Quantum theory and statistical thermodynamics : principles and worked examples

382 Pages·2017·4.493 MB·English

by  Hertel, Peter

Graduate Texts in Physics
Peter Hertel
Quantum Theory
and Statistical
Thermodynamics
Principles and Worked Examples
Graduate Texts in Physics
Series editors
Kurt H. Becker, Polytechnic School of Engineering, Brooklyn, USA
Jean-Marc Di Meglio, Université Paris Diderot, Paris, France
Sadri Hassani, Illinois State University, Normal, USA
Bill Munro, NTT Basic Research Laboratories, Atsugi, Japan
Richard Needs, University of Cambridge, Cambridge, UK
William T. Rhodes, Florida Atlantic University, Boca Raton, USA
Susan Scott, Australian National University, Acton, Australia
H. Eugene Stanley, Boston University, Boston, USA
Martin Stutzmann, TU München, Garching, Germany
Andreas Wipf, Friedrich-Schiller-Universität Jena, Jena, Germany
Graduate Texts in Physics
GraduateTextsinPhysicspublishescorelearning/teachingmaterialforgraduate-and
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physics, both pure and applied. These textbooks serve students at the MS- or
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derivations,experimentsandapplications(asrelevant)fortheirmasteryandteaching,
respectively.Internationalinscopeandrelevance,thetextbookscorrespondtocourse
syllabisufficientlytoserveasrequiredreading.Theirdidacticstyle,comprehensive-
nessandcoverageoffundamentalmaterialalsomakethemsuitableasintroductions
orreferencesforscientistsentering,orrequiringtimelyknowledgeof,aresearchfield.
More information about this series at http://www.springer.com/series/8431
Peter Hertel
Quantum Theory
and Statistical
Thermodynamics
Principles and Worked Examples
123
PeterHertel
UniversitätOsnabrückFachbereich Physik
Osnabrück
Germany
ISSN 1868-4513 ISSN 1868-4521 (electronic)
Graduate Textsin Physics
ISBN978-3-319-58594-9 ISBN978-3-319-58595-6 (eBook)
DOI 10.1007/978-3-319-58595-6
LibraryofCongressControlNumber:2017941466
©SpringerInternationalPublishingAG2017
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Preface
Thissmalltextbookattemptstopresentthevastfieldofquantumphysics,including
statisticalthermodynamics,byfocusingonafewbasicconcepts.Matterismadeof
particles which, like waves, interfere. There is a limiting case, namely classical
mechanics and the various fields of continuum physics, where the interference of
particlesplaysnorole.Althoughtheclassicaldisciplineshavebeenstudiedearlier,
their laws and rules are to be derived from quantum physics.
Hence, quantum physics provides the foundation of classical physics and ther-
modynamicsaswellasofnumerousapplicationsinelectronics,engineering,andso
on.Moreover,withtheadvent ofnanometer manipulations,usefuldevicesdirectly
based on quantum effects are already a reality or will soon become feasible.
Asidefromtouchingonfundamentalquestionsonthenatureofthings,thebook
concentrates on examples which demonstrate the principles as clearly as possible.
Allexampleshavebeenchosencarefully.Theyillustratetheprinciplesandprovide
anoverviewoverhistoricallyorintrinsicallyinterestingtopicsandproblem-solving
strategies. Statistical thermodynamics, as quantum physics at nonvanishing tem-
perature, fits quite naturally.
The mathematical level is intermediate, as low as possible and as high as nec-
essary. The mathematical framework is coherently presented in a chapter on
Mathematical Aspects, from topological spaces, measure theory, probability, gen-
eralized functions to linear operators.
Any theory should be based on carefully chosen principles which are either
stated explicitly or implicitly by embedding the new in an already established
theory. Science in general and physics in particular have to stand another test
besides logical correctness, usefulness, and simplicity. Their rules and laws must
also pass the test of experimental verification.
The presentation is neither purely axiomatic nor historic nor pedagogic, but a
mixture of these approaches. Following history would require a deeper under-
standing of what contemporary scientists would think about nature. A purely
pedagogicapproachwoulddependtoomuchontheeducationalbackgroundofthe
readers which certainly is far from homogeneous. And the problems of the axio-
matic method in the sciences are well known. Axioms can only be formulated
v
vi Preface
vaguely at first by using everyday language. They are subsequently employed to
derive statements such that the terms in which the axioms were formulated gain
precision and technical meaning. Then the wording of the axioms must be sharp-
ened, and so on: an iterative approach.
Our main goal is a readable text with a few principles as guidelines. The many
worked examples are discussed at various levels of abstraction and mathematical
rigor.
Notonlyadvancedstudentsofgeneralphysics,chemistry,molecularbiology,or
materialssciencewillprofitfromreadingthebook,butalsophilosophicalmindedor
curious people will like this treatise.
WerecommendthatthereaderstudyatleasttheIntroductionSect.1.1andSect.1.3
ontheQuantumFrameworkandSect.1.4onTimeandSpace.Sheorhemaythen
decide where to continue: reading Sect. 1.2 on the Classical Framework as well,
Chap.2withSimpleExamplesorChap.7onMathematicalAspects.Althoughthe
bookcanbestudiedasaweb,itisconceivedtobereadconsecutivelyfromthefirstto
thelastpage.Enjoy!
Osnabrück, Germany Peter Hertel
June 2017
Contents
1 Basics. .... .... .... .... ..... .... .... .... .... .... ..... .... 1
1.1 Introduction .... .... ..... .... .... .... .... .... ..... .... 2
1.1.1 The Quantum of Light ... .... .... .... .... ..... .... 2
1.1.2 Electron Diffraction.. .... .... .... .... .... ..... .... 3
1.1.3 Heisenberg Uncertainty Principle ... .... .... ..... .... 4
1.1.4 Does God Play Dice? .... .... .... .... .... ..... .... 5
1.1.5 Summary. .... ..... .... .... .... .... .... ..... .... 6
1.2 Classical Framework.. ..... .... .... .... .... .... ..... .... 6
1.2.1 Phase Space .. ..... .... .... .... .... .... ..... .... 7
1.2.2 Observables... ..... .... .... .... .... .... ..... .... 8
1.2.3 Dynamics .... ..... .... .... .... .... .... ..... .... 8
1.2.4 States ... .... ..... .... .... .... .... .... ..... .... 10
1.2.5 Properties of Poisson Brackets . .... .... .... ..... .... 10
1.2.6 Canonical Relations.. .... .... .... .... .... ..... .... 11
1.2.7 Pure and Mixed States ... .... .... .... .... ..... .... 11
1.2.8 Summary. .... ..... .... .... .... .... .... ..... .... 12
1.3 Quantum Framework . ..... .... .... .... .... .... ..... .... 12
1.3.1 q-Numbers ... ..... .... .... .... .... .... ..... .... 13
1.3.2 Hilbert Space . ..... .... .... .... .... .... ..... .... 13
1.3.3 Linear Operators.... .... .... .... .... .... ..... .... 14
1.3.4 Projectors .... ..... .... .... .... .... .... ..... .... 16
1.3.5 Normal Linear Operators.. .... .... .... .... ..... .... 17
1.3.6 Trace of an Operator. .... .... .... .... .... ..... .... 19
1.3.7 Expectation Values.. .... .... .... .... .... ..... .... 20
1.3.8 Summary. .... ..... .... .... .... .... .... ..... .... 21
1.4 Time and Space . .... ..... .... .... .... .... .... ..... .... 22
1.4.1 Measurement and Experiment.. .... .... .... ..... .... 23
1.4.2 Time Translation.... .... .... .... .... .... ..... .... 24
1.4.3 Space Translation ... .... .... .... .... .... ..... .... 25
vii
viii Contents
1.4.4 Location . .... ..... .... .... .... .... .... ..... .... 25
1.4.5 Rotation . .... ..... .... .... .... .... .... ..... .... 26
1.4.6 Orbital Angular Momentum and Spin.... .... ..... .... 27
1.4.7 Schrödinger Picture.. .... .... .... .... .... ..... .... 28
1.4.8 Summary. .... ..... .... .... .... .... .... ..... .... 29
2 Simple Examples.... .... ..... .... .... .... .... .... ..... .... 31
2.1 Ammonia Molecule .. ..... .... .... .... .... .... ..... .... 31
2.1.1 Hilbert Space and Energy Observable.... .... ..... .... 33
2.1.2 Ammonia Molecule in an External Electric Field .... .... 34
2.1.3 Dipole Moment Expectation Value.. .... .... ..... .... 35
2.1.4 Ammonium Maser .. .... .... .... .... .... ..... .... 36
2.1.5 Summary. .... ..... .... .... .... .... .... ..... .... 37
2.2 Quasi-Particles .. .... ..... .... .... .... .... .... ..... .... 38
2.2.1 Hilbert Spaces Cn and ‘2 . .... .... .... .... ..... .... 38
2.2.2 Hopping Model..... .... .... .... .... .... ..... .... 40
2.2.3 Wave Packets . ..... .... .... .... .... .... ..... .... 41
2.2.4 Group Velocity and Effective Mass.. .... .... ..... .... 42
2.2.5 Scattering at Defects. .... .... .... .... .... ..... .... 43
2.2.6 Trapping by Defects . .... .... .... .... .... ..... .... 45
2.2.7 Summary. .... ..... .... .... .... .... .... ..... .... 46
2.3 Neutron Scattering on Molecules . .... .... .... .... ..... .... 46
2.3.1 Feynman’s Approach .... .... .... .... .... ..... .... 47
2.3.2 Spherical and Plain Waves .... .... .... .... ..... .... 48
2.3.3 Neutron Scattering on a Diatomic Molecule... ..... .... 48
2.3.4 Cross Section . ..... .... .... .... .... .... ..... .... 49
2.3.5 Orientation Averaged Cross Section . .... .... ..... .... 50
2.3.6 Neutron Diffraction.. .... .... .... .... .... ..... .... 52
2.3.7 Summary. .... ..... .... .... .... .... .... ..... .... 53
2.4 Free Particles ... .... ..... .... .... .... .... .... ..... .... 53
2.4.1 Square Integrable Functions ... .... .... .... ..... .... 54
2.4.2 Location . .... ..... .... .... .... .... .... ..... .... 55
2.4.3 Linear Momentum... .... .... .... .... .... ..... .... 55
2.4.4 Wave Packets . ..... .... .... .... .... .... ..... .... 57
2.4.5 Motion of a Free Particle . .... .... .... .... ..... .... 58
2.4.6 Spreading of a Free Particle ... .... .... .... ..... .... 59
2.4.7 Summary. .... ..... .... .... .... .... .... ..... .... 60
2.5 Small Oscillations.... ..... .... .... .... .... .... ..... .... 60
2.5.1 The Hamitonian .... .... .... .... .... .... ..... .... 61
2.5.2 Ladder Operators.... .... .... .... .... .... ..... .... 62
2.5.3 Eigenstate Wave Functions.... .... .... .... ..... .... 63
2.5.4 Summary. .... ..... .... .... .... .... .... ..... .... 64
Contents ix
3 Atoms and Molecules .... ..... .... .... .... .... .... ..... .... 65
3.1 Radial Schrödinger Equation .... .... .... .... .... ..... .... 66
3.1.1 Spherical Coordinates.... .... .... .... .... ..... .... 66
3.1.2 The Laplacian. ..... .... .... .... .... .... ..... .... 67
3.1.3 Spherical Harmonics. .... .... .... .... .... ..... .... 67
3.1.4 Spherical Symmetric Potential.. .... .... .... ..... .... 69
3.1.5 Behavior at Origin and Infinity. .... .... .... ..... .... 70
3.1.6 Alternative Form.... .... .... .... .... .... ..... .... 71
3.1.7 Summary. .... ..... .... .... .... .... .... ..... .... 71
3.2 Hydrogen Atom . .... ..... .... .... .... .... .... ..... .... 72
3.2.1 Atomic Units.. ..... .... .... .... .... .... ..... .... 72
3.2.2 Non-relativistic Hydrogen Atom.... .... .... ..... .... 74
3.2.3 Orbitals.. .... ..... .... .... .... .... .... ..... .... 75
3.2.4 Relativistic Hydrogen Atom ... .... .... .... ..... .... 77
3.2.5 Classical Hydrogen Atom. .... .... .... .... ..... .... 80
3.2.6 Summary. .... ..... .... .... .... .... .... ..... .... 81
3.3 Helium Atom ... .... ..... .... .... .... .... .... ..... .... 82
3.3.1 Wave Functions .... .... .... .... .... .... ..... .... 82
3.3.2 Minimal Ground State Energy.. .... .... .... ..... .... 84
3.3.3 Sample Calculation.. .... .... .... .... .... ..... .... 86
3.3.4 The Negative Hydrogen Ion ... .... .... .... ..... .... 88
3.3.5 Summary. .... ..... .... .... .... .... .... ..... .... 88
3.4 Hydrogen Molecule .. ..... .... .... .... .... .... ..... .... 89
3.4.1 Wave Functions and Hamiltonian... .... .... ..... .... 89
3.4.2 Born–Oppenheimer Approximation.. .... .... ..... .... 91
3.4.3 The Molecular Potential .. .... .... .... .... ..... .... 91
3.4.4 Molecular Vibrations. .... .... .... .... .... ..... .... 93
3.4.5 Molecular Rotations . .... .... .... .... .... ..... .... 95
3.4.6 Summary. .... ..... .... .... .... .... .... ..... .... 95
3.5 More on Approximations ... .... .... .... .... .... ..... .... 96
3.5.1 The Minimax Theorem... .... .... .... .... ..... .... 97
3.5.2 Remarks . .... ..... .... .... .... .... .... ..... .... 98
3.5.3 Stationary Perturbations... .... .... .... .... ..... .... 100
3.5.4 Coping with Degeneracies. .... .... .... .... ..... .... 101
3.5.5 Summary. .... ..... .... .... .... .... .... ..... .... 101
3.6 Stark and Zeeman Effect.... .... .... .... .... .... ..... .... 102
3.6.1 Multipoles.... ..... .... .... .... .... .... ..... .... 102
3.6.2 Electric Dipole Moment .. .... .... .... .... ..... .... 105
3.6.3 Stark Effect... ..... .... .... .... .... .... ..... .... 106
3.6.4 Magnetic Dipole Moment. .... .... .... .... ..... .... 108
3.6.5 Zeeman Effect. ..... .... .... .... .... .... ..... .... 109
3.6.6 Summary. .... ..... .... .... .... .... .... ..... .... 110