Fundamentals of Quantum Physics: Textbook for Students of Science and Engineering

328 Pages·2012·4.92 MB·English

by  Pedro Pereyra

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Pedro Pereyra
Fundamentals of Quantum
Physics
Textbook for Students of Science
and Engineering
123
Pedro Pereyra Padilla
Física Teóricay Materia Condensada
Universidad AutónomaMetropolitana
Av.S.Pablo Del Azcapotzalco 180
02200, México D.F.
Mexico
ISSN 2192-4791 ISSN 2192-4805 (electronic)
ISBN 978-3-642-29377-1 ISBN 978-3-642-29378-8 (eBook)
DOI 10.1007/978-3-642-29378-8
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2012938636
(cid:2)Springer-VerlagBerlinHeidelberg2012
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To my daughters and son
Naira Citlalli
Jaina Erandi Mariana
Sebastián Fernando
Preface
Various reasons concurred in my decision of writing this book. Some years ago
Freeman Dyson, reasoning on the process of learning and teaching quantum the-
ory, came out with the idea that a physics student, after learning the tricks of the
quantum formalism and getting right answers, ‘‘begins to worry because he does
notunderstandwhatheisdoing’’.Thestudent,saysDyson,1‘‘hasnoclearphysical
picture in his head and tries to arrive at a physical explanation for each of the
mathematical tricks’’. He gets discouraged and after some months of unpleasant
andstrenuoustime,hesuddenlysays:‘‘Iunderstandnowthatthere isn’tanything
to be understood’’. What happens? Dyson suggests that we learn to think in
quantum-mechanical language and no longer try to explain in terms of
pre-quantum conceptions. As an undergraduate student, facing my first quantum-
mechanics textbook, I had similar feelings. I felt lost within a mathematical
formalismwithabstractobjectsandconcepts.Bras,kets,andoperators,camesea-
soned with counterintuitive statements that shook my own conceptions of nature.
Someyearslater,whenIwasreadytoacceptthattherewasnothingtounderstand,
ThomasA.Brody2organizedinMexicoagraduatestudents’seminarontheinter-
pretationsofthemathematicalformalismofquantummechanics;aseminartoshow
thatinthequantumtheory,besidesthefactsandexcellentagreementwithexperi-
mentalresults,onehastorecognizetheexistenceofopenepistemologicalproblems.
Afterthoseyears,Itaughtquantumphysicsrepeatedly,andIhavetriedtopresentthe
fundamentals ofthetheory asacoherentandcomprehensive bodyofphenomena,
using,wheneverpossible,simplemathematicaltechniques.
Thetwentiethcenturywas,tosomeextent,thecenturyofquantumtheory.The
fundamentals of the present day nanoscience and nanotechnology are closely
linkedwiththeelectronicandoptoelectronicpropertiesofphysicalsystemsinthe
quantum domain. Some systems like the repulsive square barrier and the square-
1 InnovationinPhysics,F.J.Dyson,ScientificAmerican,199,74(1958).
2 The Philosophy Behind Physics, T. A. Brody; L. de la Peña and P. E. Hodgson, editors,
Springer-Verlag,Berlin-NewYork,1993.
vii
viii Preface
well potentials, that were simple academic examples in standard quantum
mechanics courses, become systems of current interest in theoretical, experi-
mental, and applied physics. Studying these systems, fruitful approaches, models,
andtechniquesevolved.Thescatteringapproachwasoneofthem,andithasbeen
rather successful for studying transport properties. Since then I have gradually
introducedthispowerfulmethod inmylectures todiscusssomesimpleproblems.
Energy–eigenvalue equations and eigenfunctions are straightforwardly derived.
The ease with which students learn using this intuitive and algebraic method,
enticed me and enabled me to include other systems like the double-barrier
potentials, double quantum wells, and superlattices.
Thegeneralplanofthisbookissimilartothatofmostofthetextbooksdevoted
toafirstcourseofnon-relativisticquantumtheory.Inthefirsttwochapterswewill
briefly summarize the physical problems that show the limits of the classical
theories and the new ideas that explained, permanently or temporarily, those
problems. Concepts like energy quantization in the blackbody radiation, wave-
particleduality,andtheBohrpostulates,arethoroughlydiscussed.Weendupwith
a simple and intuitive derivation of the Schrödinger equation. On the problem of
the interpretation of the wave function, we keep a consistent position, but avoid
excessivediscussionofsomecontroversialissues,whicharestudiedinotherbooks
suchasIntroductiontoQuantumMechanicsbyL.delaPeña.Inthesecondpartof
this book (Chaps. 3–5), we study the usual examples: free particle; infinite
quantumwell;rectangularbarrierandthefinitequantumwell.Westudyalsomore
complex systems like the double barrier, the double quantum well, and the finite
Kronig-Penneymodel.Fullysolvingtheseexamples,wefaceadditionalproperties
like the quantum coherence. We introduce the transfer matrix method and let the
students acquaint themselves with systems that are present in industrial opto-
electronic devices. Chapter 6 is devoted to the semiclassical method of Wentzel,
Kramers, and Brillouin (the WKB approximation). This topic is discussed using
also the transfer matrixmethod. New relations are derived and applied for simple
examples. From Chap. 7 onwards, we present most of the standard content in a
regular quantum book (operators, expected values, angular momentum and spin,
matrix representations, the Pauli equation, etc.), we study the well-known har-
monic oscillator, the Hydrogen atom, and the fine structure that spurred the crisis
oftheoldquantumtheory,between1920and1925.Weendupwithasummaryof
the perturbation theory and a chapter on the distinguishable and indistinguishable
identical particles. We analyze the close relation between wave-function
symmetries, statistics and spin, and discuss properties like the Bose-Einstein
condensation and the Pauli exclusion principle.
Together with the formal presentation of the quantum theory, the main objec-
tiveofthisbookistopresentacoherentsetofconceptsandphysicalphenomena.
The most emblematic quantum properties like the tunneling effect through
potential barriers and the energy quantization in the confining potentials, appear
and reappear in more complex examples, modified or transformed. These phe-
nomena, together with the quantum version of the particle current density, the
transmission and reflection coefficients, and the splitting of the energy levels that
Preface ix
give rise to the energy bands structure, are some of the physical phenomena
studiedinthistext.Oneoftheaimsistoshowstudentsthatquantumtheory,more
than a set of axioms and differential equations, is a consistent and intelligible
theory that clarifies the quantum behavior present in an uncountable number of
realizations of the microscopic and macroscopic systems. We pursue the mathe-
maticalrigorandconceptualdepthtogetherwithsimplecalculationsandenhanced
comprehension of the quantum concepts.
The content and depth with which we discuss the fundamental issues of the
quantum theory is appropriate for a semester course or a trimester plus a com-
plementary course. In each chapter we have some solved problems, which in
certain cases complement the discussion of some topics. The second part of this
book can also be used as an introductory course to let electric and electronic
engineers get acquainted with semiconductor heterostructures, and the quantum
properties of these systems.
IthankEmilioSordoandLuisNoreñafortheirfriendshipandforthefacilities
thattheUniversidadAutónomaMetropolitanacampusAzcapotzalcoprovidesme;
to my students of different generations, especially to Fernando Zubieta, Michael
Morales,MariaFernandaAvila,andVictorIbarrafortheirsupportinanimportant
part of the transcription of my notes into the La-TeX language. I thank my stu-
dents, colleagues, and collaborators Jose Luis Cardoso and Alejandro Kunold. To
Jaime Grabinsky, Juergen Reiter, Arturo Robledo and Herbert Simanjuntak, for
their friendship and the useful comments and corrections. To Claus Ascheron,
ExecutiveEditorofSpringerVerlag,forhiscontinuedencouragementandinterest
inthepublicationofthisbook.ThefriendshipandhospitalityofDieterWeissand
the University of Regensburg, where the last steps of this English version were
taken. The support of CONACyT Mexico and the Program: Apoyo para Año
Sabático is also acknowledged. I also thank the support and love of my wife
Liuddys.
México Pedro Pereyra
Contents
1 The Origin of Quantum Concepts. . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Blackbody Radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 The Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 The Compton Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Rutherford’s Atom and Bohr’s Postulates . . . . . . . . . . . . . . . 10
1.5 On Einstein’s Radiation Theory . . . . . . . . . . . . . . . . . . . . . . 13
1.5.1 Planck’s Distribution and the Second
Postulate of Bohr . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5.2 Einstein’s Specific Heat Model . . . . . . . . . . . . . . . . 14
1.6 Solved Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.7 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Diffraction, Duality and the Schrödinger Equation . . . . . . . . . . . 19
2.1 Sommerfeld-Wilson-Ishiwara’s Quantization Rule . . . . . . . . . 20
2.1.1 The Quantization Rules and the Energies
of the Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . 20
2.1.2 The Rotator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 The De Broglie Wave-Particle Duality . . . . . . . . . . . . . . . . . 23
2.3 Diffraction Experiments and the Wave-Like
Nature of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Schrödinger’s Wave Mechanics . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Solved Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Properties of the Stationary Schrödinger Equation . . . . . . . . . . . 33
3.1 Quantization as an Eigenvalue Problem. . . . . . . . . . . . . . . . . 33
3.2 Degenerate Eigenfunctions, Orthogonality and Parity . . . . . . . 35
3.3 The Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.1 The Physical Meaning of the Free
Particle Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 42
xi
xii Contents
3.4 The Infinite Quantum Well . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5 The Particle Current Density in Quantum Mechanics . . . . . . . 48
3.6 Dirac’s Notation and Some Useful Relations. . . . . . . . . . . . . 51
3.6.1 General Properties of Bras and Kets. . . . . . . . . . . . . 52
3.6.2 Some Useful Relations . . . . . . . . . . . . . . . . . . . . . . 52
3.6.3 Momentum Representation . . . . . . . . . . . . . . . . . . . 54
3.7 Solved Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.8 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4 The Tunneling Effect and Transport Properties. . . . . . . . . . . . . . 61
4.1 The 1D Step Potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2 Scattering Amplitudes and the Transfer Matrix . . . . . . . . . . . 70
4.3 The Rectangular Potential Barrier. . . . . . . . . . . . . . . . . . . . . 73
4.3.1 Transfer Matrix of the Rectangular
Potential Barrier. . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.2 The Wave Functions in the Rectangular
Potential Barrier. . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3.3 Reflection and Transmission Coefficients
for Rectangular Potential Barriers. . . . . . . . . . . . . . . 80
4.4 The Rectangular Potential Well . . . . . . . . . . . . . . . . . . . . . . 83
4.4.1 Continuity and the Transfer Matrix
of the Rectangular Potential Well. . . . . . . . . . . . . . . 84
4.4.2 Eigenvalues and Wave Functions in Rectangular
Potential Wells. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.5 Solved Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.6 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5 Quantum Coherence and Energy Levels Splitting . . . . . . . . . . . . 99
5.1 A Rectangular Double Well Bounded by Hard Walls . . . . . . . 100
5.1.1 Continuity and the Double-Well Transfer Matrix . . . . 100
5.1.2 Energy Eigenvalues in the Double Quantum Well . . . 102
5.2 The Double Rectangular Potential Barrier. . . . . . . . . . . . . . . 104
5.2.1 Continuity and the Double-Barrier Transfer Matrix. . . 104
5.2.2 Transport Properties in the Double Rectangular
Potential Barrier. . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.3 The Finite Double Quantum Well. . . . . . . . . . . . . . . . . . . . . 107
5.3.1 Continuity and the Double-Well Transfer Matrix . . . . 108
5.3.2 Eigenvalues and Eigenfunctions in a Double
Quantum Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.3.3 Transport Properties in the Double Quantum Well. . . 113
5.4 Finite Periodic Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.4.1 Transport Properties in the Kronig–Penney Model . . . 118
5.5 Solved Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.6 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126