A Brief Tour Of Modern Quantum Mechanics

by Moshe Gitterman

Overview

This slim volume covers the traditional parts of quantum mechanics: semiclassical theories of radiation and scattering, a number of advanced problems: Feynman diagrams and relativistic quantum mechanics and a collection of modern items: superfluidity and high-temperature superconductivity. The book begins with the description of the basic principles of mechanics, electrodynamics and quantum mechanics, which are needed for understanding the subsequent chapters. Qualitative methods (analytical properties and paradoxes in quantum mechanics) are also introduced. This useful textbook also pairs the problems with their solutions.

Product Details

ISBN-13:	9789814374224
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	02/02/2012
Pages:	144
Product dimensions:	6.10(w) x 9.00(h) x 0.80(d)

Table of Contents

Preface v

1 Introduction 1

1.1 Classical mechanics 1

1.1.1 Newton's law 1

1.1.2 Principle of least action 2

1.1.3 Hamilton's equations 3

1.1.4 Hamilton-Jacobi equation 4

1.2 Electrodynamics 5

1.2.1 Maxwell's equations 5

1.2.2 Electromagnetic potentials 7

1.2.3 Lagrangian for the electromagnetic field 9

1.3 Quantum mechanics 10

1.3.1 Wave function 10

1.3.2 Dynamic behavior 11

1.3.3 Conservation laws in quantum mechanics 12

1.3.4 Different representations 14

1.3.5 Aharonov-Bohm effect 14

2 Semiclassical Theory of Radiation 17

2.1 Fermi's Golden Rule 17

2.2 Dipole transitions 19

2.3 Forbidden and strictly forbidden transitions 20

2.4 Selection rules 21

2.5 Radioactivity 21

2.6 Photoelectric effect 22

2.7 Cherenkov effect 23

2.8 Compton effect 24

2.9 Problems 25

3 Many-body Problem 29

3.1 Fermions and bosons 29

3.2 N-representation 30

3.3 Lagrangian and Hamiltonian of quantum systems 31

3.4 Second quantization 31

3.5 Hamiltonian in second quantization 32

3.6 Quantization of the electromagnetic field 34

3.7 Full quantum mechanical description of the particle-wave interaction 36

3.8 Superfluidity and superconductivity 38

3.8.1 Statistics of a bosonic gas 39

3.8.2 Microscopic theory of superfluidity 40

3.8.3 Experimental detection of superfluidity 42

3.8.4 Electron pairing in superconductors 43

3.8.5 Microscopic theory of superconductivity 45

3.8.6 High-temperature superconductivity 46

3.9 Problems 48

4 S-matrix, Green Function, Feynman Diagrams 51

4.1 S-matrix 51

4.2 Green function 52

4.3 Green function in second quantization 54

4.4 Wick theorem 55

4.5 Feynman diagrams 56

4.5.1 Electrons in an external field 58

4.5.2 Hartree and Hartree-Fock approximations 59

4.5.3 Electron gas at low and high density 60

4.6 Problems 63

5 Relativistic Quantum Mechanics 65

5.1 Klein-Gordon equation 65

5.2 Dirac equation 66

5.3 Dynamic solution of the Dirac equation 67

5.4 Electron spin 69

5.5 Dirac equation for a free particle 71

5.6 Motion in a central field 72

5.7 Nature of the physical vacuum 75

5.7.1 Lamb shift 76

5.7.2 Klein paradox 77

5.7.3 Casimir force 78

5.8 Problems 80

6 Semiclassical Approximation to Quantum Mechanics 83

6.1 Wave equation 83

6.2 Turning points 85

6.3 Energy spectrum 86

6.4 Tunneling through a potential barrier 88

6.5 Problems 90

7 Scattering 95

7.1 Phenomenological description of scattering 95

7.2 Born approximation 96

7.3 Scattering by different potentials 98

7.4 Partial waves 99

7.5 Optical theorem 101

7.6 Problems 103

8 Analytical Properties 109

8.1 Analytical properties of the wave function 109

8.2 Analytical properties of the scattering amplitude 110

8.3 Resonance scattering 111

8.4 Symmetry 112

8.4.1 Parity 112

8.4.2 Conservation laws 113

8.4.3 Degeneracy 114

8.4.4 Internal symmetries 115

8.4.5 Complex conjugation and time reversal 115

8.4.6 Gauge transformation 116

9 Paradoxes in Quantum Mechanics 119

9.1 Schrödinger's cat 119

9.2 The Einstein-Podolsky-Rosen (EPR) paradox 120

9.3 Hidden variables and Bell's inequality 122

Bibliography 127

Index 129