . . . . . . . . . . . . . . . . 65
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x Contents
4.3 Hermitian Conjugation and Hermitian Operators. . . . . . . . . . . 67
4.4 Physical Observables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5 Uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.6 The Heisenberg Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.7 Commuting Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.8 Time Evolution of Expectation Values . . . . . . . . . . . . . . . . . . 77
4.9 Ehrenfest’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.10 The Virial Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5 Basic Principles of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . 85
5.1 Basic Postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 The Hilbert Space of States . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.3 Operators in the Hibert Space . . . . . . . . . . . . . . . . . . . . . . . . 91
5.4 General Proof of the Heisenberg Inequality . . . . . . . . . . . . . . 97
5.5 The Position and Momentum Representations. . . . . . . . . . . . . 99
5.6 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6 Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.1 The Time-Evolution Operator . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2 The Schroedinger Versus the Heisenberg Picture . . . . . . . . . . 108
6.3 The Time–Energy Uncertainty Relation . . . . . . . . . . . . . . . . . 111
6.4 The Interaction Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.5 Time-Ordered Exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . 116
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7 Some More Simple Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.1 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.1.1 Energy Eigenvalues of the Harmonic Oscillator . . . . . 122
7.1.2 The fxg Versus the fNg Representation . . . . . . . . . . 126
7.1.3 Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.2 The Ammonia Molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.3 The One-Dimensional Lattice. . . . . . . . . . . . . . . . . . . . . . . . . 144
7.4 Periodic Potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.5 Other Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
8 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.1 Angular Momentum as a Quantum Observable. . . . . . . . . . . . 161
8.2 Central Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
8.3 The Angular Momentum in the fxg Representation. . . . . . . . . 164
8.4 Angular Momentum and Rotations. . . . . . . . . . . . . . . . . . . . . 166
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171