#
Quantum Mechanics I

The Fundamentals

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## Book Description

**Quantum Mechanics I: The Fundamentals** provides a graduate-level account of the behavior of matter and energy at the molecular, atomic, nuclear, and sub-nuclear levels. It covers basic concepts, mathematical formalism, and applications to physically important systems.

The text addresses many topics not typically found in books at this level, including:

- Bound state solutions of quantum pendulum
- Pöschl–Teller potential
- Solutions of classical counterpart of quantum mechanical systems
- A criterion for bound state
- Scattering from a locally periodic potential and reflection-less potential
- Modified Heisenberg relation
- Wave packet revival and its dynamics
- Hydrogen atom in
*D*-dimension - Alternate perturbation theories
- An asymptotic method for slowly varying potentials
- Klein paradox, Einstein-Podolsky-Rosen (EPR) paradox, and Bell’s theorem
- Numerical methods for quantum systems

A collection of problems at the end of each chapter develops students’ understanding of both basic concepts and the application of theory to various physically important systems. This book, along with the authors’ follow-up *Quantum Mechanics II: Advanced Topics*, provides students with a broad, up-to-date introduction to quantum mechanics.

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

## Table of Contents

**Why Was Quantum Mechanics Developed? **INTRODUCTION

BLACK BODY RADIATION

PHOTOELECTRIC EFFECT

HYDROGEN SPECTRUM

FRANCK–HERTZ EXPERIMENT

STERN–GERLACH EXPERIMENT

CORRESPONDENCE PRINCIPLE

COMPTON EFFECT

SPECIFIC HEAT CAPACITY

DE BROGLIE WAVES

PARTICLE DIFFRACTION

WAVE-PARTICLE DUALITY

**Schrödinger Equation and Wave Function**

INTRODUCTION

CONSTRUCTION OF SCHRÖDINGER EQUATION

SOLUTION OF TIME-DEPENDENT EQUATION

PHYSICAL INTERPRETATION OF ψ∗ψ

CONDITIONS ON ALLOWED WAVE FUNCTIONS

BOX NORMALIZATION

CONSERVATION OF PROBABILITY

EXPECTATION VALUE

EHRENFEST’S THEOREM

BASIC POSTULATES

TIME EVOLUTION OF STATIONARY STATES

CONDITIONS FOR ALLOWED TRANSITIONS

ORTHOGONALITY OF TWO STATES

PHASE OF THE WAVE FUNCTION

CLASSICAL LIMIT OF QUANTUM MECHANICS

**Operators, Eigenvalues, and Eigenfunctions**

INTRODUCTION

LINEAR OPERATORS

COMMUTING AND NONCOMMUTING OPERATORS

SELF-ADJOINT AND HERMITIAN OPERATORS

DISCRETE AND CONTINUOUS EIGENVALUES

MEANING OF EIGENVALUES AND EIGENFUNCTIONS

PARITY OPERATOR

ALL HERMITIAN HAMILTONIANS HAVE PARITY

SOME OTHER USEFUL OPERATORS

**Exactly Solvable Systems I: Bound States**

INTRODUCTION

CLASSICAL PROBABILITY DISTRIBUTION

FREE PARTICLE

HARMONIC OSCILLATOR

PARTICLE IN THE POTENTIAL V (x) = x2k, k = 1, 2, · · ·

PARTICLE IN A BOX

PÖSCHL–TELLER POTENTIALS

QUANTUM PENDULUM

CRITERIA FOR THE EXISTENCE OF A BOUND STATE

TIME-DEPENDENT HARMONIC OSCILLATOR

RIGID ROTATOR

**Exactly Solvable Systems II: Scattering States**

INTRODUCTION

POTENTIAL BARRIER: TUNNEL EFFECT

FINITE SQUARE-WELL POTENTIAL

POTENTIAL STEP

LOCALLY PERIODIC POTENTIAL

REFLECTIONLESS POTENTIALS

DYNAMICAL TUNNELING

**Matrix Mechanics**

INTRODUCTION

LINEAR VECTOR SPACE

MATRIX REPRESENTATION OF OPERATORS AND WAVE FUNCTION

UNITARY TRANSFORMATION

TENSOR PRODUCTS

SCHRÖDINGER EQUATION AND OTHER QUANTITIES IN MATRIX FORM

APPLICATION TO CERTAIN SYSTEMS

DIRAC’S BRA AND KET NOTATIONS

EXAMPLES OF BASIS IN QUANTUM THEORY

PROPERTIES OF KET AND BRA VECTORS

HILBERT SPACE

PROJECTION AND DISPLACEMENT OPERATORS

**Various Pictures and Density Matrix**

INTRODUCTION

SCHRÖDINGER PICTURE

HEISENBERG PICTURE

INTERACTION PICTURE

COMPARISON OF THREE REPRESENTATIONS

DENSITY MATRIX FOR A SINGLE SYSTEM

DENSITY MATRIX FOR AN ENSEMBLE

TIME EVOLUTION OF DENSITY OPERATOR

A SPIN-1/2 SYSTEM

**Heisenberg Uncertainty Principle**

INTRODUCTION

THE CLASSICAL UNCERTAINTY RELATION

HEISENBERG UNCERTAINTY RELATION

IMPLICATIONS OF UNCERTAINTY RELATION

ILLUSTRATION OF UNCERTAINTY RELATION

THE MODIFIED HEISENBERG RELATION

**Momentum Representation **INTRODUCTION

MOMENTUM EIGENFUNCTIONS

SCHRÖDINGER EQUATION

EXPRESSIONS FOR hXi AND hpi

TRANSFORMATION BETWEEN MOMENTUM AND COORDINATE REPRESENTATIONS

OPERATORS IN MOMENTUM REPRESENTATION

MOMENTUM FUNCTION OF SOME SYSTEMS

**Wave Packet **INTRODUCTION

PHASE AND GROUP VELOCITIES

WAVE PACKETS AND UNCERTAINTY PRINCIPLE

GAUSSIAN WAVE PACKET

WAVE PACKET REVIVAL

ALMOST PERIODIC WAVE PACKETS

**Theory of Angular Momentum**

INTRODUCTION

SCALAR WAVE FUNCTION UNDER ROTATIONS

ORBITAL ANGULAR MOMENTUM

EIGENPAIRS OF L2 AND Lz

PROPERTIES OF COMPONENTS OF L AND L2

EIGENSPECTRA THROUGH COMMUTATION RELATIONS

MATRIX REPRESENTATION OF L2, Lz AND L±

WHAT IS SPIN?

SPIN STATES OF AN ELECTRON

SPIN-ORBIT COUPLING

ROTATIONAL TRANSFORMATION

ADDITION OF ANGULAR MOMENTA

ROTATIONAL PROPERTIES OF OPERATORS

TENSOR OPERATORS

THE WIGNER–ECKART THEROEM

**Hydrogen Atom**

INTRODUCTION

HYDROGEN ATOM IN THREE-DIMENSION

HYDROGEN ATOM IN D-DIMENSION

FIELD PRODUCED BY A HYDROGEN ATOM

SYSTEM IN PARABOLIC COORDINATES

**Approximation Methods I: Time-Independent Perturbation Theory**INTRODUCTION

THEORY FOR NONDEGENERATE CASE

APPLICATIONS TO NONDEGENERATE LEVELS

THEORY FOR DEGENERATE LEVELS

FIRST-ORDER STARK EFFECT IN HYDROGEN

ALTERNATE PERTURBATION THEORIES

**Approximation Methods II: Time-Dependent Perturbation Theory**

INTRODUCTION

TRANSITION PROBABILITY

CONSTANT PERTURBATION

HARMONIC PERTURBATION

ADIABATIC PERTURBATION

SUDDEN APPROXIMATION

THE SEMICLASSICAL THEORY OF RADIATION

CALCULATION OF EINSTEIN COEFFICIENTS

**Approximation Methods III: WKB and Asymptotic Methods**

INTRODUCTION

PRINCIPLE OF WKB METHOD

APPLICATIONS OF WKB METHOD

WKB QUANTIZATION WITH PERTURBATION

AN ASYMPTOTIC METHOD

**Approximation Methods IV: Variational Approach**

INTRODUCTION

CALCULATION OF GROUND STATE ENERGY

TRIAL EIGENFUNCTIONS FOR EXCITED STATES

APPLICATION TO HYDROGEN MOLECULE

HYDROGEN MOLECULE ION

**Scattering Theory**

INTRODUCTION

CLASSICAL SCATTERING CROSS-SECTION

CENTRE OF MASS AND LABORATORY COORDINATES SYSTEMS

SCATTERING AMPLITUDE

GREEN’S FUNCTION APPROACH

BORN APPROXIMATION

PARTIAL WAVE ANALYSIS

SCATTERING FROM A SQUARE-WELL SYSTEM

PHASE-SHIFT OF ONE-DIMENSIONAL CASE

INELASTIC SCATTERING

**Identical Particles**

INTRODUCTION

PERMUTATION SYMMETRY

SYMMETRIC AND ANTISYMMETRIC WAVE FUNCTIONS

THE EXCLUSION PRINCIPLE

SPIN EIGENFUNCTIONS OF TWO ELECTRONS

EXCHANGE INTERACTION

EXCITED STATES OF THE HELIUM ATOM

COLLISIONS BETWEEN IDENTICAL PARTICLES

**Relativistic Quantum Theory**

INTRODUCTION

KLEIN–GORDON EQUATION

DIRAC EQUATION FOR A FREE PARTICLE

NEGATIVE ENERGY STATES

JITTERY MOTION OF A FREE PARTICLE

SPIN OF A DIRAC PARTICLE

PARTICLE IN A POTENTIAL

KLEIN PARADOX

RELATIVISTIC PARTICLE IN A BOX

RELATIVISTIC HYDROGEN ATOM

THE ELECTRON IN A FIELD

SPIN-ORBIT ENERGY

**Mysteries in Quantum Mechanics **INTRODUCTION

THE COLLAPSE OF THE WAVE FUNCTION

EINSTEIN–PODOLSKY–ROSEN (EPR) PARADOX

HIDDEN VARIABLES

THE PARADOX OF SCHRÖDINGER’S CAT

BELL’S THEOREM

VIOLATION OF BELL’S THEOREM

RESOLVING EPR PARADOX

**Numerical Methods for Quantum Mechanics**

INTRODUCTION

MATRIX METHOD FOR COMPUTING STATIONARY STATE SOLUTIONS

FINITE-DIFFERENCE TIME-DOMAIN METHOD

TIME-DEPENDENT SCHRÖDINGER EQUATION

QUANTUM SCATTERING

ELECTRONIC DISTRIBUTION OF HYDROGEN ATOM

SCHRÖDINGER EQUATION WITH AN EXTERNAL FIELD

Appendix A: Calculation of Numerical Values of h and kB

Appendix B: A Derivation of the Factor h_/(eh_/kBT − 1)

Appendix C: Bose’s Derivation of Planck’s Law

Appendix D: Distinction between Self-Adjoint and Hermitian Operators

Appendix E: Proof of Schwarz’s Inequality

Appendix F: Eigenvalues of a Symmetric Tridiagonal Matrix—QL Method

Appendix G: Random Number Generators for Desired Distributions

Solutions to Selected Exercises

Index

Concluding Remarks, Bibliography, and Exercises appear at the end of each chapter.

## 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 first volume of this course of quantum mechanics contains basic concepts of quantum mechanics, mathematical formalism, and a wide range of applications to physically important systems. The problems concerning the considered subject are included at the end of each chapter. The textbook is intended for graduate students and also as a reference book. Doubtless advantage of this tutorial is the discussion of such mysteries in quantum mechanics as the collapse of the wave function, Einstein-Podolsky-Rosen paradox, hidden variables, the paradox of Schrödinger cat, and Bell's theorem. It should be noted the presence of numerical methods in quantum mechanics."

—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 HyderabadVol I quote: "The real strength of this book lies in its scope. Each individual chapter covers the fundamentals of a topic and acts as an excellent reference for quantum researchers….I will certainly ensure that a copy remains on my bookshelf, for when I have a query on any fundamental aspect of quantum mechanics."

—Contemporary Physics(Nov 2017), review by Prof. Thomas Collier, University of ExeterVol II quote: "I found digesting the latter part of this book a most enjoyable experience. Indeed, the final chapters on broad applications of advanced quantum mechanics are entertaining to read and refrain from delving too deeply into the intricacies. The reader is given an enticing glimpse of the research directions one might apply the knowledge of quantum mechanics presented over the two volumes. As with the first volume, I will keep this book close at hand for when I require a concise and mathematically rich guide to these advanced topics of quantum mechanics."

—Contemporary Physics(Nov 2017), review by Prof. Thomas Collier, University of Exeter