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El. knyga: Quantum Mechanics for Pedestrians 1: Fundamentals

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This book, the first in a two-volume set, provides an introduction to the fundamentals of (mainly) non-relativistic quantum mechanics. This first volume chiefly focuses on the essential principles, while applications and extensions of the formalism can be found in volume 2. Including but also moving beyond material that is covered in traditional textbooks on quantum mechanics, the book discusses in detail current issues such as interaction-free quantum measurements or neutrino oscillations, as well as fundamental problems and epistemological questions, such as the measurement problem. A chapter on the postulates of quantum mechanics rounds off this first volume.

In order to quickly and clearly present the main principles of quantum mechanics and its mathematical formulation, there is a systematic transition between wave mechanics and algebraic representation in the first few chapters, in which the required mathematical tools are introduced step by step. Moreover, the appendix concisely reviews the most important mathematical tools, allowing readers to largely dispense with supplementary literature. The appendix also explores advanced topics, such as the Quantum-Zeno effect and time-delay experiments. Over 250 exercises, most of them with solutions, help to deepen the reader’s understanding of the topics discussed.

This revised second edition is expanded by an introduction to some ideas and problems of relativistic quantum mechanics. In this first volume, the Klein-Gordon and the Dirac equations are treated. Fundamentals of other areas are compiled in compact form, i.e., outlines of special relativity, classical field theory and electrodynamics.

The book is chiefly intended for student science teachers and all students of physics, majors and minors alike, who are looking for a reasonably easy and modern introduction to quantum mechanics.



This book provides an introduction into the fundamentals of non-relativistic quantum mechanics. The first part of a two-volume reference, it develops essential principles and supports learning with exercises and problems with solutions.

Recenzijos

This book is an excellent introduction to quantum mechanics suited for beginners to get first insights which may be deepened reading the appendices. The two volumes can be best recommended generally and especially for self studies. (K.-E. Hellwig, zbMATH 1445.81001, 2020)

Part I Fundamentals
1 Towards the Schrodinger Equation
3(12)
1.1 How to Find a New Theory
3(2)
1.2 The Classical Wave Equation and the Schrodinger Equation
5(7)
1.2.1 From the Wave Equation to the Dispersion Relation
5(4)
1.2.2 From the Dispersion Relation to the Schrodinger Equation
9(3)
1.3 Exercises
12(3)
2 Polarization
15(14)
2.1 Light as Waves
16(7)
2.1.1 The Typical Shape of an Electromagnetic Wave
16(1)
2.1.2 Linear and Circular Polarization
17(2)
2.1.3 From Polarization to the Space of States
19(4)
2.2 Light as Photons
23(5)
2.2.1 Single Photons and Polarization
23(2)
2.2.2 Measuring the Polarization of Single Photons
25(3)
2.3 Exercises
28(1)
3 More on the Schrodinger Equation
29(12)
3.1 Properties of the Schrodinger Equation
29(2)
3.2 The Time-Independent Schrodinger Equation
31(2)
3.3 Operators
33(6)
3.3.1 Classical Numbers and Quantum-Mechanical Operators
34(2)
3.3.2 Commutation of Operators; Commutators
36(3)
3.4 Exercises
39(2)
4 Complex Vector Spaces and Quantum Mechanics
41(14)
4.1 Norm, Bra-Ket Notation
42(2)
4.2 Orthogonality, Orthonormality
44(1)
4.3 Completeness
45(2)
4.4 Projection Operators, Measurement
47(6)
4.4.1 Projection Operators
47(4)
4.4.2 Measurement and Eigenvalues
51(1)
4.4.3 Summary
52(1)
4.5 Exercises
53(2)
5 Two Simple Solutions of the Schrodinger Equation
55(18)
5.1 The Infinite Potential Well
55(8)
5.1.1 Solution of the Schrodinger Equation, Energy Quantization
56(3)
5.1.2 Solution of the Time-Dependent Schrodinger Equation
59(1)
5.1.3 Properties of the Eigenfunctions and Their Consequences
60(2)
5.1.4 Determination of the Coefficients cn
62(1)
5.2 Free Motion
63(4)
5.2.1 General Solution
64(1)
5.2.2 Example: Gaussian Distribution
65(2)
5.3 General Potentials
67(2)
5.4 Exercises
69(4)
6 Interaction-Free Measurement
73(14)
6.1 Experimental Results
73(5)
6.1.1 Classical Light Rays and Particles in the Mach--Zehnder Interferometer
73(2)
6.1.2 Photons in the Mach--Zehnder Interferometer
75(3)
6.2 Formal Description, Unitary Operators
78(4)
6.2.1 First Approach
78(2)
6.2.2 Second Approach (Operators)
80(2)
6.3 Concluding Remarks
82(3)
6.3.1 Extensions
82(1)
6.3.2 Quantum Zeno Effect
82(1)
6.3.3 Delayed-Choice Experiments
83(1)
6.3.4 The Hadamard Transformation
83(1)
6.3.5 From the MZI to the Quantum Computer
84(1)
6.3.6 Hardy's Experiment
84(1)
6.3.7 How Interaction-Free is the `Interaction-Free' Quantum Measurement?
84(1)
6.4 Exercises
85(2)
7 Position Probability
87(12)
7.1 Position Probability and Measurements
88(5)
7.1.1 Example: Infinite Potential Wall
88(1)
7.1.2 Bound Systems
89(3)
7.1.3 Free Systems
92(1)
7.2 Real Potentials
93(2)
7.3 Probability Current Density
95(3)
7.4 Exercises
98(1)
8 Neutrino Oscillations
99(10)
8.1 The Neutrino Problem
99(1)
8.2 Modelling the Neutrino Oscillations
100(5)
8.2.1 States
100(1)
8.2.2 Time Evolution
101(1)
8.2.3 Numerical Data
102(1)
8.2.4 Three-Dimensional Neutrino Oscillations
103(2)
8.3 Generalizations
105(2)
8.3.1 Hermitian Operators
105(1)
8.3.2 Time Evolution and Measurement
106(1)
8.4 Exercises
107(2)
9 Expectation Values, Mean Values, and Measured Values
109(16)
9.1 Mean Values and Expectation Values
109(7)
9.1.1 Mean Values of Classical Measurements
109(1)
9.1.2 Expectation Value of the Position in Quantum Mechanics
110(1)
9.1.3 Expectation Value of the Momentum in Quantum Mechanics
111(2)
9.1.4 General Definition of the Expectation Value
113(2)
9.1.5 Variance, Standard Deviation
115(1)
9.2 Hermitian Operators
116(3)
9.2.1 Hermitian Operators Have Real Eigenvalues
117(1)
9.2.2 Eigenfunctions of Different Eigenvalues Are Orthogonal
118(1)
9.3 Time Behavior, Conserved Quantities
119(3)
9.3.1 Time Behavior of Expectation Values
119(1)
9.3.2 Conserved Quantities
120(1)
9.3.3 Ehrenfest's Theorem
121(1)
9.4 Exercises
122(3)
10 Stopover; Then on to Quantum Cryptography
125(14)
10.1 Outline
125(1)
10.2 Summary and Open Questions
125(5)
10.2.1 Summary
126(3)
10.2.2 Open Questions
129(1)
10.3 Quantum Cryptography
130(9)
10.3.1 Introduction
131(1)
10.3.2 One-Time Pad
131(2)
10.3.3 BB84 Protocol Without Eve
133(2)
10.3.4 BB84 Protocol with Eve
135(4)
11 Abstract Notation
139(12)
11.1 Hilbert Space
139(4)
11.1.1 Wavefunctions and Coordinate Vectors
139(2)
11.1.2 The Scalar Product
141(1)
11.1.3 Hilbert Space
142(1)
11.2 Matrix Mechanics
143(1)
11.3 Abstract Formulation
144(4)
11.4 Concrete: Abstract
148(2)
11.5 Exercises
150(1)
12 Continuous Spectra
151(14)
12.1 Improper Vectors
152(5)
12.2 Position Representation and Momentum Representation
157(4)
12.3 Conclusions
161(1)
12.4 Exercises
162(3)
13 Operators
165(22)
13.1 Hermitian Operators, Observables
166(8)
13.1.1 Three Important Properties of Hermitian Operators
167(3)
13.1.2 Uncertainty Relations
170(3)
13.1.3 Degenerate Spectra
173(1)
13.2 Unitary Operators
174(3)
13.2.1 Unitary Transformations
174(1)
13.2.2 Functions of Operators, the Time-Evolution Operator
175(2)
13.3 Projection Operators
177(4)
13.3.1 Spectral Representation
178(1)
13.3.2 Projection and Properties
179(1)
13.3.3 Measurements
180(1)
13.4 Systematics of the Operators
181(1)
13.5 Exercises
182(5)
14 Postulates of Quantum Mechanics
187(16)
14.1 Postulates
188(6)
14.1.1 States, State Space (Question 1)
188(2)
14.1.2 Probability Amplitudes, Probability (Question 2)
190(1)
14.1.3 Physical Quantities and Hermitian Operators (Question 2)
190(1)
14.1.4 Measurement and State Reduction (Question 2)
191(1)
14.1.5 Time Evolution (Question 3)
192(2)
14.2 Some Open Problems
194(5)
14.3 Concluding Remarks
199(1)
14.3.1 Postulates of Quantum Mechanics as a Framework
199(1)
14.3.2 Outlook
199(1)
14.4 Exercises
200(3)
Appendix A Abbreviations and Notations 203(2)
Appendix B Units and Constants 205(6)
Appendix C Complex Numbers 211(10)
Appendix D Calculus I 221(16)
Appendix E Calculus II 237(8)
Appendix F Linear Algebra I 245(18)
Appendix G Linear Algebra II 263(10)
Appendix H Fourier Transforms and the Delta Function 273(18)
Appendix I Operators 291(20)
Appendix J From Quantum Hopping to the Schrodinger Equation 311(6)
Appendix K The Phase Shift at a Beam Splitter 317(2)
Appendix L The Quantum Zeno Effect 319(8)
Appendix M Delayed Choice and the Quantum Eraser 327(6)
Appendix N The Equation of Continuity 333(2)
Appendix O Variance, Expectation Values 335(4)
Appendix P On Quantum Cryptography 339(6)
Appendix Q Schrodinger Picture, Heisenberg Picture, Interaction Picture 345(6)
Appendix R The Postulates of Quantum Mechanics 351(16)
Appendix S System and Measurement: Some Concepts 367(6)
Appendix T Recaps and Outlines 373(1)
T.1 Discrete - Continuous 373(2)
T.2 Special Relativity 375(13)
T.3 Classical Field Theory 388(9)
T.4 Electrodynamics 397(8)
Appendix U Elements of Relativistic Quantum Mechanics 405(1)
U.1 Introduction 405(1)
U.2 Constructing Relativistic Equations 406(7)
U.3 Plane Wave Solutions 413(5)
U.4 Covariant Formulation of the Dirac Equation 418(9)
U.5 Dirac Equation and the Hydrogen Atom 427(1)
U.6 Discussion of the Dirac Equation 428(5)
U.7 Exercises and Solutions 433(8)
Appendix V Exercises and Solutions to Chaps. 1--14 441(72)
Further Reading 513(2)
Index of Volume 1 515(4)
Index of Volume 2 519
Jochen Pade studied Physics in Freiburg, Germany, where he received his PhD in Theoretical Physics in 1978. Since 1980, he has been a lecturer at the Carl von Ossietzky University of Oldenburg, Germany. His main research interests are in theoretical physics, the didactics and popularization of science.
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