About Book

Modern quantum mechanics refers to the study of processes of photons and atoms. It is based on quantum field theory. It deals with the behaviour of matter and light on the atomic and subatomic scale. It attempts to describe and account for the properties of molecules and atoms and their constituents—electrons, protons, neutrons, and other more esoteric particles such as quarks and gluons. These properties include the interactions of the particles with one another and with electromagnetic radiation. The behaviour of matter and radiation on the atomic scale often seems peculiar and the consequences of quantum theory are accordingly difficult to understand and to believe. Its concepts frequently conflict with common-sense notions derived from observations of the everyday world. This book attempts to understand the multiple branches that fall under the discipline of quantum mechanics and how such concepts have practical application. It deals with measurement in quantum mechanics, supersymmetric quantum mechanics, statistical mechanics, scale relativity theory, finite differences and bound state calculations. Such selected concepts that redefine the field of modern quantum mechanics have been presented in this book. It studies, analyses and upholds the pillars of the subject and its utmost significance in modern times. This book attempt to assist those with a goal of delving into this field.

Preface............................................................................................................v

Chapter 1 Exact Finite Differences for Quantum Mechanics..............................1

Abstract...................................................................................................1

1. Introduction............................................................................................1

2. Exact first-order finite differences derivatives of functions.........2

3. Exact first-order finite differences derivative for the exponential

function...................................................................................................3

4. The matrix associated to the exact finite differences derivative.7

5. Quantum mechanical momentum and time operators...............14

6. The particle in a linear potential.....................................................15

Chapter 2 New System-Specific Coherent States by Supersymmetric Quantum Mechanics for Bound State Calculations............................................18

1. Introduction..........................................................................................18

2. Theoretical formulation......................................................................20

3. Computational results........................................................................29

4. Discussion and perspectives.............................................................36

Chapter 3 Emergent un-Quantum Mechanics.......................................................40

1. Introduction..........................................................................................40

2. Complexity and symmetry induce dynamics...............................40

3. Dynamics..............................................................................................47

4. A World without Planck’s constant................................................54

5. Observables, quantization and bracket relations..........................63

6. Quantum probability..........................................................................67

7. Applications.........................................................................................78

8. Concluding remarks...........................................................................83

Acknowledgements.............................................................................85

Chapter 4 Correspondences of Scale Relativity Theory with Quantum

Mechanics...................................................................................................88

1. Introduction..........................................................................................88

2. Hydrodynamic model of quantum mechanics

(Madelung’s model)............................................................................89

3. DeBroglie’s theory of double solution. The need for introducing

the model of Bohm and Vigier........................................................90

4. Comments.............................................................................................92

5. The motion equation of the physical object in the fractal

approximation of motion. The Eulerian separation of motion on

resolution scales..................................................................................93

6. Rotational motions and flow regimes of a fractal fluid..............95

7. Irotational motions of a fractal fluid. The incorporation of the

associate wave corpuscle through the solidarity of movements

and generation of Schrodinger equation........................................97

8. Extended hydrodynamic model of scale relativity and

incorporation of associated wave corpuscle through fractal

potential. The correspondence with Madelung model................99

9. “Mechanisms” of duality through coherence in corpuscle-wave

phase....................................................................................................101

10. Wave-corpuscle duality through flowing stationary regimes of a

coherent fractal fluid in phase. The potential well....................105

11. Wave-corpuscle duality through non-stationary regimes of an

incoherent fractal fluid.....................................................................106

12. The corpuscle as a horologe and its incorporation in the

associated wave. Consequences.....................................................108

13. Informational energy through the fractal potential of complex

scalar speed field. The generation of forces................................111

14. Lorenz type mechanism of wave-corpuscle duality in non

stationary systems.............................................................................113

15. Conclusions........................................................................................125

Chapter 5 Complementarity in Quantum Mechanics and Classical Statistical

Mechanics.................................................................................................131

1. Introduction........................................................................................131

2. Complementarity in quantum mechanics:

A general overview..........................................................................133

3. Relevant inequalities in classical probability theory..................144

4. Complementarity in classical statistical mechanics....................149

5. Final remarks.....................................................................................157

Chapter 6 Generalized Non-Relativistic Supersymmetric Quantum

Mechanics.................................................................................................160

1. Introduction........................................................................................160

2. Introduction to supersymmetric quantum mechanics in one

dimension...........................................................................................162

3. Computational examples.................................................................167

4. Generalization to multi-dimensions..............................................174

5. Clusters of distinguishable particles.............................................182

6. Electronic structure of atoms: Hydrogen and helium...............196

7. Aufbau approach for excited states...............................................200

8. Conclusions and perspectives.........................................................202

Chapter 7 Measurement in Quantum Mechanics: Decoherence and the

Pointer Basis.............................................................................................210

1. Introduction........................................................................................210

2. Measurement in quantum mechanics...........................................212

3. Decoherence.......................................................................................216

4. Decoherence and the pointer basis...............................................225

5. Conclusions........................................................................................227

Citation......................................................................................................233

Index...........................................................................................................235

About Author

Kate Kazantsev pursued her Ph.D. in Theoretical and Applied from State University, Novosibirsk. She is a Senior Guest Lecturer at Technical University, Dnipro. She has attended numerous scientific conferences and symposiums; and has presented 10 papers on quantum physics. She is a veteran in the field of quantum mechanics with numerous revolutionizing texts, articles and research papers to her credit. Kazantsev has won a “Distinguished Faculty Lecturer” award for her outstanding contribution towards the academic community especially in the undergraduate programs.