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Quantum Mechanics II

Advanced Topics

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## Book Description

**Why the Quantum Field Theory?**

**Quantum Mechanics II: Advanced Topics **uses more than a decade of research and the authors’ own teaching experience to expound on some of the more advanced topics and current research in quantum mechanics. A follow-up to the authors introductory book **Quantum Mechanics I: The Fundamentals**, this book begins with a chapter on quantum field theory, and goes on to present basic principles, key features, and applications. It outlines recent quantum technologies and phenomena, and introduces growing topics of interest in quantum mechanics. The authors describe promising applications that include ghost imaging, detection of weak amplitude objects, entangled two-photon microscopy, detection of small displacements, lithography, metrology, and teleportation of optical images. They also present worked-out examples and provide numerous problems at the end of each chapter.

Establishes a Need for the Quantum Field Theory

Consisting of ten chapters, this illuminating text:

- Covers the basic ideas of both classical and quantum field theories
- Highlights path integral formalism, supersymmetric quantum mechanics, coherent and squeezed states, Berry's phase, Aharonov-Bohm and Sagnac effects, and Wigner function
- Addresses basic principles, salient features, and applications
- Describes basic concepts of quantum computers, some of the quantum algorithms, and features of quantum computation
- Explores advances made in the field of quantum cryptography
- Provides a brief and compact introduction to topics of growing interest including quantum versions of theory of gravity, Zeno effect, teleportation, games, cloning, diffusion, and chaos
- Focuses on the theoretical aspects of various advanced topics
- Outlines some of the quantum technologies and/or technological applications of quantum phenomena
- Presents the basic principles and salient features of ghost imaging, detection of weak amplitude object and small displacements, entangled two-photon microscopy, quantum lithography, metrology, and teleportation of optical images
- Contains several worked-out problems at the end of each chapter
- Includes material that can be covered in an advanced course on quantum mechanics

**Quantum Mechanics II: Advanced Topics** addresses the basic principles and current research on various topics in quantum mechanics, and is a valuable resource for advanced undergraduate and graduate students in physics, chemistry, and engineering with an interest in quantum mechanics.

Print Versions of this book also include access to the ebook version.

## Table of Contents

**Quantum Field Theory**

INTRODUCTION

WHY QUANTUM FIELD THEORY?

WHAT IS A FIELD?

CLASSICAL FIELD THEORY

QUANTUM EQUATIONS FOR FIELDS

QUANTIZATION OF NONRELATIVISTIC WAVE EQUATION

ELECTROMAGNETIC FIELD IN VACUUM

INTERACTION OF CHARGED PARTICLES WITH ELECTROMAGNETIC FIELD

SPONTANEOUS EMISSION

QUANTIZATION OF KLEIN–GORDON EQUATION

QUANTIZATION OF DIRAC FIELD

GAUGE FIELD THEORIES

CONCLUDING REMARKS

BIBLIOGRAPHY

EXERCISES

Path Integral Formulation

INTRODUCTION

TIME EVOLUTION OF WAVE FUNCTION AND PROPAGATOR

PATH INTEGRAL REPRESENTATION OF PROPAGATOR

CONNECTION BETWEEN THE PROPAGATOR AND THE CLASSICAL ACTION

SCHRÖDINGER EQUATION FROM PATH INTEGRAL FORMULATION

TRANSITION AMPLITUDE OF A FREE PARTICLE

SYSTEMS WITH QUADRATIC LAGRANGIAN

PATH INTEGRAL FOR HARMONIC OSCILLATOR

PATH INTEGRAL VERSION OF EHRENFEST’S THEOREM

CONCLUDING REMARKS

BIBLIOGRAPHY

EXERCISES

Supersymmetric Quantum Mechanics

INTRODUCTION

SUPERSYMMETRIC POTENTIALS

RELATIONS BETWEEN THE EIGENSTATES OF TWO SUPERSYMMETRIC HAMILTONIANS

HIERARCHY OF SUPERSYMMETRIC HAMILTONIANS

APPLICATIONS

CONCLUDING REMARKS

BIBLIOGRAPHY

EXERCISES

Coherent and Squeezed States

INTRODUCTION

THE UNCERTAINTY PRODUCT OF HARMONIC OSCILLATOR

COHERENT STATES: DEFINITION AND UNCERTAINTY PRODUCT

PHYSICAL MEANING OF COHERENT STATES

GENERATION OF COHERENT STATES

PROPERTIES OF COHERENT STATES

SQUEEZED STATES

DEFORMED OSCILLATORS AND NONLINEAR COHERENT STATES

CONCLUDING REMARKS

BIBLIOGRAPHY

EXERCISES

Berry’s Phase, Aharonov–Bohm and Sagnac Effects

INTRODUCTION

DERIVATION OF BERRY’S PHASE

ORIGIN AND PROPERTIES OF BERRY’S PHASE

CLASSICAL ANALOGUE OF BERRY’S PHASE

EXAMPLES FOR BERRY’S PHASE

EFFECTS OF BERRY’S PHASE

APPLICATIONS OF BERRY’S PHASE

EXPERIMENTAL VERIFICATION OF BERRY’S PHASE

PANCHARATNAM’S WORK

THE AHARONOV–BOHM EFFECT

SAGNAC EFFECT

CONCLUDING REMARKS

BIBLIOGRAPHY

EXERCISES

Phase Space Picture and Canonical Transformations

INTRODUCTION

SQUEEZE AND ROTATION IN PHASE SPACE

LINEAR CANONICAL TRANSFORMATIONS

WIGNER FUNCTION

TIME EVOLUTION OF THE WIGNER FUNCTION

APPLICATIONS

ADVANTAGES OF THE WIGNER FUNCTION

CONCLUDING REMARKS

BIBLIOGRAPHY

EXERCISES

Quantum Computers

INTRODUCTION

WHAT IS A QUANTUM COMPUTER?

WHY IS A QUANTUM COMPUTER?

FUNDAMENTAL PROPERTIES

QUANTUM ALGORITHMS

FEATURES OF QUANTUM COMPUTATION

QUANTUM COMPUTATION THROUGH NMR

WHY IS MAKING A QUANTUM COMPUTER EXTREMELY DIFFICULT?

CONCLUDING REMARKS

BIBLIOGRAPHY

EXERCISES

QUANTUM CRYPTOGRAPHY

INTRODUCTION

STANDARD CRYPTOSYSTEMS

QUANTUM CRYPTOGRAPHY–BASIC PRINCIPLE

TYPES OF QUANTUM CRYPTOGRAPHY

MULTIPARTY QUANTUM SECRET SHARING

APPLICATIONS OF QUANTUM CRYPTOGRAPHY

IMPLEMENTATION AND LIMITATIONS

FIBER-OPTICAL QUANTUM KEY DISTRIBUTION

CONCLUDING REMARKS

BIBLIOGRAPHY

EXERCISES

Some Other Advanced Topics

INTRODUCTION

QUANTUM THEORY OF GRAVITY

QUANTUM ZENO EFFECT

QUANTUM TELEPORTATION

QUANTUM GAMES

QUANTUM CLONING

QUANTUM DIFFUSION

QUANTUM CHAOS

CONCLUDING REMARKS

BIBLIOGRAPHY

EXERCISES

Quantum Technologies

INTRODUCTION

QUANTUM ENTANGLEMENT

QUANTUM ENTANGLED PHOTONS

GHOST IMAGING

DETECTION OF WEAK AMPLITUDE OBJECT

ENTANGLED TWO-PHOTON MICROSCOPY

DETECTION OF SMALL DISPLACEMENTS

QUANTUM LITHOGRAPHY

QUANTUM METROLOGY

QUANTUM TELEPORTATION OF OPTICAL

IMAGES

CONCLUDING REMARKS

BIBLIOGRAPHY

EXERCISES

**Solutions to selected exercises**

**Index**

## Author(s)

### Biography

**S. Rajasekar** received his B.Sc. and M.Sc. in physics both from the St. Joseph’s College, Tiruchirapalli. In 1987, he received his M.Phil. in physics from Bharathidasan University, Tiruchirapalli. He was awarded a Ph.D. in physics (nonlinear dynamics) from Bharathidasan University in 1992. In 2005, he became a professor at the School of Physics, Bharathidasan University. His recent research focuses on nonlinear dynamics with a special emphasis on nonlinear resonances. He has coauthored a book, and authored or coauthored more than 80 research papers in nonlinear dynamics.

**R. Velusamy **received his B.Sc. in physics from the Ayya Nadar Janaki Ammal College, Sivakasi in 1972 and M.Sc. in physics from the P.S.G. Arts and Science College, Coimbatore in 1974. He received an M.S. in electrical engineering at the Indian Institute of Technology, Chennai in the year 1981. In the same year, he joined in the Ayya Nadar Janaki Ammal College as an assistant professor in physics. He was awarded an M.Phil. in physics in 1988. He retired in 2010. His research topics are quantum confined systems and wave packet dynamics.

## Reviews

"The second volume of this textbook contains many advanced topics of current research in quantum mechanics. The problems concerning the considered subject are included at the end of any chapter. The textbook is intended for graduate students and also as a reference book. Doubtless advantage of this tutorial is to have material on current research, such as supersymmetric quantum mechanics, coherent and squeezed states, Sagnac effect, quantum computers, quantum cryptography, and so on. A separate chapter is devoted to discussing some of the issues that are at the forefront of current research: quantum gravity, quantum Zeno effect, quantum teleportation, quantum games, quantum cloning, quantum diffusion, and quantum chaos."

—Zentralblatt MATH1318"… excellent, up-to-date … can be used as either a two-to-three-semester graduate text or as a standalone reference book.

Quantum Mechanics I: The Fundamentalscovers the canonical basics andQuantum Mechanics II: Advanced Topicscovers a range of modern developments from introductory quantum field theory through quantum information theory and other quantum technologies, such as quantum metrology and imaging, that are not discussed in other sources … I recommend this set highly."

—Dr. Jonathan P. Dowling, Hearne Professor of Theoretical Physics and Co-Director, Hearne Institute for Theoretical Physics, Louisiana State University, and Author ofSchrödinger's Killer App: Race to Build the World's First Quantum Computer"Be assured … these two books by Rajasekar and Velusamy will definitely tell you how to do quantum mechanics."

—Dr. K.P.N. Murthy, Professor, School of Physics and Director, Centre for Integrated Studies, University of Hyderabad