Elements of Classical and Quantum Physics

391 Pages·2018·3.05 MB·English

by  Michele Cini

UNITEXT for Physics
Michele Cini
Elements of
Classical and
Quantum
Physics
UNITEXT for Physics
Series editors
Paolo Biscari, Milano, Italy
Michele Cini, Roma, Italy
Attilio Ferrari, Torino, Italy
Stefano Forte, Milano, Italy
Morten Hjorth-Jensen, Oslo, Norway
Nicola Manini, Milano, Italy
Guido Montagna, Pavia, Italy
Oreste Nicrosini, Pavia, Italy
Luca Peliti, Napoli, Italy
Alberto Rotondi, Pavia, Italy
UNITEXTforPhysicsseries,formerly UNITEXT CollanadiFisicae Astronomia,
publishestextbooksandmonographsinPhysicsandAstronomy,mainlyinEnglish
language, characterized of a didactic style and comprehensiveness. The books
published in UNITEXT for Physics series are addressed to graduate and advanced
graduate students, but also to scientists and researchers as important resources for
their education, knowledge and teaching.
More information about this series at http://www.springer.com/series/13351
Michele Cini
Elements of Classical
and Quantum Physics
123
Michele Cini
Universitàdi RomaTor Vergata
Rome
Italy
ISSN 2198-7882 ISSN 2198-7890 (electronic)
UNITEXTfor Physics
ISBN978-3-319-71329-8 ISBN978-3-319-71330-4 (eBook)
https://doi.org/10.1007/978-3-319-71330-4
LibraryofCongressControlNumber:2017963983
©SpringerInternationalPublishingAG,partofSpringerNature2018
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor
for any errors or omissions that may have been made. The publisher remains neutral with regard to
jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
Printedonacid-freepaper
ThisSpringerimprintispublishedbytheregisteredcompanySpringerInternationalPublishingAG
partofSpringerNature
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
To my wife Anna and to my son Massimo
Preface
This book originates out of many years of teaching different courses at the
University of Roma Tor Vergata, although it includes so many applications for
diversebranchesofPhysicsthatIhopethatmanyprofessionalphysicistswillfindit
ofinterestaswell.Itcanserveasatextbookforsecond-andthird-yearstudentsof
PhysicsandrelateddisciplinesthatrequirebeingintroducedtoTheoreticalPhysics
at the level of the short degree. I develop all the mathematical methods, but the
main focus is on the physical meaning of the formalisms. I tried to make the book
funandinteresting bystimulatingthereader’scuriosityabout manymore physical
effectsthanisusualintextbooks.Theserangefromthemeasurementofstellarradii
to anyons, quantum pumping, entanglement, frame dragging, teleportation, black
holes,superconductivityandmore.Physicsistheonlyscienceinwhichtheoryand
experiment have comparable importance. Like in any other natural science,
experiment discerns what is true and what is false; however, in countless cases,
important discoveries are due to theoreticians making detailed predictions that
guided experimenters to verify new effects—antimatter was discovered theoreti-
cally by Dirac; many gravitational effects were predicted by Einstein; in quantum
electrodynamics,theprogressofboththeoryandexperimentpushedtheagreement
tomorethan10significantdigits.Thankstotheory,Physicsisnotjustanimmense
inventoryofdata,buthasacoherentlogicwhichcanonlybeunderstoodintermsof
ingenious and beautiful Mathematics.
More exercises can be found in a book by Michele Cini, Francesco Fucito and
Mauro Sbragaglia, Solved Problems in Quantum and Statistical Physics, Springer
VerlagItalia2012.AmoreadvancedtreatmentofSolidStateTheoryispresentedin
Michele Cini, Topics and Methods in Condensed Matter Theory, Springer 2007.
Rome, Italy Michele Cini
October 2017
vii
Contents
1 Theoretical Physics and Mathematics . . . . . . . . . . . . . . . . . . . . . . . 1
Part I Classical Physics-Complements
2 Analytical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Galileo’s Revolution and Newton’s F ¼ma. . . . . . . . . . . . . . . 7
2.2 Lagrangian Formalism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Rotating Platform. . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.2 Kepler Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 The Path Between t and t and the Action Integral . . . . . . . . . 19
1 2
2.3.1 Principle of Least Action . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Legendre Transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 The Hamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5.1 Reduced Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5.2 Canonical Transformations . . . . . . . . . . . . . . . . . . . . . 27
2.5.3 Hamilton–Jacobi Equation . . . . . . . . . . . . . . . . . . . . . 31
2.5.4 Point Charge in an Electromagnetic Field . . . . . . . . . . 32
2.5.5 Poisson Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6 Delaunay Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.7 Noether Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.8 Chaos. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Dirac’s Delta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1 Definition of the d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.1 Volume of the Hypersphere in N Dimensions . . . . . . . 48
3.1.2 Plancherel Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4 Some Consequences of Maxwell’s Equations. . . . . . . . . . . . . . . . . . 51
4.1 Fields and Potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Green’s Function of the Wave Equation and Retarded
Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 Lienard–Wiechert Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 55
ix
x Contents
4.4 Geometrical Optics and Fermat’s Principle. . . . . . . . . . . . . . . . 57
4.5 Coherent Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.5.1 The Measurement of Stellar Diameters . . . . . . . . . . . . 58
5 Thermal Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1 The Principles of Thermodynamics . . . . . . . . . . . . . . . . . . . . . 61
5.2 Black Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3 Statistical Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.4 Gibbs Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.5 Microcanonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.5.1 Entropy of the Perfect Gas . . . . . . . . . . . . . . . . . . . . . 79
5.6 Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.6.1 Canonical Temperature. . . . . . . . . . . . . . . . . . . . . . . . 82
5.6.2 Information Entropy, Irreversible Gates
and Toffoli Gate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.7 Canonical Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.7.1 Maxwell Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.7.2 Perfect Gas in the Canonical Ensemble
and Boltzmann Statistics. . . . . . . . . . . . . . . . . . . . . . . 86
5.7.3 Thermodynamic Magnitudes in the Canonical
Ensemble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.7.4 Theorem of Equipartition . . . . . . . . . . . . . . . . . . . . . . 91
5.8 Grand-Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.8.1 Monte Carlo Methods. . . . . . . . . . . . . . . . . . . . . . . . . 94
6 Special Relativity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.1 Galileo’s Ship. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2 From Thought Experiments to the Lorentz Transformation . . . . 99
6.2.1 Relativistic Addition of Velocities. . . . . . . . . . . . . . . . 107
6.3 The Geometry of Special Relativity: Minkowsky
Chronotope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.3.1 Cartesian Tensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3.2 Action of the Free Field . . . . . . . . . . . . . . . . . . . . . . . 114
6.3.3 Doppler–Fizeau Effect and Aberration of Light . . . . . . 114
6.3.4 Relativistic Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 117
6.3.5 Field Lagrangian and Hamiltonian;
Energy-Momentum Tensor . . . . . . . . . . . . . . . . . . . . . 120
7 Curvilinear Coordinates and Curved Spaces . . . . . . . . . . . . . . . . . 123
7.1 Parallel Transport, Affine Connection and Covariant
Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.2 Geodesics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Contents xi
8 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.1 Principle of Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.2 The Principle of General Covariance and the Curved
Space-Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.2.1 Space Geometry in Stationary Problems . . . . . . . . . . . 140
8.2.2 Curved Space in a Rotating Frame . . . . . . . . . . . . . . . 140
8.2.3 Generalized Equation of Motion . . . . . . . . . . . . . . . . . 141
8.2.4 Generalized Maxwell Equations . . . . . . . . . . . . . . . . . 142
8.3 Einstein Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.3.1 Linearized Field Equations . . . . . . . . . . . . . . . . . . . . . 144
8.4 Schwarzschild Solution and Black Holes . . . . . . . . . . . . . . . . . 145
8.5 Relativistic Delay of Signals in a Gravity Field . . . . . . . . . . . . 148
8.6 Clocks in a Gravity Field, GPS and Gravitational
Red Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.7 Bending and Gravitational Lensing of Light. . . . . . . . . . . . . . . 150
8.8 Shift of the Perihelion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.9 Geodetic Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.10 Frame Dragging and Gravitomagnetic Field . . . . . . . . . . . . . . . 155
8.11 Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
8.12 The Standard Model of Cosmology . . . . . . . . . . . . . . . . . . . . . 159
Part II Quantum Mechanics
9 The Wave Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
9.1 Corpuscles and Waves in Classical Physics . . . . . . . . . . . . . . . 169
9.2 Dualism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
9.3 Which Way Did the Electron Pass Through? . . . . . . . . . . . . . . 173
9.3.1 Bohm-Aharonov Effect. . . . . . . . . . . . . . . . . . . . . . . . 173
9.3.2 Experiments by Deutsch on Photons . . . . . . . . . . . . . . 174
9.3.3 Plane Wave, Superposition Principle
and Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
9.4 The Copenhagen Interpretation . . . . . . . . . . . . . . . . . . . . . . . . 179
9.4.1 The Bohm Formulation of Quantum Theory . . . . . . . . 181
9.5 Quantum Eraser Experiments and Delayed Choice
Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
9.5.1 Is the Interpretation a Closed Issue? . . . . . . . . . . . . . . 183
10 The Eigenvalue Equation and the Evolution Operator . . . . . . . . . . 185
10.1 Stationary State Equation and Its Resolvent . . . . . . . . . . . . . . . 188
10.2 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
10.3 Schrödinger and Heisenberg Formulations of Quantum
Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190